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- Herausgeber: John Wiley & Sons
- Kategorie: Wissenschaft und neue Technologien
- Sprache: Englisch
Integrating latest research results and characterization techniques, this book helps readers understand and apply fundamental principles in nonlinear polymer rheology. The author connects the basic theoretical framework with practical polymer processing, which aids practicing scientists and engineers to go beyond the existing knowledge and explore new applications. Although it is not written as a textbook, the content can be used in an upper undergraduate and first year graduate course on polymer rheology. * Describes the emerging phenomena and associated conceptual understanding in the field of nonlinear polymer rheology * Incorporates details on latest experimental discoveries and provides new methodology for research in polymer rheology * Integrates latest research results and new characterization techniques like particle tracking velocimetric method * Focuses on the issues concerning the conceptual and phenomenological foundations for polymer rheology * Has a companion website for readers to access with videos complementing the content within several chapters
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Veröffentlichungsjahr: 2018
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Table of Contents
Cover
Title Page
Copyright
Preface
References
Acknowledgments
Introduction
1 Rheology: The Definition
2. Molecular Approach of Tube Model and Continuum-Mechanical Constitutive Modeling Versus a Phenomenology-Based Treatment
3. Linear Versus Nonlinear Responses: Characterization Tool Versus Science of Rheology
4. Shear Thinning, Stress Plateau, and Yielding
5. Is There Always Homogeneous Deformation?
6. Rheology Versus Fluid Mechanics
7. Emerging Trends
8. Summary
References
About the Companion Website
Part I: Linear Viscoelasticity and Experimental Methods
Chapter 1: Phenomenological Description of Linear Viscoelasticity
1.1 Basic Modes of Deformation
1.2 Linear Responses
1.3 Classical Rubber Elasticity Theory
References
Chapter 2: Molecular Characterization in Linear Viscoelastic Regime
2.1 Dilute Limit
2.2 Entangled State
2.3 Molecular-Level Descriptions of Entanglement Dynamics
2.4 Temperature Dependence
References
Chapter 3: Experimental Methods
3.1 Shear Rheometry
3.2 Extensional Rheometry
3.3
In Situ
Rheostructural Methods
3.4 Advanced Rheometric Methods
3.5 Conclusion
References
Chapter 4: Characterization of Deformation Field Using Different Methods
4.1 Basic Features in Simple Shear
4.2 Yield Stress in Bingham-Type (Yield-Stress) Fluids
4.3 Cases of Homogeneous Shear
4.4 Particle-Tracking Velocimetry (PTV)
4.5 Single-Molecule Imaging Velocimetry
4.6 Other Visualization Methods
References
Chapter 5: Improved and Other Rheometric Apparatuses
5.1 Linearly Displaced Cocylinder Sliding for Simple Shear
5.2 Cone-Partitioned Plate (CPP) for Rotational Shear
5.3 Other Forms of Large Deformation
5.4 Conclusion
References
Part II: Yielding – Primary Nonlinear Responses to Ongoing Deformation
Chapter 6: Wall Slip – Interfacial Chain Disentanglement
6.1 Basic Notions of Wall Slip in Steady Shear
6.2 Stick–Slip Transition in Controlled-Stress Mode
6.3 Wall Slip during Startup Shear – Interfacial Yielding
6.4 Relationship between Slip and Bulk Shear Deformation
6.5 Molecular Evidence of Disentanglement during Wall Slip
6.6 Uncertainties in Boundary Condition
6.7 Conclusion
References
Chapter 7: Yielding during Startup Deformation: From Elastic Deformation to Flow
7.1 Yielding at
Wi
< 1 and Steady Shear Thinning at
Wi
> 1
7.2 Stress Overshoot in Fast Startup Shear
7.3 Nature of Steady Shear
7.4 From Terminal Flow to Fast Flow under Creep: Entanglement–Disentanglement Transition
7.5 Yielding in Startup Uniaxial Extension
7.6 Conclusion
7.A Experimental Estimates of Rouse Relaxation Time
References
Chapter 8: Strain Hardening in Extension
8.1 Conceptual Pictures
8.2 Origin of “Strain Hardening”
8.3 True Strain Hardening in Uniaxial Extension: Non-Gaussian Stretching from Finite Extensibility
8.4 Different Responses of Entanglement to Startup Extension and Shear
8.5 Conclusion
88.A Conceptual and Mathematical Accounts of Geometric Condensation
References
Chapter 9: Shear Banding in Startup and Oscillatory Shear: Particle-Tracking Velocimetry
9.1 Shear Banding After Overshoot in Startup Shear
9.2 Overcoming Wall Slip during Startup Shear
9.3 Nonlinearity and Shear Banding in Large-Amplitude Oscillatory Shear
References
Chapter 10: Strain Localization in Extrusion, Squeezing Planar Extension: PTV Observations
10.1 Capillary Rheometry in Rate-Controlled Mode
10.2 Instabilities at Die Entry
10.3 Squeezing Deformation
10.4 Planar Extension
References
Chapter 11: Strain Localization and Failure during Startup Uniaxial Extension
11.1 Tensile-Like Failure (Decohesion) at Low Rates
11.2 Shear Yielding and Necking-Like Strain Localization at High Rates
11.3 Rupture-Like Breakup: Where Are Yielding and Disentanglement?
11.4 Strain Localization Versus Steady Flow: Sentmanat Extensional Rheometry Versus Filament-Stretching Rheometry
11.5 Role of Long-Chain Branching
11.A Analogy between Capillary Rheometry and Filament-Stretching Rheometry
References
Part III: Decohesion and Elastic Yielding After Large Deformation
Chapter 12: Nonquiescent Stress Relaxation and Elastic Yielding in Stepwise Shear
12.1 Strain Softening After Large Step Strain
12.2 Particle Tracking Velocimetry Revelation of Localized Elastic Yielding
12.3 Quiescent and Uniform Elastic Yielding
12.4 Arrested Wall Slip: Elastic Yielding at Interfaces
12.5 Conclusion
References
Chapter 13: Elastic Breakup in Stepwise Uniaxial Extension
13.1 Rupture-Like Failure during Relaxation at Small Magnitude or Low Extension Rate (
Wi
R
< 1)
13.2 Shear-Yielding-Induced Failure upon Fast Large Step Extension (
Wi
R
> 1)
13.3 Nature of Elastic Breakup Probed by Infrared Thermal-Imaging Measurements
13.4 Primitive Phenomenological Explanations
13.5 Step Squeeze and Planar Extension
References
Chapter 14: Finite Cohesion and Role of Chain Architecture
14.1 Cohesive Strength of an Entanglement Network
14.2 Enhancing the Cohesion Barrier: Long-Chain Branching Hinders Structural Breakup
References
Part IV: Emerging Conceptual Framework and Beyond
Chapter 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 Nonmonotonicity
15.5 Metastable Nature of Shear Banding
References
Chapter 16: Molecular Networks as the Conceptual Foundation
16.1 Introduction: The Tube Model and its Predictions
16.2 Essential Ingredients for a New Molecular Model
16.3 Overcoming Finite Cohesion after Step Deformation: Quiescent or Not
16.4 Forced Microscopic Yielding during Startup Deformation: Stress Overshoot
16.5 Interfacial Yielding via Disentanglement
16.6 Effect of Long-Chain Branching
16.7 Decohesion in Startup Creep: Entanglement–Disentanglement Transition
16.8 Emerging Microscopic Theory of Sussman and Schweizer
16.9 Further Tests to Reveal the Nature of Responses to Large Deformation
16.10 Conclusion
References
Chapter 17: “Anomalous” Phenomena
17.1 Essence of Rheometric Measurements: Isothermal Condition
17.2 Internal Energy Buildup with and without Non-Gaussian Extension
17.3 Breakdown of Time–Temperature Superposition (TTS) during Transient Response
17.4 Strain Hardening in Simple Shear of Some Polymer Solutions
17.5 Lack of Universal Nonlinear Responses: Solutions versus Melts
17.6 Emergence of Transient Glassy Responses
References
Chapter 18: Difficulties with Orthodox Paradigms
18.1 Tube Model Does Not Predict Key Experimental Features
18.2 Confusion About Local and Global Deformations
18.3 Molecular Network Paradigm
References
Chapter 19: Strain Localization and Fluid Mechanics of Entangled Polymers
19.1 Relationship between Wall Slip and Banding: A Rheological-State Diagram
19.2 Modeling of Entangled Polymeric Liquids by Continuum Fluid Mechanics
19.3 Challenges in Polymer Processing
References
Chapter 20: Conclusion
20.1 Theoretical Challenges
20.2 Experimental Difficulties
References
Symbols and Acronyms
Subject Index
End User License Agreement
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Guide
Cover
Table of Contents
Preface
Part I: Linear Viscoelasticity and Experimental Methods
Begin Reading
List of Illustrations
Chapter 1: Phenomenological Description of Linear Viscoelasticity
Figure 1.1 (a) Depiction of simple shear in three dimensions, showing two parallel surfaces at y = 0 (fixed) and H (displaced by X over time
t
). The force F required to hold the bottom surface stationary can be measured to define shear stress in Eq. (1.6) where Σ is the total area of the shearing surface. (b) Displacement X of the upper surface as a function of time at a constant shear rate V/H. (c) Startup shear shown by the step function of V versus time
t
. (d) Homogeneous continuous simple shear, produced by either the moving plate at speed V or a force F on the plate. (e) Step strain realized by the displacement of the moving surface by X over a period of
t
0
for a gap distance of H, so that γ = X/H.
Figure 1.2 Sketches of various stress responses (liquid, solid, or viscoelastic liquid) during startup deformation in the linear response regime, that is, in the limit of
Wi
≪ 1, where the Weissenberg number
Wi
is defined in Section 1.2.7.
Figure 1.3 (a) An abstract representation of the Maxwell element, made of a dashpot of viscosity η and spring of modulus G in series. (b) A shear setup made a layer of solid (or liquid) over a layer of liquid (or solid) with thickness H
s
and H
l
, which mimics the Maxwell model. A top surface displacement of X is a sum of the displacements x
l
and x
s
associated with the liquid and solid layers, respectively.
Figure 1.4 (a) Illustration of Eq. (1.23a). (b) Maxwell model's depiction of stress growth and relaxation (dashed curve) according to Eqs. (1.23b) and (1.23c), respectively.
Figure 1.5 Schematic illustration on double-log of the Maxwell model description of the storage and loss moduli G′′ and G″ as a function of the oscillation frequency
ω
. For
ω
τ ≪ 1, G′ ∼
ω
2
and G″ ∼
ω
and G′ = G″ at
ω
τ = 1.
Figure 1.6 (a) The Kelvin–Voigt element with a spring of modulus G and dashpot of viscosity η in parallel. (b) Experimental version of the Kelvin–Voigt model made of a solid block adjacent to a liquid block in parallel.
Figure 1.7 Illustration of transformation in the end-to-end distance vector from
R
to
R′
during affine simple shear, where γ = X/H (Figure 1.1(a)).
Figure 1.8 Illustration of uniaxial extension along Z-axis, involving a tensile force F over the shaded cross-sectional area Σ for a specimen of length L.
Figure 1.9 Microscopic view of a chain network where the basic building block has a coil size of R. In the case of simple shear, the areal density of strands is denoted by ψ
×
of Eq. (1.56b), which would be unchanged in the presence of cross-linking.
Figure 1.10 Illustration of the concept of geometric condensation that emerges during uniaxial extension where strand areal density in the XY plane increases from the initial ψ
x
to λψ
x
upon extension by a factor of λ. A detailed conceptual and mathematical description of the geometric condensation concept can be found in the Appendix in Chapter 8.
Chapter 2: Molecular Characterization in Linear Viscoelastic Regime
Figure 2.1 Illustration of a subchain in a bead-spring model.
Figure 2.2 Storage and loss moduli G′ and G″ according to the Rouse and Zimm models where the Flory exponent ν is 1/2 for theta solutions and 3/5 for good solvents.
Figure 2.3 Zero-shear viscosity η
0
versus molecular weight M on a log–log plot. Here, the critical molecular weight M
c
separates Rouse melts (after iso-free volume corrections) from entangled melts. The dashed line showing exponent 3 is the idealized scaling behavior, to be discussed in Section 2.3.2. Due to finite chain length effects, η
0
is always smaller than its ideal value.
Figure 2.4 Illustration of elastic recovery under (a) simple shear and (b) uniaxial extension where the dashed lines are used to show the equilibrium positions. Full recoil corresponds to the deformed sample returning to the shapes indicated by the dashed lines. The thin lines reveal the positions of the samples after (partial) elastic recoil in (a).
Figure 2.5 Time-dependent compliance of a viscoelastic material during and after creep in the linear response regime, where the elastic is comparable to the recoverable compliance and approximately given by the reciprocal elastic modulus G. Chain entanglement produces the creep plateau before flow that begins beyond the time marked by the vertical dashed line. So the response to the imposed stress in such a creep test is initially elastic and eventually viscous; the transition occurs at the vertical dashed line.
Figure 2.9 Transient network picture showing a mesh size
l
ent
and effective junctions with an average lifetime τ.
Figure 2.6 Relaxation modulus G(
t
) of an entangled polymer melt as a function of time
t
, showing an elastic plateau modulus and two characteristic times, the disengagement time τ and tube confinement time τ
e
.
Figure 2.7 Storage and loss moduli G′ and G″ as a function of the oscillation frequency, typically obtained from frequency sweep of small-amplitude oscillatory shear. For monodisperse entangled solutions or melts, the crossover frequency ω
c
can be taken as the reciprocal terminal relaxation time or reptation time or disengagement time. The magnitude G
c
at ω
c
is about a factor of 3.5 smaller than G
pl
for monodisperse well-entangled polymer solutions and melts. The Maxwell model (solid lines) may capture the behavior at low frequencies. The complete curves are depicted by adding the dashed lines.
Figure 2.8 A snapshot of an entangled melt, magnified 1 million times – a photo of a bowl of crystal noodle.
Figure 2.10 (a) Linear dependence of entanglement spacing
l
ent
on packing length
p
. (b) Quadratic dependence of
N
e
on
p
/
l
K
based on the same data as those in (a).
Figure 2.11 Dependence of the areal density ψ
ent
of entanglement strands in Eq. (2.13) on the packing length
p
as well as that of (right Y axis) for various polymers.
Figure 2.12 Wool's depiction of chain entanglement in his percolation model. It shows that a minimum chain length is as long as being able to return to a flat surface (denoted by the horizontal line) three times.
Figure 2.13 The number
n
c
of backbone lengths corresponding to a subchain of molecular weight M
c
.
Figure 2.14 Chain movement from the position given by the solid line to that indicated by the dashed line amounts to diffusion of chain's center-of-mass by a distance comparable to its size R.
Figure 2.15 Normalized storage and loss moduli according to the tube model given by Eq. (2.48a).
Figure 2.16 Self-diffusion constant D
s
as a function of the reduced molecular weight M/M
e
on double-logarithmic scales. The dashed line showing exponent −2 is the idealized scaling behavior. Due to the finite chain length effects, D
s
is higher than the ideal values. For illustrative purpose, M/M
e
= 1 is used to denote the borderline between Rouse chain and entangled chain diffusions.
Figure 2.17 (a) Scaling behavior of the product of self-diffusion coefficient D
s
and molecular weight M squared for hydrogenated polybutadiene at 175 °C, on double-logarithmic scales in the unit of g
2
(cm
2
/s) mol
−2
. The open symbols represent trace-diffusion and confirm the asymptotic scaling behavior. (b) The product of zero-shear viscosity η
0
and diffusion coefficient D
s
, normalized by that for a Rouse melt, for hydrogenated polybutadiene at 175 °C. The dependence of the product is noticeably strongly than indicated by the slope of unity (straight line).
Figure 2.18 (a) Disengagement time τ
d
as a function of concentration φ for different binary mixtures involving a common “parent” 1,4-polybutadiene of M = 410 kg/mol and various polybutadienes of lower molecular weights. (b) Terminal relaxation time given by η
0
/G
c
for the same set of binary mixtures as those in (a).
Chapter 3: Experimental Methods
Figure 3.1 (a) Schematic illustration of a shear cell (with gap distance H) with linear displacement of the upper plate, either by a step-motor with a force transducer, or by a constant force (air cylinder) along with a linear-variable-differential transformer. (b) XZ view of the parallel-plate shear cell where a load cell can be installed to measure the local shear stress at the wall, as done in the literature.[1]
Figure 3.2 Illustration of a cone-plate rotational shear apparatus in terms of the plate radius
R
and cone angle θ. The stationary fixture can also be the cone instead of the plate. Also indicated is the meniscus, which in this system is a free surface surrounding the gap in the cone-plate arrangement.
Figure 3.3 Illustration of a Couette device, made of inner cylinder of radius
R
concentrically placed inside a cylindrical cup with radius (
R
+ Δ), where the inner cylinder is shown to rotate at an angular velocity Ω. Depending on its design to connect to the step-motor, the outer cylinder can be rotational instead of the inner cylinder.
Figure 3.4 Pressure-driven shear through extrusion, by either displacement of the piston at speed V
p
or pressure P, where the die can be either capillary or slit, as shown in Figure 3.5.
Figure 3.5 (a) Capillary die with diameter D and (b) slit die with width w and thickness H.
Figure 3.6 Rheo-optical setup for simple shear to measure strain-induced birefringence with the polarization P of the incident laser either (a) parallel to or (b) at 45
o
with the shearing direction (X axis). The analyzer A is always perpendicular to P.
Figure 3.7 Michel-Levy birefringence chart from http://www.olympusmicro.com/primer/techniques/polarized/michel.html.
Source:
Reproduced with permission of Olympus America Inc.
Figure 3.8 Depiction of a general protocol involving either rate or stress switching or a combination of rate and stress switching.
Chapter 4: Characterization of Deformation Field Using Different Methods
Figure 4.1 (a) Sketch of simple shear based on homogeneous deformation assumption. (b) Illustration of an arbitrarily chosen layer of thickness Δ in the interior of the shear cell that experiences zero net force after the inertia effect vanishes. (c) “Flow curve” or constitutive curve showing one-to-one correspondence between shear stress σ and shear rate , so that the relation is invertible, rendering support for the existence of a constant shear rate across the gap.
Figure 4.2 Comparison between the Bingham equation and the alternative expression given in Eqs. (4.3a)-(4.3c), where the rates are merely the nominal rates.
Figure 4.3 A rheomicroscopic setup involving a rotational rheometer such as Bohlin CVO over a confocal microscope, where the maximum velocity could be 500 µm/s, corresponding to a shear rate of 10 s
−1
.
Figure 4.4 (a) Illustration of a cone-plate apparatus coupled to particle-tracking velocimetry (PTV).
Source:
Wang 2007 [4]. Reproduced with permission of Wiley. (b) PTV setup for a sliding-plate shear cell.
Source:
Boukany and Wang 2007 [8]. Reproduced with permission of American Institute of Physics.
Figure 4.5 Three frames from PTV video recording of the illuminated tracking particles at various times during startup at a nominal shear rate of 10 s
−1
.
Source:
Tapadia and Wang 2006 [9]. Reproduced with permission of American Physical Society.
Figure 4.6 Particle-tracking velocimetric setup for observation of strain localization at the slit die entry and inland. Here the laser passes along X-axis through thin openings to illuminate a thin plane of PTV particles in the sample. The objective lens along with a video camera is placed along the Y-axis.
Chapter 5: Improved and Other Rheometric Apparatuses
Figure 5.1 Depiction of customer-made device for linear displacement, comprising an inner cylindrical rod and an outer cylinder that are two half cylinders.
Figure 5.2 Cone-partitioned plate (CPP) device is made of a large cone (of radius
R
2
) against a partitioned plate that is made of a smaller disk of radius
R
1
and a ring of width approximately equal to (
R
2
−
R
1
). (a) Side view showing the rotating cone, the stationary upper plate of radius
R
1
and a ring of inner radius
R
1
+ Δ and outer radius
R
2
where Δ is the gap distance between the disk and ring and (b) top view of the partitioned plate.
Figure 5.3 Comparison of shear stress growth upon startup shear at a shear rate of 1.0 s
−1
between C/P and CPP device.
Figure 5.4 One-dimensional squeeze setup, made of a trench with width w and length L and a matching plunger that is moving downward along the Z-axis, where the sample thickness is 2
h
. The bottom of the trench is stationary. If the plunger is pushed down, the sample is expelled from the ends of the trench at x = ±L/2.
Figure 5.5 Plot of the z-component velocity profile depicted in Eq. (5.17).
Chapter 6: Wall Slip – Interfacial Chain Disentanglement
Figure 6.1 Different states at various polymer/wall interfaces: (a) in quiescence with strong polymer adsorption or under weak shear (corresponding either to slow shear rate or to low shear stress) where the depiction is somewhat misleading: the adsorbed chains are actually still inter-penetrating into the unbound chains (see Figure 16.6 for a more realistic illustration of the interfacial layer) and unbound chains are disentangling in the sense that there is significant sliding between adsorbed and free chains; (b) full disentanglement between adsorbed chains and unbound (bulk) chains during complete wall slip; (c) chain desorption in the presence of shear at weakly adsorbing interfaces.
Figure 6.2 (a) Schematic illustration of wall slip in simple steady shear, produced by the displacement of the upper surface with velocity V while fixing the lower surface at zero speed. The effective shear rate in the bulk is smaller than the apparent rate V/H because of wall slip, characterized by either the slip velocity V
s
or the extrapolation length
b
. Here
b
is termed the
extrapolation length
because it is not a physical dimension of or length scale in the system. For example, it is not the thickness of the interfacial slip layer. Rather, it is defined by extrapolating along the bulk velocity gradient to where the velocity would be zero outside the sample of thickness H as shown. (b) Interfacial layer of thickness
a
i
characterized by presence of high shear rate and low viscosity η
i
, where the long-dashed line shows the bulk shear rate.
Figure 6.3 Velocity profiles in the cases of a capillary or a slit (channel) die extrusion before and after the stick–slip transition. Because the stress level is the same at the transition, the velocity profiles in the bulk are identical.
Figure 6.4 Spurt flow of a linear polyethylene from pressure-controlled capillary rheometry at 190 °C, based on a die of diameter D = 0.5 mm and length L = 4.5 mm. As indicated in the original paper, the extrudate is
wavy
(denoted by the triangles) just below the spurt transition at P
c
and turns
rough
on the slip branch upon the transition, as indicated by the squares.
Figure 6.5 Volumetric output versus applied pressure (shear stress at the wall) for a series of relatively monodisperse polybutadiene melts, from capillary extrusion with a die of diameter D = 0.96 mm and length L = 24.8 mm. The numbers from 1 to 8 label the molecular weights of 38, 68, 102, 151, 204, 240, 320, 580 kg/mol.
Figure 6.6 Stick–slip transition of 1,4-polybutadiene melts of four different molecular weights, M
w
= 64, 119, 244, and 437 kg/mol, respectively, using a Monsanto capillary rheometer.
Figure 6.7 Removal of the stick–slip transition by a surface coating that induces desorption at high shear stress at the capillary wall.
Figure 6.8 Stick–slip transition of a high-density polyethylene at five temperatures, expressed in terms of the die output versus the wall shear stress, obtained with a Monsanto (pressure-controlled) capillary rheometer. The critical stress for the transition is seen to increase with rising temperature. In the original literature, this temperature dependence of σ
c
was invoked to justify the conclusion that the stick–slip transition is not caused by chain desorption.
Figure 6.9 (a) Stick–slip transition observed in force-controlled mode using the apparatus depicted in Figure 5.1, for four 1,4-polybutadiene melts of different molecular weights. (b) Maximum extrapolation length
b
max
determined from (a) according to Eq. (6.16).
Figure 6.10 Stick–slip transition observed at three different gap distances at T = 25 °C. The inset shows that the abrupt jump, given by (V
s
/V
c
− 1), varies linearly with 1/H in agreement with Eq. (6.16).
Figure 6.11 Comparison of the stick–slip transition between capillary extrusion and simple shear of the same sample (PB207K) in terms of the wall stress σ.
Figure 6.12 Stick–slip transition of linear and star-like PB melts in simple shear (H = 0.2 mm) and capillary extrusion (D = 1 mm) at T = 25 °C.
Figure 6.13 PTV determination of the velocity profile in a sliding-plate setup (cf. Figure 4.4(b)) to reveal wall slip at the stick–slip transition.
Figure 6.14 Storage and loss moduli G′ and G″ of PB400K melt at reference temperature T
ref
= 40 °C. According to the Williams–Landel–Ferry shift factor, the chain dynamics are slower by a factor of 2.1 at 25 °C.
Figure 6.15 Velocity profiles at different times during startup shear with apparent rate V/H (a) 1 s
−1
and (b) 4 s
−1
for an entangled DNA solution. The insets show the corresponding stress versus time curves, with stress rising to a maximum (overshoot) and then decreasing to its steady-state value.
Figure 6.16 (a) Stress responses upon a startup shear at V/H = 4 s
−1
. The startup shear is terminated at 0.5 s. The stress peaks at
t
iy
, before the shear is terminated, and then has a steep drop after 0.5 s. (b) Corresponding particle-tracking velocimetric observations reveal interfacial failure at 0.43 s during shear and recoil-like motions after the shear ceases.
Figure 6.17 Illustration of simple shear (a) in absence of wall slip, which is usually guaranteed for
Wi
app
< 1, (b) in presence of measurable wall slip in the stress plateau between
Wi
app
> 1 and
Wi
app
<
Wi
ws-bnl
, (c) at the maximum of
b
, that is, the upper bound of
Wi
app
where the bulk shear rate still locks in at with
Wi
c
∼ 1, and (d) when the bulk undergoes strong shear, that is, > or
Wi
> 1.
Figure 6.18 Stress plateau at a level nearly independent of the applied rate for a entangled DNA solution.
Figure 6.19 Interfacial yield stress σ
iy
and steady-state stress σ
ss
as a function of shear rate V/H according to Figure 6.18.
Figure 6.20 Interfacial yield strain γ
iy
versus shear rate V/H according to Figure 6.18.
Figure 6.21 Linear viscoelastic behavior of PB1M(13%)-1.5K and PB1M(13%)-10K, as revealed by small amplitude oscillatory shear measurements.
Figure 6.22 Rheometric and particle-tracking velocimetric measurements during startup shear of PB1M(13%)-1.5K, based on a 20-mm-diameter parallel-disk shear cell having a 50-µm gap. (a) Shear stress response to a startup shear of 0.4 s
−1
(
Wi
app
= 14). (b) Evolution of the velocity profile at different times after applying the startup shear (
t
= 1, 30, and 75 s). Before the shear stress maximum is reached, the velocity profile is linear across the gap. At long times, massive slip emerges.
Figure 6.23 (a) Stress response upon startup shear at shear rates 1.6, 3.0, and 8.0 s
−1
respectively, where the vertical arrows indicate the moments when the particle-tracking velocimetric observations were made. (b) Velocity profiles at the indicated times and Weissenberg numbers. As also shown in Figure 6.22(b), the profiles are completely plug-flow like due to the massive wall slip. While the startup shear at
Wi
app
= 57 and 108 involved H = 50 µm, the startup test with
Wi
app
= 288 was made for H = 40 µm. (c) Plot of the bulk Weissenberg number
Wi
versus the apparent Weissenberg number
Wi
app
, based on the information from Figure 6.22(b) and 6.23(b), where the vertical dashed line marks the onset of strong nonlinear responses of the bulk. The inset shows how the long-time stresses vary with
Wi
, measured at moments indicated by the vertical arrows in Figure 6.22(a) and 6.23(a).
Figure 6.24 (a) Stress response to startup shear at an apparent rate 0.3 s
−1
(
Wi
app
= 20). (b) The velocity profile showing wall slip for PB1M(13%)-10K where
Wi
∼
Wi
c
∼ 1, obtained from the 20-mm-diameter parallel-disk shear cell with separation of 50 µm.
Figure 6.25 (a) Illustration of the stress plateau: over a wide range of applied shear rates, with
Wi
app
changing from
Wi
c
∼ 1 to
Wi
ws-bnl
, the shear stress only varies in the narrow range between σ
h
and σ
c
. (b) Dependence of the slip velocity on
Wi
app
for the two PB solutions.
Figure 6.26 (a) Dependence of the extrapolation length
b
on the apparent Weissenberg number
Wi
app
during rate-controlled steady shear of entangled polymeric liquids. (b) Dependence of
b
on the applied shear stress σ, with a stick–slip transition at σ
c
.
Figure 6.27 Plot of Eq. (6.28) for bulk Weissenberg number
Wi
versus
Wi
app
.
Figure 6.28 Illustration of the effect of doubling the gap distance in the marginal slip regime: Upon increasing the gap distance from H to 2H as shown in (a), if the slip velocity would stay the same, then shear rate in the bulk would double as shown in (b), resulting in a higher shear stress level. Such gap dependence of shear stress does not occur in the marginal state because the slip velocity V
s
also doubles, so that the bulk shear rate and stress would remain unchanged as shown in (c). This behavior is a property of the marginal state: Multiple values for V
s
, including its upper bound , exist at a common shear stress σ
c
.
Figure 6.29 Transient shear viscosity η as a function of strain at different gap distances for three shear rates for a 5% 1,4-polybutadiene solution. The polybutadiene has an ultrahigh molecular weight 4 × 10
6
g/mol. The solvent is Escmo110 (from Kraton). With a decade of change in the applied shear rate, the resulting steady-shear stress is nearly the same because over the shear rate change from 1 to 10 s
−1
the shear viscosity also drops nearly a decade as indicated by the vertical bar. This stress plateau and the extreme shear thinning are caused by the massive wall slip in steady state. It reveals the existence of a marginal slip state depicted in Figure 6.25(a).
Source
: Unpublished data taken by Mengchen Wang at University of Akron.
Figure 6.30 Bulk Weissenberg number
Wi
as a function of the gap distance H/H
i
according to Eq. (6.32). In the limit
Wi
≪ 2
b
max
/H, Eq. (6.32) simply reduces to (
Wi
)
n
(H
i
/H) = 1, as shown by the dashed line which is also computed for
n
= 0.1. The two horizontal lines indicate the limiting values of 30 and 300 respectively for
Wi
.
Figure 6.31 (a) Stress versus time of an aqueous entangled DNA solution upon startup shear at an apparent rate 0.5 s
−1
corresponding to
Wi
∼ 7, where the inset indicates no slip prior to the stress overshoot and slip afterward according to the method depicted in Figure 4.3 in Chapter 4. (b) Conformations of individual DNA molecules in the slip layer at different times during the startup shear depicted in (a), where the shearing direction is indicated with the arrow. (c) Conformations of DNA molecules across the gap in steady state that show coiled state except for the one at 0 µm from the shearing wall.
Figure 6.32 Slip velocity V
s
as well as H, which is the velocity increase from the stationary plate to the moving place due to the bulk shear, along with the depiction by Eq. (6.26), showing linear variation of V
s
with
Wi
app
.
Figure 6.33 (a) Steady state velocity field at V/H = 5 s
−1
with the inset showing the stress versus time curve. Here the bulk shear rate 0.56 s
−1
is only one tenth of the nominal rate 5 s
−1
. (b) Tumbling of DNA molecule at the wall during steady shear with considerable slip at the same time as the moment examined by SMIV (cf. Figure 4.3).
Chapter 7: Yielding during Startup Deformation: From Elastic Deformation to Flow
Figure 7.1 (a) Typical transient viscosity of entangled polymer solutions and melts in response to startup shear, starting from zero, where the dashed line is the zero-shear envelope depicting the stress response in the limit of
Wi
≪ 1. The continuous line for
Wi
> 1 is at every moment below the dashed curve. (b) Corresponding shear stress versus time for the same two conditions depicted in (a) where the initial linear growth ceases, indicating the onset of partial yielding or “strain softening,” before the arrival of the stress maximum, which we call “yield point.” The maximum represents a transition from (initial) elastic deformation to flow. (c) Shear stress versus strain upon startup shear at four discrete apparent shear rates from 0.04 to 10 s
−1
, along with a curve at rate 0.02 s
−1
near the end of the terminal region.
Figure 7.2 (a) Illustration of steady shear viscosity of monodisperse (entangled) polymer melts as a function of stress showing a transition-like feature around a critical stress level σ
c
where η steeply declines with σ. (b) Stress versus apparent shear rate curves obtained in both controlled-rate and controlled-stress modes, along the small-amplitude oscillatory shear data |G
*
| versus the oscillation frequency ω, where the shear thinning exponent is as small as 0.08.
Figure 7.8 (a) Linear dependence of the normalized peak shear stress on the yield strain γ
y
for the seven samples listed in Table 7.1. (b) Scaling behavior of γ
y
against
Wi
R
for three entangled PB solutions of different Z (filled symbols) and four SBR melts of different molecular weights (open symbols). (c) Combination of (a) and (b) shows the scaling dependence of the normalized peak stress on
Wi
R
.
Figure 7.3 Shear stress overshoot as a function of the elapsed strain for
Wi
R
< 1, for a 10% polybutadiene solution with Z = 40 entanglements per chain.
Figure 7.4 Shear stress as a function of the elapsed strain for
Wi
R
> 1, where two lines of slopes G
pl
and G
coh
are drawn to, respectively, indicate the initial perfect elastic response and a common modulus G
coh
= G
pl
/2.2 shared by all the curves at the stress maxima.
Figure 7.5 Collapse of all the curves from Figure 7.4 upon normalizing the strain and stress with the coordinates of (γ
y
, σ
y
).
Figure 7.6 Master curves based on the normalized stress versus strain curves from two different PB solution in different regimes.
Figure 7.7 Shear stress overshoot as a function of the elapsed strain for an SBR500K melt at
Wi
R
> 1, where two slopes are defined in the same way as in Figure 7.4.
Figure 7.9 Shear stress overshoot at different applied rates, ranging from 0.08 to 90 s
−1
in a PB1M(10%)-15K solution.
Figure 7.10 Depiction of two elastic recovery tests in which the shear stress is set to zero either before or after reaching the peak stress at γ
y
during startup shear. There is then elastic recovery, the sample returning to or toward its original shape. Characteristically, the elastic recovery is nearly complete for γ < γ
y
and incomplete for γ > γ
y
.
Figure 7.11 Elastic recovery experiments, after a startup shear at = 20 s
−1
for a duration of either 0.16 or 0.23 or 0.56 s, presented in double Y-axes. The up-pointing triangles refer to the time-dependent shear stress (left-hand-side Y-axis) as a function of time up to
t
= 0.56 s. The strain (right-hand Y-axis) increases linearly with time, the increase terminating at the three different times when the elastic recoil begins. If elastic recoil begins prior to
t
y
, there is complete elastic recoil.
Figure 7.12 Elastic recovery experiments from startup shear, made at a shear rate 0.2 s
−1
much smaller than the rates used in Figure 7.11, for durations of 1.6, 6, 9.2, and 14 s, respectively, on the same double Y-axes as used in Figure 7.11. Here the inertia also takes place to show appearance of overstrain recovery.
Figure 7.13 (a) Shear stress as a function of time upon startup shear at = 1 s
−1
for a PB1M(5%) solution, where the vertical arrows indicate the different stages (i.e., different values of
t
1
) from which discrete elastic recovery experiments start. (b) Elastic recovery measurements after shearing for the different amounts of strain γ
1
=t
1
as indicated in (a). (c) Recoverable strain γ
r
= γ
1
− γ(t
2
≫ τ) plotted against the imposed strain γ
1
read from the data in (b). The inset indicates the location of the peak value of γ
r
denoted by γ
nwbd
in the stress versus strain curve. (d) Similar Figure to (c) involving a smaller
Wi
R
of 0.135, based on the data (0.3 s
−1
) presented in Figure 7.14.
Source:
Data taken from Wang
et al
. 2012 [30]. (e) Variation of G
eff
in Eq. (7.7b) as a function of strain γ
1
, showing its decrease until the elapsed strain of γ
nwbd
when it grows again. The inset shows how G
eff
makes a U-shaped turn with σ. (f) Recoverable strain γ
r
versus the driving force, namely, the stress σ at the beginning of the elastic recovery test.
Source:
Data taken from Wang
et al
. 2013 [31].
Figure 7.14 Elastic recovery data summarized in the form of the normalized recoverable strain versus the normalized elapsed strain for two PB solutions. The inset shows, in the case of the 1M(5%)-10K solution, the shear stress as a function of the elapsed strain γ
1
for the five discretely applied rates. The 1M(5%)-10K solution involves the same two components as those are listed in Table 12.1 in Chapter 12 from another study, and the 0.7M(5%)-1.8K solution is listed in Table 7.3.
Figure 7.15 (a) Stress overshoot upon the rate switch from 0.3 to 3.0 s
−1
at different stages during startup shear with = 0.3 s
−1
for the 1M(5%)-10K solution. (b) The peak stress σ
2y
associated with subsequent rate of 3.0 s
−1
, along with the stress versus time curve due to the first startup at 0.3 s
-1
. Also plotted is the initial modulus that characterizes the stress growth upon the rate switch, defined as G
int
= (dσ
2
/dγ
2
)|t
2→0
, where γ
2
= t
2
.
Figure 7.16 Sketches of the storage and loss moduli G′
eq
and G″
eq
for a quiescent entangled polymer solution (blue lines) as well as G′ and G″ extracted from SAOS superimposed onto steady shear at rate (red lines).
Figure 7.17 (a) Frequency dependences of G′ and G″ from superimposed SAOS of a 0.5% DNA solution in steady shear at different steady shear rates ranging from 0 to 0.2 s
−1
.
Source:
Data taken from Boukany and Wang 2009 [48]. (b) G′ and G″ curves from imposed SAOS of a 3% entangled PB solution, showing the low-frequency crossover at a common value of 1/τ
eff
at three different temperatures of 10, 25, and 40 °C.
Source:
Data taken from Boukany and Wang 2009 [48]. As indicated in Table 7.4, the relaxation time τ
e
∼ 1/ω
e
shifts with temperature as follows: from 40 to 10 °C, τ
e
increases by a factor of 4.6 and from 40 to 25 °C, τ
e
increases by a factor of 2.1.
Figure 7.18 Master curves showing that under steady shear the crossover frequency ω
c
of both entangled DNA and PBD solutions increases linearly with increasing
Wi
.
Figure 7.19 Two protocols that are used to probe the nature of steady shear: (a) rate switching, which has been applied in Figure 7.15(a,b); (b) superimposing a small step strain on the background of steady shear.
Figure 7.20 Stress change as a function of elapsed strain upon switching from steady-state shear at = 0.04 s
−1
to steady-state shear at 0.045, 0.05, 0.07, and 0.1 s
−1
, respectively.
Figure 7.21 Stress relaxation due to the superimposed small strain of magnitude 0.1 (produced at 1 s
−1
) as a function of the rescaled time, showing that the relaxation is dictated by the underlying shear rate (0.004, 0.01, and 0.04 s
−1
).
Figure 7.22 Stress relaxation from the superimposed small strain (0.1), at three different temperatures, in steady shear at 0.04 s
−1
.
Figure 7.23 Characteristics of a “flow curve,” namely, steady shear stress σ versus bulk shear rate , showing a stress plateau around σ
1
(left axis) as well as the corresponding steady shear viscosity η (right-hand axis) as a function of , displaying shear thinning. Upon creep at stress level σ
1
, over time the viscosity of an entangled polymer must undergo a change in viscosity from η
0
to η
1
. This transition is an entanglement–disentanglement transition because the initial large viscosity η
i
is understood to originate from polymer entanglement, so the decline to the much reduced viscosity η
1
must imply loss of chain entanglement, namely, disentanglement.
Figure 7.24 Nominal (apparent) shear rate versus the elapsed strain γ at different applied shear stresses, based on the cone-partitioned plate device (Figure 5.2). Figure 9.9 in Chapter 9 shows that at the three times indicated by the vertical arrows the velocity profile is not linear, showing apparent wall slip at 101 and 308 s and shear banding at 350 s. Therefore, the bulk shear rate is different from , forcing us not to use as the Y-axis label.
Figure 7.25 (a) Startup shear of PB1M(5%)-10K in creep mode, namely, at different applied stresses, in the absence of any shear strain localization, showing entanglement–disentanglement transition like behavior, represented as /σ versus
t
, which allows all the curves to collapse at short times when the responses are elastic. (b) Corresponding more conventional probe of nonlinear rheological responses based on rate-controlled startup shear in terms of the transient viscosity σ(
t
)/ where the steady shear stresses are also indicated. (c) Inverse of the data in (a), namely, transient viscosity in creep mode, showing stronger overshoot. (d) Steady shear stress versus steady rate in both creep- and rate-controlled modes.
Figure 7.26 Depiction of the limiting responses of entangled polymeric fluids upon startup: (a) simple shear and (b) uniaxial extension, showing initial elastic deformation and eventual flow. Here the illustration does not include the phenomenon of melt rupture observed at sufficiently high rates (cf. Section 11.3).
Figure 7.27 Normalized engineering stress σ
engr
as a function of Hencky strain ϵ for three Hencky rates of 0.004, 0.008, and 0.016 s
−1
, where the open symbols are calculated using Eq. (7.11).
Figure 7.28 Engineering stress σ
engr
as a function of strain, expressed as (λ − 1/λ
2
), for a wide range of Hencky rate from 0.003 to 10 s
−1
. The final data points in each curve represent the onset of visible tensile strain localization.
Figure 7.29 Depiction of fast startup uniaxial extension of entangled melts at different stages before and after the yield point (the engineering stress maximum), up to the point where the extension is still homogeneous. Well before the yield point, a non-cross-linked melt acts like a permanent network as depicted. Thus, the number of extended strands per unit cross-sectional area increases from state (a) to (b) at the expense of the non-load-bearing strands such that the mass density remains constant. Clearly, during the affine-like extension, there is geometric condensation (from (a) to (b)) of the active entanglement strands (EntS), namely, ψ increases in the plane whose normal is along the direction of extension.[70] A detailed mathematical account of the geometric condensation concept is provided in Appendix 8.A in Chapter 8. There emerges partial yielding when the stress response softens, namely, no longer increases according to Eq. (1.52). As depicted, the total number of EntS decreases upon reaching the yield point. Further loss of EntS across the sample's cross section accompanies the smooth decline of σ
engr
. This “dissolution” of the entanglement network eventually reaches a point where uniform extension cannot be sustained, and the sample nucleates a weak segment where uneven extension occurs due to the localized yielding of the entanglement network. The specimen eventually fails. More discussion on this subject is given in Chapter 11.
Figure 7.30 Yield Hencky strain ϵ
y
, which is the strain at the peak of σ
engr
, read from Figure 7.28, as a function of
Wi
. The continuous curve is given by the Maxwell model in Eq. (7.12).
Figure 7.31 Master curve of ϵ
y
versus
Wi
R
, where the extension tests at smaller
Wi
R
in filled symbols end up in tensile decohesion and tests at larger
Wi
R
in open symbols eventually break up through necking-like tensile strain localization. These extensional failures are the subject of Chapter 11. The borderline between tensile failure and shear-yielding-induced necking is denoted
Wi
R
*
. Around
Wi
R
= 1 and beyond, there appears to be a scaling regime with exponent 1/3, as given by Eq. (7.21), reminiscent of the scaling behavior seen in Figure 7.8(b). The plus-containing squares are from Eq. (16.21).
Figure 7.32 Normalized yield engineering stress σ
engr(y)
as a function of the yield strain, expressed as (λ
y
− 1/), where the open and filled symbols have the same meanings as in Figure 7.31.
Figure 7.33 Depiction of two elastic recovery tests involving fast startup uniaxial extension of entangled melts and setting the tensile stress to zero either before or after reaching the peak engineering stress at λ
y
. Characteristically, the elastic recovery is nearly complete for λ < λ
y
and incomplete for λ > λ
y
.
Figure 7.34 Recoverable strain ϵ
rec
versus the elapsed strain ϵ, for eight different extension rates, at different stages of uniaxial extension. Here the vertical arrows indicate the locations (i.e., ϵ
y
) of the yield point at three extension rates 0.8, 1.6, and 4.0 s
−1
that satisfies
Wi
R
> 1. Beyond the yield points at the respective extension rates, the recovered strains are denoted with filled symbols.
Figure 7.35 Engineering stress σ
engr
as a function of Hencky strain ϵ for Hencky rates 0.4, 0.08, 0.04, and 0.008 s
−1
. These rates produce peak stresses at Hencky strains of ϵ
y
= 0.92, 0.75, 0.69, and 0.48, respectively.
Figure 7.36 Recoverable strain, defined by the ratio of initial (L
0
) to final length (L
∞
) of the specimen, from different stages in startup uniaxial extension of an SBR241K melt at a Hencky rate of 1 s
−1
. The inset shows engineering (large squares) and Cauchy (small diamonds) stresses σ
engr
and σ
E
as functions of the stretching ratio λ. The eight vertical arrows indicate the strains at which the extended specimen is released to commence elastic recovery.
Chapter 8: Strain Hardening in Extension
Figure 8.1 Sketch of transient shear viscosity η
+
= σ(
t
)/ at different rates where the thick curve expresses the zero-shear limit.
Figure 8.2 Sketch of transient extensional viscosity = σ
E
(
t
)/ at different rates where the thick curve expresses the zero-rate limit.
Figure 8.3 States of entanglement in the low-rate (
Wi
≪ 1) regime I and high-rate (
Wi
> 1) regime II, respectively, before and after yielding. Analogous to Eq. (1.55), the shear stress σ is determined by the product of the areal number density ψ of entanglement strands (dots) and the retraction force borne by each strand. In regime II, shear stress decline takes place as the total number of active entanglement strands decreases, leading to the reduced ψ.
Figure 8.4 States of entanglement in regimes I, I′, and II for
Wi
≪ 1,
Wi
> 1, and
Wi
≫ 1, respectively. Unlike simple shear, there can be a larger areal number density of entanglement strands than is found in an equilibrium system because the loading area, that is, the XY cross section, continuously shrinks during uniaxial extension, as shown in the Figure for regime II under the condition of
Wi
≫ 1.
Figure 8.5 Transient viscosity at a vanishing effective rate (
Wi
≪ 1) according to the Maxwell model of Eq. (8.5) (circles) and at
Wi
= 10 (diamonds) and 100 (squares), respectively, according to Eq. (8.6) for rubber elasticity, resembling Figure 8.2. The diamonds and squares are also plotted against the Henckystrain on the upper X axes.
Figure 8.6 (a) Transient viscosity of a low-density polyethylene (LDPE) melt at 130
o
C and Hencky rates ranging from 0.03 to 3 s
−1
from startup uniaxial extension tests, based on an SER apparatus on ARES-G2, where the line with dots is the reference representing the zero-rate envelope. The inset shows the storage and loss moduli of this LDPE. (b) Engineering stress versus stretching ratio, replotted from the identical data in (a).
Figure 8.7 Engineering stress versus strain (stretching ratio λ) of a dendritic 3,4-polyisoprene melt, at four Hencky rates from 0.03 to 0.75 s
−1
, where the top curve is the plot of the rubber elasticity formula. Also plotted is a curve according to the Maxwell model given in Eq. (7.11), with
Wi
= 0.1 and G
pl
= 0.53 MPa, which apparently cannot depict the rheological response well after the yield point.
Figure 8.8 Transient viscosity at Hencky rate 0.3 s
−1
of SBR1M along with its binary mixtures with SBR20K at volume fractions of φ = 0.8, 0.6, and 0.4, respectively, showing “strain hardening” becoming stronger with decreasing φ. All four samples have comparable Rouse time. But the reptation time smoothly decreases from 11,000 to 4,100 s with lowering φ at room temperature.
Figure 8.9 Transient viscosity of SBR1M(40%) at three rates 0.03, 0.1, and 0.3 s
−1
as well as the corresponding representation in terms of the engineering stress σ
engr
versus strain (λ) in the inset.
Figure 8.10 Transient extensional viscosity versus time of PS285K melt and PS1.76M(18%)-4K mixture, obtained from filament stretching rheometric measurements at 130
o
C, at the equivalent rates defined as the product of Hencky rate and a characteristic relaxation time similar to τ
e
.
Figure 8.11 (a) Engineering stress σ
engr
versus strain, expressed as (λ − 1/λ
2
), for a wide range of Hencky rate from 0.001 to 15 s
−1
, covering tensile failure and melt rupture across the spectrum of different responses (cf. Chapter 11). The last data points in each curve represent the onset of visible tensile strain localization.
Source
: Data taken from Zhu
et al
., 2013 [16]. (b) Transient extensional viscosity as a function of time, based on the data in (a).
Figure 8.12 (a) Entanglement network, denoted by straight lines that cross one another at the entanglement points, undergoes affine uniaxial extension along Z direction. (b) Viewed “head-on” along Z-axis in the XY-plane, in the affine deformation limit, the same number of entanglement strands (filled circles) is seen to condense to a smaller cross-sectional area by a factor of λ = 3. Each entanglement strand has a molecular cross-sectional area of
s
=
pl
K
, as shown in Eq. (2.38a). A cross section in the XY-plane is occupied by both entanglement strands (filled circles) and strands (open circles) that do not contribute to the entanglement network. Open circles denote the fact that an XY-plane bisects one entanglement strand multiple times as shown in (c). With increasing λ in the affine extension, the fraction ξ(λ) of a cross-sectional area occupied by the filled circles increases linearly with λ as ξ(λ) = λξ
i
, where the initial value ξ
i
at λ = 1 is given by ξ
i
=
s
ψ
ent
, with ψ
ent
given by Eq. (2.31) and
s
by Eq. (2.38a). The inverse of this fraction is exactly the number
q
of times an entanglement strand folds back and forth within its physical volume across an area of π(
l
ent
)
2
according to Eq. (2.39a):
q
(
n
e
) = 1/ξ
i
. (c) In the undeformed state, the Gaussian strand has a coil size
l
ent
with
N
e
Kuhn segments (Eq. 2.29). From the coiled state with an areal number density ψ
ent
of entanglement strands, in the affine extension limit, an entanglement strand stretches until it is fully extended at defined in Eq. (8.A1). In this limit of full extension of the entanglement strand, ξ() = ξ
i
= 1 so that the cross section is all occupied by entanglement strands. The mass density conservation means the pervaded volume Ω
e
at λ = 1 should be the same as the volume given by piling up these straightened strands (of length
N
e
l
K
and cross-sectional area
s
) at , namely, (
N
e
l
K
)
s
Q
e
=
N
e
l
K
×
pl
K
(Ω
e
/ν
e
) ≡ Ω
e
because the physical volume of an entanglement strand is ν
e
≡ (
l
ent
)
2
p
, where use is made of Eqs. (2.29) and (2.38a).
Chapter 9: Shear Banding in Startup and Oscillatory Shear: Particle-Tracking Velocimetry
Figure 9.1 (a) Particle-tracking velocimetric measurements of velocity profiles showing the velocity v(y) at each height y along the gap distance, at
Wi
app
= 0.5 and 1, respectively, of a 10% polybutadiene (PB) solution, using data from Figure 2(b) in Ref. [29]. In Ref. [29] both axes were mislabeled, in incorrect units of centimeters and centimeters per second. (b) Velocity profiles at different times at rather high
Wi
app
of 10 and 140 of a 5% PB solution, using data from Figure 1(b) in Ref. [18]. With the solution viscosity η
0
= 21,062 Pa s and η
s
= 97 Pa s, the slip length
b
max
of PB0.7M(5%)-1.8K is estimated to be (21,062/97)
l
ent
(φ = 1)0.05
−0.6
< 0.01 mm where
l
ent
(φ = 1) is ca. 3.7 nm.
Figure 9.2 Strain field in simple shear in the presence of “internal slip,” depicted by a discontinuity of magnitude V
s
in the velocity file, so that the local shear rates deviate from the nominal rate or averaged rate of V/H.
Figure 9.5 Normalized velocity profiles, where
H
is the gap distance and V is the velocity of the moving surface, showing how the system transforms from shear banding to the approximate shear homogeneity as V/H increases. The particle-tracking velocimetric video
1
shows that the shear banding seems to prevail for 2 s
−1
at long times, namely, after 350 s.
Figure 9.8 Velocity profiles at different times at three respective apparent rates at V/H = (a) 1 s
−1
, (b) 5 s
−1
, and (c) 6 s
−1
. Corresponding rheological data are given in the insets.
Figure 9.3 (a) Time-dependent shear stress at V/H = 5.0 s
−1
, showing stress overshoot of a well-entangled DNA solution. (b) The corresponding velocity profiles at different times.
Figure 9.4 Entanglement–disentanglement transition (EDT) of a well-entangled DNA solution at an applied shear stress of 22 Pa, where the inset shows the PTV measurements, revealing the time evolution of the actual velocity profiles.
Figure 9.6 Velocity profiles of startup shear at V/H = 1 s
−1
at various stages indicated by the vertical arrows in the inset, which shows the transient stress response in the form of an overshoot.
Figure 9.7 Storage and loss moduli G′ and G″ from small amplitude oscillatory shear of 1M(10%)-9K solution, revealing a terminal relaxation time of τ = 63 s.
Figure 9.9 PTV determination of velocity profiles at various stages during creep at 2400 Pa corresponding to the time-dependent rise of the apparent shear rate in Figure 7.21, where the symbols at y = 0 and y = H are values of the speeds at the two walls.
Figure 9.10 (a) Stress responses to startup shear of PB1M(13%)-10K solution at nominal rates of 1.5, 3, and 6 s
−1
. (b) Velocity profile at V/H = 1.5 s
−1
, which shows three bands of rates different from 1.5 s
−1
. (c) Velocity profiles at the other two rates of 3 and 6 s
−1
. (d) Different local shear rates at the four different applied nominal rates (open squares). Here, the open circles represent the level of rate in the slip band at the moving plate plotted against the right-hand-side Y axis.
Figure 9.11 (a) Velocity profiles of an entangle DNA solution at different stages during startup shear at an apparent rate of 5 s
−1
. (b) The corresponding stress versus time curve where the vertical arrows indicate the stages for the PTV observations in (a).
Figure 9.12 Velocity profiles at three different moments during startup shear at an apparent rate of 1 s
−1
, showing apparent wall slip.
Source
: Data taken from Ravindranath and Wang 2008 [19].
Figure 9.13 Long-time velocity profiles at three different shear rates during startup shear of PB1M(10%)-5K, showing shear banding and apparent wall slip. Here the open symbols denote the speeds of the walls in the absence of wall slip.
Source
: Data taken from Ravindranath and Wang 2008 [19].
Figure 9.14 (a) Velocity profiles at four different moments during startup shear at 0.5 s
−1
along with the corresponding stress versus time plot in the inset. (b) Long-time velocity profiles at V/H = 5 s
−1
from two different loadings along with the stress versus time plot in the inset.
Figure 9.15 (a) Linear velocity profiles for PB1M(10%)-15K upon startup shear at apparent rates 0.2, 1, and 2 s
−1
. (b) Linear velocity profiles upon startup creep at 2000 Pa at different stages of the same solution where the inset shows the entanglement–disentanglement transition, namely, the rise of the shear rate over time during the creep.
Figure 9.16 Apparent storage and loss moduli G′ and G″ of 1M(15%)-9K solution at room temperature from LAOS at strain amplitude γ
0
= 70% and 100% and frequency ω = 1 rad/s, based on 25 mm parallel-plate disks where γ
0
is taken as the strain at a radial distance of 4 mm from the edge.
Figure 9.17 Velocity profiles at the instant of maximum plate speed for different times at a radial distance of 4 mm from the edge in 25 mm parallel-plate disks. The sample at this radial distance experienced 100% strain amplitude and the applied frequency was 1 rad/s.
Figure 9.18 Velocity profiles at three different frequencies of 0.25, 0.5, and 1 rad/s at 63, 50, and 50 s from the start of the LAOS, respectively.
Chapter 10: Strain Localization in Extrusion, Squeezing Planar Extension: PTV Observations
Figure 10.1 Pressure P at the die entry versus volumetric flow rate Q
f
from the die in steady state, calculated from constant piston speed and constant pressure, for SBR241K melt whose characteristics are provided in Tables 7.5 and 7.6. At 70 °C, the terminal polymer dynamics are faster than at room temperature by a factor of 24. For the no-slip branch under constant Q
f
, a die of L = 32.0 mm and D = 2.0 mm is used while the rest of the constant-Q
f
data is based on a die of L = 16.0 mm and D = 1.0 mm. The controlled-pressure data (circles) were obtained from a Monsanto capillary rheometer while the controlled-speed data (squares and diamonds) were obtained from a Rosand capillary rheometer.
Source:
Reproduced from unpublished research results based on the doctoral thesis of Xiangyang Zhu.
Figure 10.2 Pressure responses expressed in terms of wall shear stress σ = PD/4L, where P is the piston pressure, at seven different applied volumetric throughputs, determined by the piston speed according to Eq. (10.1), Q
b
= 0.4, 0.5. 0.6, 0.7, 0.8, 1.5, and 2.0 mm
3
/s respectively.
Figure 10.3 Schematic illustration of particle-tracking velocimetric setup for observations at the die entry A and die inland B from a slit die.
Figure 10.4 Piston pressure P (on the left-hand-side Y axis) and slip velocity in the slit die (on the right-hand-side Y axis) according to PTV measurements, against γ
w
=
t
, which is proportional to the elapsed time, at constant throughput Q
b
= 4 mm
3
/s, based on a Rosand capillary rheometer. The onset at γ
w
= 0 is defined as the moment when the piston pressure P has built up to a level around 9 MPa. P reaches a maximum at γ
max
= 325, that is,
t
= 69 s. At around γ
w
= 200, P reaches the same level as its steady-state value, around 18 MPa. Also indicated is the steady-state slip velocity of V
s(ss)
= 0.625 mm/s = Q
b
/wH.
Figure 10.5 Entrance pressure loss P
entrance
and total pressure P versus γ
w
, both showing a maximum of different origins, where the dashed lines are drawn to “fill in” the missing data prior to the emergence of the visible wall slip. On the dashed lines, the extrusion involves the no-slip branch, characterized by high piston pressure at negligible Q
f
as shown in Figure 10.1. As in Figure 10.4, γ
w
is defined to be zero when V
s
= 0. Thus, during the period of no-slip, γ
w
is negative.
Figure 10.6 Streak images of PTV particles in the convergence region of a microfluidic device, showing (a) 1% DNA solution at an effective
Wi
= 212 evaluated from Q
f
= 20 µL/h and (b) 0.5% DNA solution at
Wi
= 717 (Q
f
= 1000 µL/h). In (a), there is a zone where the stream lines involve high velocity near the entry, with the two corners relatively stagnant, as shown in the magnified image. In contrast, the DNA solution of lower concentration is incapable of developing shear banding and cannot avoid vortex formation.
Figure 10.7 Streak images of PTV particles in the convergence region of a microfluidic device, showing (a) 0.7% water-based DNA solution at an effective
Wi
= 3000 (Q
f
= 800 µL/h) and (b) 0.7% DNA solution based on 40% sucrose and 60% water at
Wi
= 5000 (Q
f
= 800 µL/h), showing strain localization (shear banding) and vortex flow, respectively. The blow-up are the two images on the first row.
Figure 10.8 Snapshot of an extrudate prepared by prefilling the die, followed by resting before application of a high piston pressure to drive the capillary extrusion well beyond the stick–slip transition.
Figure 10.9 Deformation field during the “steady” extrusion revealed by PTV measurements in the entry region for SBR536K using a slit die shown in Figure 10.3 mounted onto a Rosand capillary rheometer. The volumetric throughput Q
b
= 12 mm
3
/s and P is around 20 MPa.
Figure 10.10 Velocity profile for SBR536K according to the particle-tracking velocimetric observation, corresponding to Figure 10.9.
Chapter 11: Strain Localization and Failure during Startup Uniaxial Extension
Figure 11.1 Master curve of Malkin and Petrie,[4] schematically illustrating the dependence of the limiting (failure) strain ϵ
*
on Hencky strain rate , in uniaxial extension of polymer liquids, drawn according to Ref. [4]. Regime I is a flow-dominant domain with
Wi
≪ 1 where there is little ongoing elastic process. Malkin and Petrie labeled regime II as the transition zone, involving a “superposition of elastic deformation and viscous elongational flow.” Regime III was termed “rubbery,” where “the entire deformation is completely elastic.” Finally, regime IV was identified as the “glassy-like” regime, although the reason to use such a term is not obvious.
Figure 11.2 Engineering stress versus strain (λ − 1/λ
2
) of SBR325K at room temperature.
Figure 11.3 “Phase diagram” depicting four regimes from I and IV as a function of either
Wi
