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Carnegie Mellon 1
Rheological Behavior and Polymer Properties
G. C. Berry
Department of Chemistry
Carnegie Mellon University
Colloids, Polymers and Surfaces
e-mail: [email protected]
web site: http://www.chem.cmu.edu/berry
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Carnegie Mellon 2
• Introduction 3 (12)
• Rheological methods 16 (19)
• Linear elastic parameters 26 (5)
• Linear viscoelastic functions 33 (12)
• Several viscoelastic experiments 44 (16)
• Relations among linear viscoelastic functions 62 (10)
• Examples of linear viscoelastic functions 73 (9)
• Time-temperature equivalence 83 (9)
• The glass transition temperature 93 (13)
• The viscosity 107 (26)
• Effects of polydispersity 134 (4)
• Network formation 139 (13)
• Isochronal Behavior 153 (6)
• Examples from the literature 160 (45)
Branched and linear metallocene polyolefins 161 (10)
Colloidal dispersions 172 (9)
Wormlike Micelles 182 (4)
Deformation of rigid materials 187 (4)
Nonlinear shear behavior 192 (16)
Linear and nonlinear bulk properties 209 (6)
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Carnegie Mellon 3
• Introduction
• Rheological methods
• Linear elastic parameters
• Linear viscoelastic functions
• Several viscoelastic experiments
• Relations among linear viscoelastic functions
• Examples of linear viscoelastic functions
• Time-temperature equivalence (Thermo-rheological simplicity)
• The glass transition temperature
• The viscosity
• Effects of polydispersity
• Network formation
• Isochronal Behavior
• Examples from the literature
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Carnegie Mellon 4
PROTEINS
POLYNUCLEOTIDES
POLYSACCHARIDES
GUMS
RESINS
ELASTOMERS
THERMOPLASTIC THERMOSETTING
NATURAL SYNTHETIC
POLYMERS
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Carnegie Mellon 5
Some Common Elastomers, Plastics and Fibers
ELASTOMERS PLASTICS FIBERS
Polyisoprene polyethylene
polyisobutylene polytetrafluoroethylene
poybutadiene polystyrene
poly(methyl methacrylate)
Phenol-formaldehyde
Urea-formaldehyde
Melamine-formaldehyde
←⏐⏐⏐⏐⏐⏐⏐⏐ Poly(viny l chlorid )e ⏐⏐⏐⏐⏐⏐⏐⏐⏐⏐⏐→
←⏐⏐⏐⏐⏐⏐⏐⏐ Polyurethanes⏐⏐⏐⏐⏐⏐⏐⏐⏐⏐⏐⏐⏐⏐⏐→
←⏐⏐⏐⏐⏐⏐⏐⏐ Polysiloxanes ⏐⏐⏐⏐⏐⏐⏐⏐⏐⏐⏐⏐⏐⏐⏐→
←⏐⏐⏐⏐⏐⏐⏐⏐⏐⏐ Polyamide⏐⏐⏐⏐ ⏐⏐→
←⏐⏐⏐⏐⏐⏐⏐⏐⏐⏐ Polyester ⏐⏐⏐⏐⏐ ⏐⏐→
←⏐⏐⏐⏐⏐⏐⏐⏐⏐⏐ Polypropylen e ⏐⏐⏐ ⏐→
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Carnegie Mellon 6
Molecular Weight M
Mn
Mw
Mz
Fraction of Molecules With
Molecular Weight M
A Schematic Illustration of a Typical Distribution
of Molecular Weights, showing Mn, Mw, and Mz
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Carnegie Mellon 7
A generalized Average of molecular weights:
wµ is the weight fraction of polymer with molecular weight Mµ:
M(α) = ⎝⎜⎛
⎠⎟⎞Σ
µ wµMµ
α 1/α
Specia l Case :s
Numb er average:
Mn = M(0)/M−1
(−1) = 1/⎝⎜⎛
⎠⎟⎞Σ
µ wµM µ
−1
Weight avera ge:
Mw = M(1)/M(0) = Σµ
wµMµ
z-averag e:
Mz = M 2(2)/M(1) = ⎝⎜
⎛⎠⎟⎞Σ
µ wµMµ
2 /Σµ
wµMµ
G. C. Berry "Molecular Weight Distribution" Encyclopedia of Materials
Science and Engineering, ed. M. B. Bever, Pergamon Press, Oxford, 3759-68 (1986)
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Carnegie Mellon 8
Temperature
Specific Volume
Tm
A schematic v-T diagram for a typical nonpolymeric material.
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Carnegie Mellon 9
Temperature
Specific Volume
TmTg
A schematic v-T diagram for a typicalsemi-crystalline polymeric material.
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Carnegie Mellon 10
Temperature
Specific Volume
Tg
A schematic v-T diagram for a typical
noncrystalline polymeric material.
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Carnegie Mellon 11
Rigid Plastic
Flexible Plastic
Elastomer
Strain
Stress
Typical Stress-Strain Behavior for Plastics and Elastomers
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Carnegie Mellon 12
F. W. Billmeyer Jr. (1976):J. Polym. Sci.: Symp. (1976) 55: 1-10
"…characterization of polymers is inherently more
difficult than that of other materials. Polymers are
roughly equivalent in complexity to, if not more complex
than, other materials, at every physical level of
organization from microscopic to macroscopic…"
"We would wish, ideally, to characterize all aspects of a
polymer structure in enough detail to predict its
performance from first principles. I seriously doubt that
this will ever be possible, and I am sure that even if it
were, it would never be economically feasible."
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Carnegie Mellon 13
Microscopic Characterization Needed at Many Resolutions:
A
A
B
A
BB
A A B A BB
A
A
B
A
B B
C C C
C C
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Carnegie Mellon 14
2-D projection of a random arrangement of a chain
with 1000 non-overlapping bonds, each step
otherwise randomly selected
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Carnegie Mellon 15
Mean chain dimensions:
For a linear chain with contour length L
(without excluded volume effects):
Mean square-end-to-end dimension:
R2L = 2âL
â is the persistence length (2â is the Kuhn length)
for a flexible chain, â << L.
Mean square-radius of gyration:
R2G = R2
L/6 = âL/3
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Carnegie Mellon 16
• Introduction
• Rheological methods
• Linear elastic parameters
• Linear viscoelastic functions
• Several viscoelastic experiments
• Relations among linear viscoelastic functions
• Examples of linear viscoelastic functions
• Time-temperature equivalence (Thermo-rheological simplicity)
• The glass transition temperature
• The viscosity
• Effects of polydispersity
• Network formation
• Isochronal Behavior
• Examples from the literature
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Carnegie Mellon 17
Computer System
for
Data Acquisition
and
Instrument Control
Rheometer
Torque
Transducer
Position
Transducer
Temperature
Transducer
Force
Transducer
Shape
Transducer
Output
Interfaces
Shear Stress
vs
Time (Frequency)
Shear Strain
vs
Time (Frequency)
Normal Force
vs
Time (Frequency)
Temperature
vs
Time
Schematic of Rheometer System
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Carnegie Mellon 18
CONTROLLED STRESS
IN TENSION
Sample
"Frictionless"
Bearing
Removable
Weight
Position
Transducer
Tare
Device Input Output
Removable Weight Controlled weight Controlled force
Position Transducer Measure of shaft position Voltage (current)
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Carnegie Mellon 19
CONTROLLED DEFORMATION
IN TENSION
Sample
Position
Transducer
Crosshead
Drive Screws
Device Input Output
Crosshead Drive Controlled Drive Controlled force
Position Transducer Measure of shaft position Voltage (current)
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Carnegie Mellon 20
CONTROLLED STRESS RHEOMETER
Sample
Shaft
"Frictionless mount"
Controlled
Torque
Drive
Fixed Shaft
(Alternate: controlled rotation)
Fixtures
Angle
Position
Transducer
Device Input Output
Controlled Torque Drive Controlled voltage Controlled torque
Angle Position Transducer Measure of shaft angle Voltage (current)
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Carnegie Mellon 21
CONTROLLED DEFORMATION RHEOMETER
Sample
Shaft
"Frictionless mount"
Controlled
Rotation
Drive
Fixtures
Angle
Position
Transducer
Torque Transducer
(Force Transducer)
Device Input Output
Controlled Deformation Drive Controlled voltage Controlled shaft rotation
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Carnegie Mellon 22
Electromagnetic Coils
I : A-F
II : a-f
,
AB
a
b
C
D
c
d
EF
e
f
G
H
g
h
Aluminum Cylinder
Attached to Rotor
Iron Core
• Phasing of the currents in Coils I and II can produce a time-
dependent torque:
◊ Constant torque amplitude
◊ Sinusoidal torque amplitude
• Torque amplitude may readily be varied over a factor of 106.
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Carnegie Mellon 23
Sample
2R
Height h
Ω
Fixtures
Parallel Plates
Sample
Fixtures
Angle α
2R
Ω
& Cone Plate
Sample
Fixtures
Δ R
2R
h
Ω Concentric Cylinders
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Carnegie Mellon 24
Geometric Factors in Rheometry
Geometry Measured Calculateda
Translational geometriesParallel Plate Force: F Stress: σ = F /wb
widt ,hw; breadt h b; separati on h Displacemen :t D Strain: γ= D/h
Concentric Cylinders Force: F Stress: σ = F /2πRh
inner radius R; gap Δ; heigh th Displacemen :t D Strain: γ= D/Rln(1 + Δ/R)
Rotationa l geometries
Parallel Plate Torque: M Stress: σ = (2 /r R)M /πR3
outer radius R; separation h Rotation: Ω Strain: γ(r) = ( /r h) Ω
Cone & Plate Torque: M Stress: σ = (3/2)M /πR3
outer radius R; cone angle π - α Rotation: Ω Strain: γ= (1/α) Ω
Concentric Cylinders Torque: M Stress: σ = (R/2h)M /πR3
inner radius R; gap Δ; heigh th Rotation: Ω Strain: γ(r) = (R/ΔR) Ωf(R,r)
f(R,r) = ( /R r)21 + Δ/R1 + Δ/2R
a σ and γ are the shea r stress and strai ,n respectively
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Carnegie Mellon 25
Functions and Parameters Used
Function/Parameter Symbol Units
Time t T
Frequency ω T-1
Stra in Component εij---
Elongationa l strain ε ---Shear strain γ ---Rat eof shear γ, ε T-1
Stre ss Component SijML-1T-2
Shear stress σ ML-1T-2
Modulus G, K, E ML-1T-2
Compliance J , B, D M-1LT2
Viscosity η ML-1T-1
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Carnegie Mellon 26
• Introduction
• Rheological methods
• Linear elastic parameters
• Linear viscoelastic functions
• Several viscoelastic experiments
• Relations among linear viscoelastic functions
• Examples of linear viscoelastic functions
• Time-temperature equivalence (Thermo-rheological simplicity)
• The glass transition temperature
• The viscosity
• Effects of polydispersity
• Network formation
• Isochronal Behavior
• Examples from the literature
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Carnegie Mellon 27
Linear elastic phenomenology
Shear stress σShea r strain γ
γ = Jσ = (1/G)σ
Elongati onal stress σΤ
Elongati onal strain ε
ε = DσΤ = (1/E)σΤ
Pressure ∆PVolume cha nge ∆V
∆V/V = B∆ P = (1/K)∆P
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Carnegie Mellon 28
Linear Elastic Functions
Shear Compliance
Shear Modulus
J
G
Bulk Compliance
Bulk Modulus
B
K
Tensile Compliance
Tensile Modulus
D = J/3 + B/9
1/E = 1/3G + 1/9K
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Carnegie Mellon 29
Linear elastic phenomenology
εij = 12 ⎝
⎜⎛
⎠⎟⎞∂ui
∂xj +
∂uj∂xi
; u is the displacement
vector
2εij = J [Sij – 13 δij Sαα] + (2/9)δij B Sαα
Sij = 2G [εij – 13 δij εαα ] + δij K εαα
δij = 1 if i = j, and δij = 1 if i ≠ j
In this notation,
Shear stress σ = S12
Shear strain γ = 2ε12
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Carnegie Mellon 30
Relations Among Linear Elastic Constants
K, G E, G K, E K, ν E, ν G, ν
K K EG3[3G – E]
K K E3[1 – 2ν]
2G[1 + ν]3[1 – 2ν]
E 9KG3K + G
E E 3K(1 – 2ν) E 2G(1 + ν)
G G G 3KE9K – E
3K[1 – 2ν]2[1 + ν]
E2[1 + ν]
G
ν 3K – 2G6K + 2G
E2G – 1
3K – E6K
ν ν ν
J = 1/G, = 1/B K, =D 1/E
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Carnegie Mellon 31
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Carnegie Mellon 32
• Introduction
• Rheological methods
• Linear elastic parameters
• Linear viscoelastic functions
• Several viscoelastic experiments
• Relations among linear viscoelastic functions
• Examples of linear viscoelastic functions
• Time-temperature equivalence (Thermo-rheological simplicity)
• The glass transition temperature
• The viscosity
• Effects of polydispersity
• Network formation
• Isochronal Behavior
• Examples from the literature
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Carnegie Mellon 33
Linear Elastic Functions
Shear Compliance
Shear Modulus
J
G
Bulk Compliance
Bulk Modulus
B
K
Tensile Compliance
Tensile Modulus
D = J/3 + B/9
1/E = 1/3G + 1/9K
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Carnegie Mellon 34
Linear Viscoelastic Functions
Shear Compliance
Shear Modulus
J(t)
G(t)
Bulk Compliance
Bulk Modulus
B(t)
K(t)
Tensile Compliance
Tensile Modulusa
D(t) = J(t)/3 + B(t)/9
1/E(s) = 1/3G(s) + 1/9K(s)
a. The superscript " " denotes a Laplace transform.
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Carnegie Mellon 35
Linear viscoelastic phenomenology—
Stress Controlled
γ(t) = ∑i=1
Ν
J(t – ti) Δσi = ∫0
σ(t)
d[σ(u)] J(t – u)
γ(t) = ∫-∞ tduJ(t – u)
∂σ(u)∂u
γ(t) = Joσ(t) + ∫0 ∞
du σ(t – u) ∂J(u)
∂u
γ (t)
Δσ 1
t
t 1 t 2
γ (t)
Δσ 2
σ (t)
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Carnegie Mellon 36
Linear viscoelastic phenomenology—
Strain Controlled
σ(t) = ∑i=1
Ν
G(t – ti) Δγi = ∫0
γ(t)
d[γ(u)]G(t – u)
σ(t) = ∫-∞ tduG(t – u)
∂γ(u)∂u
σ(t) = Goγ(t) + ∫0 ∞
du γ(t – u) ∂G(u)
∂u
γ (t)
Δγ 1
t
t 1 t 2
σ (t)
Δγ 2
σ (t)
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Carnegie Mellon 37
Linear elastic phenomenology
εij = 12 ⎝
⎜⎛
⎠⎟⎞∂ui
∂xj +
∂uj∂xi
; u is the displacement
vector
2εij = J [Sij – 13 δij Sαα] + (2/9)δij B Sαα
Sij = 2G [εij – 13 δij εαα ] + δij K εαα
δij = 1 if i = j, and δij = 1 if i ≠ j
In this notation,
Shear stress σ = S12
Shear strain γ = 2ε12
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Carnegie Mellon 38
Linear viscoelastic phenomenology
εij = 12 ⎝
⎜⎛
⎠⎟⎞∂ui
∂xj +
∂uj∂xi
; u is the displacement vector
2εij(t) = ∫-∞ tds{J(t – s)[
∂Sij(s)∂s –
13 δij
∂Sαα (s)∂s ]
+ (2/9)δij B(t – s) ∂Sαα (s)
∂s }
Sij(t) = ∫-∞ tds{2G(t – s)[
∂εij(s)∂s –
13 δij
∂εαα (s)∂s ]
+ δij K(t – s) ∂εαα (s)
∂s } In this notation,
Shear stress σ(t) = S12 (t)
Shear strain γ(t) = 2ε12(t)
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Carnegie Mellon 39
Relation between G(t) and J(t)
1t ∫
0 tdu G(t – u) J(u) = 1
with Laplace transform:
s2G(s)J(s) = 1
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Carnegie Mellon 40
Shear Compliance J(t) andRecoverable Shear Compliance R(t)
R(t) = J(t) – t/η = J∞ – [J∞ – Jo]α( ) tα(t): Retardation Function
Shear Modulu s ( )G t
(G )t =Ge + [Go – Ge]ϕ( )t ϕ(t): Relaxation Function
η the (linear) viscosity, with 1/η = 0 for a solid,
Ge the equilibrium modulus, with Ge = 0 for a fluid,
Go the "instantaneous" modulus, with JoGo = 1, and
J∞ the limit of R(t) for large t:
Solid: J∞ = Je = 1/Ge; equilibrium compliance
Fluid: J∞ = Js; steady-state recoverable compliance
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Carnegie Mellon 41
Creep Shear Compliance J(t)
R(t) = J(t) – t/η = J∞ – [J∞ – Jo]α(t)
Shea r Modulus G( )t
(G t) = Ge + [Go – Ge]ϕ(t)
• Linear elastic solid: 1/η = 0, J∞ = Je = 1/Ge, α(t) = ϕ(t) = 0
• Linear viscous fluid: 1/η > 0, Go = 0, α(t) = ϕ(t) = δ(t)
• Linear viscoelastic solid: 1/η = 0, J∞ = Je = 1/Ge, 0 < α(t) < ϕ(t) ≤ 1
• Linear viscoelastic fluid: 1/η > 0, J∞ = Js (= Joe), 0 < α(t) < ϕ(t) ≤ 1
Bulk Compliance B(t) = Be – [Be – Bo]β(t)
Bulk Modulus K(t) = Ke + [Ko – Ke]κ(t)
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Carnegie Mellon 42
Retardation function α(t)
R(t) = J( )t – /tη = J∞ – [J∞ – Jo]α(t)
Relaxation function ϕ(t)
(G t) = Ge + [Go – Ge]ϕ(t)
ϕ (t)
α (t)
Time
Viscoelastic Fluid or Solid
0
1
Linear Viscous Fluid
Linear Elastic Solid
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Carnegie Mellon 43
Simple example of the relation between G(t) and J(t)
Maxwell fluid:
G(t) = Goexp(- t/τ); τ = η/Go
( )J t = J s + /t η; J s = J o = 1/Go
( )R t = J s
Note: ϕ(t) = exp(-t/τ) and α (t) = 0 for this model.
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Carnegie Mellon 44
Often used relations for ϕ(t) and α(t)
A weight se t of exponentials wit h N relaxation times:
α(t) = ∑m
N-1
α i exp(–t/λ i) = 1
J∞ – Jo ∫
-∞ ∞
d(ln λ) L(λ)exp(–t/λ)
ϕ(t) = ∑1
N
ϕ i exp(–t/τi) = 1
Go – Ge ∫
-∞ ∞
d(ln τ) H(τ)exp(–t/τ)
Notes: Σα i = Σϕ i = 1, and
m is equal to 0 or 1 for a solid and fluid, resp.
(1 − m)λ0 > τ1 > λ1 > … > λi > τi > λi+1 > … > λN-1 > τN
(The contribution λ0 is absent for a fluid)
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Carnegie Mellon 45
• Introduction
• Rheological methods
• Linear elastic parameters
• Linear viscoelastic functions
• Several viscoelastic experiments
• Relations among linear viscoelastic functions
• Examples of linear viscoelastic functions
• Time-temperature equivalence (Thermo-rheological simplicity)
• The glass transition temperature
• The viscosity
• Effects of polydispersity
• Network formation
• Isochronal Behavior
• Examples from the literature
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Carnegie Mellon 46
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Carnegie Mellon 47
Creep and recovery with a step shear stress
Stress history: σ(t) = 0 t < 0
σ(t) = σo 0 ≤ t≤ Teσ(t) = 0 t > Te
q = t - T e
Time
0
0
t
t = T e
γ (t)
γ R ( θ ) = γ (t = T e ) - γ (t)
Stress
Strain
γ (t) = a + bt
σ (t) = σo
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Carnegie Mellon 48
The strain in creep for t ≤ Te:
γ(t) = σo∫0 tdu (J t – u) δ( u - 0)
= σo (J t) = σo[R(t) + t/η]
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Carnegie Mellon 49
The strain in creep for t ≤ Te:
γ(t) = σo∫0 tdu (J t – u) δ( u - 0)
= σo (J t) = σo[R(t) + t/η]
The strain fo r ϑ = t– Te > 0 in recovery:
γ(t) = σo∫0 Tedu (J t – u) δ( u - 0) – σo∫Te
tdu (J t – u) δ( u - Te)
γ(ϑ) = σo[ (J ϑ + Te) – (J ϑ)] = σo[R(ϑ + Te) – R(ϑ) + Te/η]
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Carnegie Mellon 50
The strain in creep for t ≤ Te:
γ(t) = σo∫0 tdu (J t – u) δ( u - 0)
= σo (J t) = σo[R(t) + t/η]
The strain fo r ϑ = t– Te > 0 in recovery:
γ(t) = σo∫0 Tedu (J t – u) δ( u - 0) – σo∫Te
tdu (J t – u) δ( u - Te)
γ(ϑ) = σo[ (J ϑ + Te) – (J ϑ)] = σo[R(ϑ + Te) – R(ϑ) + Te/η]
The recoverable strain γR(ϑ) = γ(Te) – γ(t) for ϑ > 0:
γR(ϑ) = σo{ J(Te) – [ (J ϑ + Te) – (J ϑ)]}
= σo{R(ϑ) + R(Te) – R(ϑ + Te)}
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Carnegie Mellon 51
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Carnegie Mellon 52
The stress response for t > 0:
σ(t) = γo∫0 t
duG( t– )u δ(u - 0) = γoG(t)
= γo{Ge + (Go – Ge)ϕ(t)}
σ(∞) = γoGe
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Carnegie Mellon 53
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Carnegie Mellon 54
The stress response for t ≤ Te :
σ(t) = γ ∫0 t du (G –t u) = γ [Get + (Go – Ge)∫0
tds ϕ( )s ]
For a fluid in steady-state deformat , ion σ = ηγ, or
η = σ(∞)/γ = Go ∫0 ∞ ds ϕ( )s
The strain fo r t > Te:
σ(t) = 0 = γ ∫0 Te du (G –t u) + ∫Te
tdu G(t – u)
∂γ(u)∂u
For large Te and t, (full recoil after steady flow) it can be shown
that fo r a fluid thi s gives:
τc = ηJs = ∫0 ∞ds sϕ( )s /∫0
∞ds ϕ( )s
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Carnegie Mellon 55
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Carnegie Mellon 56
The strain response for t > 0:
γ(t) = ωσo∫0 t
duJ( – t u)c (osωu)
In the st -eady st ate limit with l arge t:
γ(t) = σo{ J '(ω)sin(ωt) – J ''(ω)c (osωt)}
I -n pha se(or rea l or storage) dynam iccomplianc :e
J'(ω) = J∞ – ω[J∞ – Jo]∫0 ∞ds α( )s sin(ωs)
Out-of-phas e( orimaginar y orl ) oss dynami ccompliance
J"(ω) = (1/ωη) + ω[J∞ – Jo]∫0 ∞ds α( )s c (osωs)
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Carnegie Mellon 57
Alternatively
γ(t) = σo |J* (ω)|sin [ωt – δ(ω)]
"Dynamic compliance":
|J* (ω)|2 = [ (J' ω)]2 + [ "(J ω)]2
Phas ea ngleδ(ω):tan δ(ω) = "(J ω)/ (J' ω)
For sma ll ω:
J (' ω) ≈ J∞, J"(ω) ≈ 1/ωη, and J"(ω) – 1/ωη ≈ ω
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Carnegie Mellon 58
Oscillation with a sinusoid shear strain
Strain history: γ(t) = 0 < 0t
γ(t) = γosin(ωt) t≥ 0
T hestre ssrespons ef or t> 0 is give n by
σ(t) = ωγo ∫0 t
duG( t– )uc (osωu)
I n thestea -dystat elim itw ithlar ,ge t
σ(t) = γo{G'(ω)sin(ωt) + G''(ω)c (osωt)}
I -n pha se(or rea l or storage) dynam iccomplianc :e
G'(ω) = Ge + ω[Go – Ge]∫0 ∞ds ϕ(s)sin(ωs)
Out-of-phas e( orimaginar y orl ) oss dynami ccompliance
G''(ω) = ω[Go – Ge]∫0 ∞ds ϕ(s)cos(ωs)
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Carnegie Mellon 59
Alternatively
σ(t) = γo |G* (ω)|sin [ωt + δ(ω)]
"Dynamic compliance":
|G*(ω)|2 = [G (' ω)]2 + [G"(ω)]2
Phas ea ngleδ(ω):tan δ(ω) = "(G ω)/ (G'ω)
For sma ll ω:
(G'ω) ≈ Ge + ω2[Go – Ge]∫0
∞ds sϕ(s) fluid⇒ (ωη)2Js
G''(ω) = ω[Go – Ge]∫0 ∞ds ϕ( )s fluid⇒ ωη
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Carnegie Mellon 60
Exact relations among the dynamic moduli and compliances:
|G*(ω)||J* (ω)| = 1
(J' ω) = (G'ω)/| *G (ω)|2
"(J ω) = "(G ω)/| *G (ω)|2
(G'ω) = (J' ω)/|*J (ω)|2
"(G ω) = "(J ω)/|*J (ω)|2
tan δ(ω) = "(J ω)/ (J' ω) = "(G ω)/ (G'ω)
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Carnegie Mellon 61
The dynamic viscosity:
In-phase with the strain rate:
η (' ω) = "(G ω)/ω
Out-o -f phase wi th the strain rate:η"(ω) = (G'ω)/ω
For sma ll ω:
η (' ω) = [Go – Ge]∫0 ∞ds ϕ( )s fluid⇒ η
η (''ω) ≈ Ge/ω + ω[Go – Ge]∫0 ∞ds sϕ(s) fluid⇒ ωη2Js
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Carnegie Mellon 62
• Introduction
• Rheological methods
• Linear elastic parameters
• Linear viscoelastic functions
• Several viscoelastic experiments
• Relations among linear viscoelastic functions
• Examples of linear viscoelastic functions
• Time-temperature equivalence (Thermo-rheological simplicity)
• The glass transition temperature
• The viscosity
• Effects of polydispersity
• Network formation
• Isochronal Behavior
• Examples from the literature
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Carnegie Mellon 63
Linear Viscoelastic Functions
Shear Compliance
Shear Modulus
J(t)
G(t)
Bulk Compliance
Bulk Modulus
B(t)
K(t)
Tensile Compliance
Tensile Modulusa
D(t) = J(t)/3 + B(t)/9
1/E(s) = 1/3G(s) + 1/9K(s)
a. The superscript " " denotes a Laplace transform.
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Carnegie Mellon 64
Relation between G(t) and J(t)
1t ∫
0 tdu G(t – u) J(u) = 1
with Laplace transform:
s2G(s)J(s) = 1
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Carnegie Mellon 65
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Carnegie Mellon 66
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Carnegie Mellon 67
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Carnegie Mellon 68
An often used relation between G(t) and J(t)
A weight set of exponentials with N relaxation times:
α(t) = ∑m
N-1
α i exp(–t/λi) = 1
J∞ – Jo ∫
-∞ ∞
d(ln λ) L(λ)exp(–t/λ)
ϕ(t) = ∑1
N
ϕ i exp(–t/τi) = 1
Go – Ge ∫
-∞ ∞
d(ln τ) H(τ)exp(–t/τ)
Notes: Σα i = Σϕ i = 1, and
m is equal to 0 or 1 for a solid and fluid
λ0 > τ1 > λ1 > … > λ i > τi > λ i+1 > … > λN-1 > τN
(λ0 absent for a fluid)
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Carnegie Mellon 69
Determination of L(λ) (or the αi-λi )set from ( )J t
(Simila r considerations apply to the determination of
(H τ) (or the ϕ i-τi )set from G( )t )
Derivative methods for L(λ):
1st Approx.: L(λ) ≈ ( )M m [∂ ( )R t /∂ln ]t t = λ
( )M m = ∂lnL(λ)/∂ln λ (interative)
2nd Approx.: L(λ) ≈ [∂ ( )J t /∂ln t – ∂2 ( )J t /∂(ln )t 2] t = 2λ
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Carnegie Mellon 70
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Carnegie Mellon 71
Determination of L(λ) (or the αi-λi )set from ( )J t
(Simila r considerations apply to the determination of
(H τ) (or the ϕ i-τi )set from G( )t )
Invers e transform methods for αi-λi:
Th e inver se transform is "ill-posed", and a stable
solution s require s constraints (e. .g , αi ≥ 0)
In a n ofte n use d strategy, a set of logarithmically spaced
λ i ar e chosen suc h that the spa n in 1/λI does no t exceed
the tim e span in the experimental data. A constrained
nonlinea r le ast squares analysis is then use d to
determin e the αi. Commercia l package s are available for
thi s transform.
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Carnegie Mellon 72
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Carnegie Mellon 73
• Introduction
• Rheological methods
• Linear elastic parameters
• Linear viscoelastic functions
• Several viscoelastic experiments
• Relations among linear viscoelastic functions
• Examples of linear viscoelastic functions
• Time-temperature equivalence (Thermo-rheological simplicity)
• The glass transition temperature
• The viscosity
• Effects of polydispersity
• Network formation
• Isochronal Behavior
• Examples from the literature
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Carnegie Mellon 74
Low Molecular Weight Glass Formerslope = 1/3H(τ)(L λ) = 1slope( )G t( )J tη tan δ(ω)t/ηGoJolog λ or log τ log t log ωt/ηη (' ω) = (G''ω)
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Carnegie Mellon 75
Low Molecular Weight Glass Formerslope = 1/3H(τ)(L λ) = 1slope( )G t( )J tη tan δ(ω)t/ηGoJolog λ or log τ log t log ωt/ηη (' ω) = (G''ω) < Polymeric Fluid with M Me = 1/3slope(H τ)(L λ) = 1slope = -1/2slope( )G t( )J tη tan δ(ω)GoJolog λ or log τ log t log ωt/ηη (' ω) = (G''ω)
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Carnegie Mellon 76
Polymeric Fluid with M >> Meslope = 1/3H(τ)(L λ)log λ or log τ = -1/2slope = 1slopeJ NGN( )G t( )J t log tη (' ω) = (G''ω) tan δ(ω) log ωGoJoηt/η Low Molecular Weight Glass Former = 1/3slope(H τ)(L λ) = 1slope( )G t( )J tη tan δ(ω)t/ηGoJolog λ or log τ log t log ωt/ηη (' ω) = (G''ω) < Polymeric Fluid with M Me = 1/3slope(H τ)(L λ) = 1slope = -1/2slope( )G t( )J tη tan δ(ω)GoJolog λ or log τ log t log ωt/ηη (' ω) = (G''ω)
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Carnegie Mellon 77
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Carnegie Mellon 78
Peak I with L(λ) linear in λ1/3 before the peak decreases sharply tozer .o
• The behavior ascribed to peak I, first reported by Andrade, isseen in a variety of materials, such as metals, ceramics,crystall ine and glassy polymers and small organic molecules;the decrease of L(λ) to zero being evident in examples of the
latter.
• The area under peak I provides the contribution JA – Jo to the
total recoverable compliance Js.
• It seems likely that the mechanism giving rise to peak I may be
distinctly different from the largely entropic origins of peaks IIand III described in the following.
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Carnegie Mellon 79
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Carnegie Mellon 80
Peak II that increases in peak area with increasing M until reaching
a certain level, beyond which the peak is invariant with increasing
M, both in area and position in λ
• Peak II is ascribed to Rouse-like modes of motion, either fluid-
like for low molecular weight in the range for which the area
increases with M, or pseudo-solid like (on the relevant time
scale) in the range of M after peak II I develops.
• For low molecular weight, the Rouse model gives the area of
peak II as
Js – (JA + Jo) = (2M/5ϕRT).
• For the pseudo-solid like behavior, obtaining when peak III has
developed, reflecting the effects of intermolecular
entanglement, the area of peak II becomes invariant with M and
given by
JN – (JA + Jo) = (Me/ϕRT).
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Carnegie Mellon 81
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Carnegie Mellon 82
Peak III that develops as peak II area ceases to increase with
increasing ϕM, wi th peakIII deve loping an area invariant with
ϕM, anda maximum a tλMAX that move s to larger λ asλMAX ∝
(ϕM/Mc)3.4 fo r ϕM > Mc
• The area under peak III, also invariant with M, ascribed to theeffects of chain entanglements is given by
Js – (JN + JA + Jo) = (kMe/ϕ2+sRT),
where k is in the range 6-8 in most cases, and
s ≈ 2(ε – 1)/(3ε – 2) ≈ 0 to 1/4 with ε = ∂ln R2G/∂ln M
• Overall,
Js – (JA + Jo) = (2M/5ϕRT)[1 + (ϕ1+sM/kMc)ε]−1/ε
1.0 1.5 2.0 � 2.5 3.0
Log (X)
~
3.5
1.0
1.2
1.4
1.6
1.8
2.0
2.2
Log (J ϕ ) + .Cst
S
2
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Carnegie Mellon 83
• Introduction
• Rheological methods
• Linear elastic parameters
• Linear viscoelastic functions
• Several viscoelastic experiments
• Relations among linear viscoelastic functions
• Examples of linear viscoelastic functions
• Time-temperature equivalence (Thermo-rheological simplicity)
• The glass transition temperature
• The viscosity
• Effects of polydispersity
• Network formation
• Isochronal Behavior
• Examples from the literature
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Carnegie Mellon 84
Consider the following reduced expressions:
[J(t/τc) – Jo]/Js = [R( /t τc) – Jo]/ Js + t/η Js
[ (J t/τc) – Jo]/ Js = [R( /t τc) – Jo]/ Js + t/τc
τc = Jsη'(0) (= Jsη)
T he"tim –e temperatur eequivalence" approximation:
[ (J t/τc) – Jo]/Js is asingl -e value d func tion o f t/τc ove r ara nge of
temperature.
A lthoughrare , ly i f eve , r t rulyaccurat efo r a ll temperatur , e itisnever-the-les s a usefu l andwide ly use d approxima tionf or us ewithmateria lsexhibiting no phas etransition ove r th etemperaturerang e ofinterest.
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Carnegie Mellon 85
Since τc may not be know n over the ra nge o f temperature of
interest, it i s often useful to "reduce" data to a common reference
temperature TREF. Formally, this may be accomplished with
[ (J t) – Jo]/bTJs(TREF) = [R(t) – Jo]/bTJs(TREF) + /tηbTJs(TREF)
[ (J t) – Jo]/bTJs(TREF) = [R(t) – Jo]/bTJs(TREF) + t/hTbTτc(TREF)
[ (J t/aT) – Jo]/bT = [R(t/aT) – Jo]/bT + /thTbTη(TREF)
[ (J t/aT) – Jo]/bT = [R(t/aT) – Jo]/bT + /taTη(TREF)
bT = b(T, TREF) =Js(T)/Js(TREF)
hT = h(T, TREF) =η (' 0)[T]/η'(0)[TREF] {=η(T)/η(TREF)}
aT = bT hT
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Carnegie Mellon 86
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Carnegie Mellon 87
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Carnegie Mellon 88
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Carnegie Mellon 89
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Carnegie Mellon 90
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Carnegie Mellon 91
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Carnegie Mellon 92
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Carnegie Mellon 93
• Introduction
• Rheological methods
• Linear elastic parameters
• Linear viscoelastic functions
• Several viscoelastic experiments
• Relations among linear viscoelastic functions
• Examples of linear viscoelastic functions
• Time-temperature equivalence (Thermo-rheological simplicity)
• The glass transition temperature
• The viscosity
• Effects of polydispersity
• Network formation
• Isochronal Behavior
• Examples from the literature
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Carnegie Mellon 94
Temperature
Specific Volume
Tg
A schematic v-T diagram for a typical
noncrystalline polymeric material.
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Carnegie Mellon 95
A Free Volume Model:
(vf)i = (v – vo)
i at a certain position ri,
v = (specific) volume
vf = free volume
vo
= occupied volume
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Carnegie Mellon 96
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Carnegie Mellon 97
The glass transition temperature Tg
Tg depends on both intramolecular conformation and
intermolecular interactions.
Various Models/Treatments:
• Iso Free Volume: f(Tg) = constant
• Iso Viscous: η(Tg) = constant
• Iso Entropic: ∆S(Tg) = constant
None of these are fully satisfactory are free of arbitraryassumptions, and all contain parameters that can not beindependently evaluated.
The free volume and entropic models provide similar expectations
re the dependence of Tg on chain length and diluent.
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Carnegie Mellon 98
40
60
80
100
120
0 0.2 0.4 0.6 0.8 1
Syndiotactic fraction
T (°C)g
PMMA
40
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Carnegie Mellon 99
Estimation of Tg and Tm via Group Contributions
Tg ≈ M-1ΣYg,i
Tm ≈ M-1ΣYm,i
• The Yx,i represent molar group contributions to the relevant property
• Higher order approximations are available for both cases
D. W. van Krevelen, Properties of polymers : their correlation with chemical structure, theirnumerical estimation and prediction from additive group contributions, 3rd Ed., Elsevier;Amsterdam ; New York, 1990.
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Carnegie Mellon 100
Group Yg,i Group Yg,i Group Yg,i
1. Polyisobutylene only
Group contributions: gY (K·g/mol)
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Carnegie Mellon 101
Group Ym,i Group Group
Group contributions:Y (K·g/mol) m
Y m,iYm,i
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Carnegie Mellon 102
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Carnegie Mellon 103
1.2
1.4
1.6
1.8
2.0
2.2
2.4
200 300 400 500 600 700
boyer
krevelen avg
krevelen calc
T /Km
T /Tg
m
D.W. Van Krevelen, op citR. F. Boyer, Rubber Reviews 36:1303-421
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Carnegie Mellon 104
Both free volume and entropic models give results that may be cast
in the forms:
Tg(M) ≈ Tg (∞){1 + kM/Mn}
1Tg(w, Μ) =
⎝⎜⎛
⎠⎟⎞w
Tg(Mn) + R
1 - wTg;DIL
⎝⎜⎜⎛
⎠⎟⎟⎞
1w + R (1 - )w
Both KM and R are model specific parameters, bes tevaluated
experimental .ly
For example, in thefree volume model, KM and R aris e from theextr afre e vol ume provided by chain ends and diluent, respectively:typical , ly R i s in therange 0.5 to 1.5.
Note, t hat i f Tg;DIL > Tg(Mn), then Tg(w, Mn) is increased bythe
diluent.
[ .G C. Berry J. Phys. Che .m 70:1194-8 (1966)]
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Carnegie Mellon 105
60
80
100
120
T (°C)g
0 1 2 3 4 5
10 /Mn4
p(Syndio) ~ 0.50
Free Radicalp(Syndio) ~ 0.76
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Carnegie Mellon 106
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Carnegie Mellon 107
• Introduction
• Rheological methods
• Linear elastic parameters
• Linear viscoelastic functions
• Several viscoelastic experiments
• Relations among linear viscoelastic functions
• Examples of linear viscoelastic functions
• Time-temperature equivalence (Thermo-rheological simplicity)
• The glass transition temperature
• The viscosity
• Effects of polydispersity
• Network formation
• Isochronal Behavior
• Examples from the literature
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Carnegie Mellon 108
The temperature dependence of the viscosity:
η(T) ≈ ηLOC(T) F(large scale structure, T)
≈ ηLOC(T) F(large scale structure)
"Arrhenius" form:
ηLOC(T) ∝ exp[W/T] if T > (1.5-2)Tg
For melts of crystalline polymers, Tm > (1.5-2)Tg, permitting use
of this simple form.
"Vogel-Fulcher" form:
For amorphous polymers with 0 ≤ (T – Tg)/K < ≈ 200:
ηLOC(T) ∝ exp[C/(T – To)] if T < (1.5-2)Tg
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Carnegie Mellon 109
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Carnegie Mellon 110
The temperature dependence of the viscosity:
η(T) ≈ ηLOC(T) F(large scale structure, T)
≈ ηLOC(T) F(large scale structure)
For amorphous polymers with 0 ≤ (T – Tg)/K < ≈ 200:
ηLOC(T) ∝ exp[C/(T – To)] if T < (1.5-2)Tg
"WLF form":
ηLOC(T)/ηLOC(TREF) = exp[C/(T – To) – C/(TREF – To)]
= exp⎝⎜⎜⎛
⎠⎟⎟⎞
– C(T – TREF)
ΔREF(T – TREF + ΔREF)
with C and To being constants, and ∆REF = TREF – To.
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Carnegie Mellon 111
If TREF = Tg then
ηLOC(T)/ηLOC(Tg) = exp⎝⎜⎜⎛
⎠⎟⎟⎞
– K (T – Tg)T – Tg + Δ
where Δ = Tg – To and K = C/Δ.
For many polymers:
K = 2300 K and Δ = 57.5 K
These parameters may be interpreted in terms of the "free-volume"model
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Carnegie Mellon 112
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Carnegie Mellon 113
Viscosity of Polymers and Their Solutions
η( , M c, T) ≈ ηLOC(T) F(M, c, T)
Dilute solutions
ηLOC(T) ≈ ηSolvent(T)
F(M, c, T) ≈ 1 + [η]c + …
[η] = πNAKR2GRH/M
G. C. Berry J. Rheology 40:1129-54 (1996)
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Carnegie Mellon 114
F(M, c, T) ≈ 1 + [η] + …c
[η] = πNAKR2GRH/M
Spherica l Particles
R = RH = (5/3)1/2RG; K = 50/9
[η]c = (5/2)ϕ
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Carnegie Mellon 115
F(M, c, T) ≈ 1 + [η]c + …
[η] = πNAKR2GRH/M
Flexi ble Chain Linea r Polymers
R2G = (âL/3)α2; α the chain expansion factor
â the persistence length
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Carnegie Mellon 116
F(M, c, T) ≈ 1 + [η] + …c
[η] = πNAKR2GRH/M
Flexibl eC hainLinea r Polymers
R2G = (âL/3)α2; α th echa in expa nsionfactor
â th epersistenc e length
H :igh M
3RH/2 ≈ RG ∝ L1/2; K ≈ 10/3
ML[η] = πNA(20/9)(â/3)3/2α3L1/2 = Φ'(â/3)3/2α3L1/2
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Carnegie Mellon 117
F(M, c, T) ≈ 1 + [η] + …c
[η] = πNAKR2GRH/M
Flexibl eC hainLinea r Polymers
R2G = (âL/3)α2; α th echa in expa nsionfactor
â th epersistenc e length
H :igh M
3RH/2 ≈ RG ∝ L1/2; K ≈ 10/3
ML[η] = πNA(20/9)(â/3)3/2α3L1/2 = Φ'(â/3)3/2α3L1/2
Low :M
RH ≈ L; K ≈ 1
ML[η] = πNA(â/3)L (Debye)
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Carnegie Mellon 118
Flexible Chain Branched Polymers
ML[η] = πNAKR2GRH/L
g = R2G/(R
2G)LIN; calculated α = 1
High M:h = RH/(RH)LIN; K ≈ KLIN (f g, shap )eh ≈ g1/2
[η] = (f g, shap )e g3/2[η]LIN
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Carnegie Mellon 119
Flexible Chain Branched Polymers
ML[η] = πNAKR2GRH/L
g = R2G/(R
2G)LIN; calculated α = 1
High M:h = RH/(RH)LIN; K ≈ KLIN (f g, shap )eh ≈ g1/2
[η] = (f g, shap )e g3/2[η]LIN
Sta :r [η] = g1/2[η]LIN
Comb: [η] = g3/2[η]LIN
Random: [η] = g[η]LIN
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Carnegie Mellon 120
Flexible Chain Branched Polymers
ML[η] = πNAKR2GRH/L
g = R2G/(R
2G)LIN; calcula tedα = 1
H :igh M
h = RH/(RH)LIN; K ≈ KLIN (f ,g shap )eh ≈ g1/2
[η] = (f ,g shap )e g3/2[η]LIN
Star: [η] = g1/2[η]LIN
Com :b [η] = g3/2[η]LIN
Ra :ndom [η] = g[η]LIN
Low :M
[η] = πNAKR2GRH/LML
RH ≈ L; K ≈ 1
[η] = g[η]LIN
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Carnegie Mellon 121
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Carnegie Mellon 122
Viscosity of Polymers and Their Solutions
η(M, ,c T) ≈ ηLOC(T) F(M, c, T)
Concentrated solutions and undiluted linear flexiblechain polymers
ηLOC(T) ≈ ηLOC(Tg)exp{ –K(T – Tg)/(T – Tg +∆)}
F(M, c, T) ≈ 1 + [η](c)c
Low M (Rouse behavior; α = 1):
F(M, c, T) ≈ 1 + ~X ≈
~X
~X = [η](c)c; a modified Fox parameter
ML[η](c) = πNA(â/3)L; ([η](c) independent of c in thisrange)
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Carnegie Mellon 123
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Carnegie Mellon 124
High M (Entanglement regime)
F(M, c, T) ≈ 1 + ~XE(~X/~Xc ) ≈ ~XE(~X/~Xc )
E(~X/~Xc ) = {1 + (~X/~Xc )4.8}1/2
~Xc = πNA(â/3)ρMc ≈ 100 f or ma ny polymers
Mc = ~Xc/πNA(â/3)ρ ≈ 100/πNA(â/3)ρ
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Carnegie Mellon 125
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Carnegie Mellon 126
The dependence of Tg on the diluent concentration must be
considered for polymer solutions:
ηLOC(T)/ηLOC(Tg) = exp⎝⎜⎜⎛
⎠⎟⎟⎞
– K (T – Tg)T – Tg + Δ
where Δ = Tg – To and K = C/∆.
For many polymers:
K = 2300 K and Δ = 57.5 K
∆ is approximately independent of the polymer concentration
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Carnegie Mellon 127
0 0.2 0.4 0.6 0.8 1
Volume Fraction Polymer
To
Tg
T – Tog
0
100
200
300
400
Temperature/K
Polystyrene/Dibenzyl ether
G. C. Berry and T. G Fox Adv. Polym. Sci. 5:261-357 (1968)
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Carnegie Mellon 128
3 4 5 6
log( ϕ )Mw
0
2
4
-2
(log η / )Pa·s
1
3
5
-1
1.0
0.75
0.50
0.25
0.125
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Carnegie Mellon 129
Viscosity of Polymers and Their Solutions
η(M, c, T) ≈ ηLOC(T) F(M, c, T)
Branched C hain Po lymer s (Concen tra ted or und iluted)
ηLOC(T) ≈ [ηLOC(T)]LIN; Rare exceptions to this known
F(M, c, T) ≈ 1 + [η](c)c
ML[η](c) = πNA(â/3) gL
F(M, c, T) ≈ 1 + ~XE(
~X/
~Xc );
~X = [η](c)c
E(~X/
~Xc ) = {1 + B(g, MBR/Mc)(
~X/
~Xc )
4.8}1/2
B(g, MBR/Mc) ≈ 1 unless the branch molecular MBR > Mc
~Xc = πNA(â/3)ρMc ≈ 100 for many polymers
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Carnegie Mellon 130
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Carnegie Mellon 131
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Carnegie Mellon 132
Moderately Concentrated Solutions
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Carnegie Mellon 133
Viscosity of Polymers and Their Solutions
η(M, c, T) ≈ ηLOC(T) F(M, c, T)
Modera te ly Concentra ted Solutions
ηLOC(T) ≈ [ηLOC(T)]1 - µc = 0 [ηLOC(T)]µ
c = ρ; µ ≈ ϕ = c/ρ
F(M, c, T) ≈ 1 + [η](c)c
ML[η](c) = πNA(â/3)α(c)2(RH(c)/L)L
F(M, c, T) ≈ 1 + H(c)~XE(~X/
~Xc );
~X = [η](c)c
E(~X/
~Xc ) = {1 + (
~X/
~Xc )
4.8}1/2
~Xc = πNA(â/3)ρMc ≈ 100 for many polymers
[G. C. Berry J. Rheology 40:1129-54 (1996)]
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Carnegie Mellon 134
• Introduction
• Rheological methods
• Linear elastic parameters
• Linear viscoelastic functions
• Several viscoelastic experiments
• Relations among linear viscoelastic functions
• Examples of linear viscoelastic functions
• Time-temperature equivalence (Thermo-rheological simplicity)
• The glass transition temperature
• The viscosity
• Effects of polydispersity
• Network formation
• Isochronal Behavior
• Examples from the literature
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Carnegie Mellon 135
Molecular Weight Polydispersity
• ηLOC(T) scales with Mn through Tg
• η/ηLOC(T) scales with Mw, except perhaps for unusual
distributions
• Peak I in L(λ) is essentially unaffected by molecular weight
dispersion
• Peak II in L(λ) may comprise two pieces:
i) an area proportional to ϕLMzMz+1/Mw, with the averages
calculated for chains with M < Me at volume fraction ϕL, and
ii) an area proportional to (1 – ϕL)Me for chains with M > Mc at
volume fraction 1 – ϕL
• Peak III in L(λ) has an area proportional to
(1 – ϕL)-2(Mz/Mw)2.5
• The maxima for peaks II and III separate in λ as (1 – ϕL)Mw
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Carnegie Mellon 136
Theoretical treatments are usually cast in terms of G(t), often in the
form:
G(t) = {Σi wiGi(t)
ν}1/ν
Gi(t) = shea r modulu s for chains with Mi
at weight fraction wi
For example:ν = 1 in the "repta tion modelν = 1/2 in the "doubl -e reptation" model
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Carnegie Mellon 137
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Carnegie Mellon 138
Theoretical treatments are usually cast in terms of G(t), often in the
form:
G(t) = {Σi wiGi(t)
ν}1/ν
Gi(t) = shea r modulu s for chains with Mi
at weight fraction wi
For example:ν = 1 in the "repta tion modelν = 1/2 in the "doubl-e reptation" model
The effects o f increased dispersity of molecula r specie s is usuallymost promi nent in Peak III in L(λ), followed by effect s in Pe akIIin L(λ). This i sseen in (L λ) fo r a polymer undergoing
crosslinking to form a branched polymer, le ading to anetworkpolymer
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Carnegie Mellon 139
• Introduction
• Rheological methods
• Linear elastic parameters
• Linear viscoelastic functions
• Several viscoelastic experiments
• Relations among linear viscoelastic functions
• Examples of linear viscoelastic functions
• Time-temperature equivalence (Thermo-rheological simplicity)
• The glass transition temperature
• The viscosity
• Effects of polydispersity
• Network formation
• Isochronal Behavior
• Examples from the literature
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Carnegie Mellon 140
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Carnegie Mellon 141
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Carnegie Mellon 142
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Carnegie Mellon 143
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Carnegie Mellon 144
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Carnegie Mellon 145
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Carnegie Mellon 146
Power-law behavior
G(t) = [Go – Ge]ϕ(t) + Ge
J ( )t = Jo + ψ(t) + t/η
ψ(t) = (Js – Jo)[1 – α(t)]
Suppose that for all t (note, this involves permissible, but peculiarbehavior for large t):
ψ(t) = (t/λ)μ
With this expression, and 1/η = 0:
[J'(ω) – Jo]/Jo =µΓ(µ)cos(µπ/2) (ωλ)-µ
J"(ω)/Jo =µΓ(µ)sin(µπ/2) (ωλ)-µ
Use of the convolution integral relating J(t) and G(t) gives
ϕ(t) =Eµ(-kµ(t/λ)µ)
with Ge = 0 and 1/η = 0, where kµ = µΓ(µ) and
Eµ(x) =∑n=0
∞
xn
Γ(nµ + 1) : The Mittag-Leffler function
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Carnegie Mellon 147
For small µ,G(t) ≈ Go{1 + ( /tλ)µ}−1
For a nyµ, for large t/λ
G(t) ≈Gos (in µπ)/µπ( /tλ)µ
G(t)J( )t ≈s (in µπ)/µπ < 1
(G'(ω) – Ge)/(Go – Ge) ≈Γ(1-μ) si [(n 1-μ)π/2] (ωλ)μ
"G (ω)/(Go – Ge) ≈Γ(1-μ) c [os(1-μ)π/2] (ωλ)μ
[J '(ω) – Jo]/Jo =µΓ(µ)cos(µπ/2) (ωλ)-µ
J"(ω)/Jo =µΓ(µ)s (in µπ/2) (ωλ)-µ
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Carnegie Mellon 148
Bounded power-law behavior for ϕ(t) might be obtained in th eform
ϕ(t) =1; for t ≤ τΝ=(τΝ/t)μ; for τΝ < t ≤ τ1, with 0 < µ < 1
=(τq/t)m; for > t τ1, with m> 1
where τq = τ1(τΝ/τ1)μ/m.The ,n
G'(ω) – Ge ∝ ω2 andG"(ω) ∝ ω for ω << 1/τ1;
G'(ω) = Go andG"(ω) = 0 forω >> 1/τΝ;
(G'(ω) – Ge)/(Go – Ge) ≈Γ(1-μ) si [(n 1-μ)π/2] (ωτΝ)μ
"G (ω)/(Go – Ge) ≈Γ(1-μ) c [os(1-μ)π/2] (ωτΝ)μ
for the interva 1/l τ1 < ω < 1/τΝ.
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Carnegie Mellon 149
An alternative relation that also exhibits partial power-law behavior is givenby:
ϕ(t) =⎝⎜⎜⎛
⎠⎟⎟⎞∑
i = 1
Ν
(τ1/τi) n/m e (– /xp tτi) /⎝⎜⎜⎛
⎠⎟⎟⎞∑
i = 1
Ν
(τ1/τi) n/m
where τi = τ1/im; m = 2 a nd n = 0 in the Rouse mode .l
For the intermediat e interva 1l /τ1 < ω < 1/τΝ,
(G'(ω) – Ge)/(Go – Ge) ≈{π/2m sin[(1-μ)π/2]} (ωτ1)μ
"G (ω)/(Go – Ge) ≈{π/2m sin[(2-μ)π/2]} (ωτ1)μ
where µ = (1 + n)/m ( =µ 1/2, for the Rouse mode )l .
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Carnegie Mellon 150
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Carnegie Mellon 151
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Carnegie Mellon 152
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Carnegie Mellon 153
• Introduction
• Rheological methods
• Linear elastic parameters
• Linear viscoelastic functions
• Several viscoelastic experiments
• Relations among linear viscoelastic functions
• Examples of linear viscoelastic functions
• Time-temperature equivalence (Thermo-rheological simplicity)
• The glass transition temperature
• The viscosity
• Effects of polydispersity
• Network formation
• Isochronal Behavior
• Examples from the literature
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Carnegie Mellon 154
ISOCHRONAL BEHAVIOR
• In some cases, the temperature is scanned while the dynamic
properties are determined at fixed frequency; such experiments
might typically be reported as G'(ω;T) an d tan δ(ω; T) or η'(ω;T)
versus T, depending on the application.
• Insofar a s G'( ωτc(T)) a nd G "(ωτc(T)) a s func tions of ωτc(T) a re
inde pen dent of T, the isoc hronal plots a re s e en t o be mapp ings
in which ωτc(T) increases with dec reas ing tem pera ture with:
τc(T) ∝ exp⎝⎜⎜⎛
⎠⎟⎟⎞
KT - (Tg - Δ)
• For a reference temperature equal to the glass temperature Tg,
so that aΤ = τc(T)/τc(Tg):
ln aΤω = ln ω – KΔ2
⎝⎜⎜⎛
⎠⎟⎟⎞
T - Tg1 + (T - Tg)Δ ≈ k1 + k2(T - Tg) +
…
with the linear approximation valid for (T - Tg) << Δ; k1 = ln ω and
k2 = K /Δ2.
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Carnegie Mellon 155
1
0
-1
-2
-3
420-2-4
− ( log a ω )
-10 0 10 20
- T T g
- T T g = 0
ω = 1 s
-1
1
0
-1
-2
-3
/Log G' G o and Log tan δ
Log tan δ
/Log G' Go
/Log G' Go
Log tan δ
T
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Carnegie Mellon 156
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Carnegie Mellon 157
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Carnegie Mellon 158
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Carnegie Mellon 159
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Carnegie Mellon 160
• Introduction
• Rheological methods
• Linear elastic parameters
• Linear viscoelastic functions
• Several viscoelastic experiments
• Relations among linear viscoelastic functions
• Examples of linear viscoelastic functions
• Time-temperature equivalence (Thermo-rheological simplicity)
• The glass transition temperature
• The viscosity
• Effects of polydispersity
• Network formation
• Isochronal Behavior
• Examples from the literature
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Carnegie Mellon 161
Examples from the literature
• Branched and linear metallocene polyolefins
• Colloidal dispersions
• Wormlike Micelles
• Deformation of rigid materials
• Nonlinear shear behavior
• Linear and nonlinear bulk properties
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Carnegie Mellon 162
log ω-1012012345log G'(ω)log G''(ω)Unmodified Linear
Metallocene polyethylenesClaus Gabriel and Helmut Münstedt Rheol. Acta 38: 393-403 (1999)
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Carnegie Mellon 163
log ω-1012012345log G'(ω)log G''(ω)Unmodified Linear2345log G'(ω)log G''(ω)Modified Branched 1
Metallocene polyethylenesClaus Gabriel and Helmut Münstedt Rheol. Acta 38: 393-403 (1999)
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Carnegie Mellon 164
log ω-1012 (log G'ω) (log G''ω)2345M 2odified Branched2345 (log G'ω) (log G''ω)M 1odified Branched
Metallocene polyethylenesClaus Gabriel and Helmut Münstedt Rheol. Acta 38: 393-403 (1999)
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Carnegie Mellon 165
-2-1012-4-5log ω (log J' ω) log η (' ω) Unmodified Linear 2Modified Branched34 1Modified Branched
Metallocene polyethylenesClaus Gabriel and Helmut Münstedt Rheol. Acta 38: 393-403 (1999)
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Carnegie Mellon 166
23456-4-5log J'(ω) log η (' ω)/η (' 0) Unmodified Linear 1Modified Branched 2Modified Branched-10 log ω η(' 0)
Metallocene polyethylenesClaus Gabriel and Helmut Münstedt Rheol. Acta 38: 393-403 (1999)
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Carnegie Mellon 167
log ω η(' 0) b23456-4-5 (log J' ω)/b log η (' ω)/η (' 0) Unmodified Linear 1Modified Branched 2Modified Branched-10 U 1M 2M
log η (' 0) 3.28 3.68 4.00
log b -0.7 0 0
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Carnegie Mellon 168
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Carnegie Mellon 169
From creep/recovery
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Carnegie Mellon 170
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Carnegie Mellon 171
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Carnegie Mellon 172
Examples from the literature
• Branched and linear metallocene polyolefins
• Colloidal dispersions
• Wormlike Micelles
• Deformation of rigid materials
• Nonlinear shear behavior
• Linear and nonlinear bulk properties
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Carnegie Mellon 173
Colloidal dispersions: Linear and nonlinear
viscoelastic behavior.
Dilute dispersion of spheres interacting via a hard-core
potential:
η = ηLOC{1 + (5/2)ϕ + k'(5/2)2ϕ 2 + …}
ϕ = volum efracti on = c/ρ(5/2)ϕ = [η]c
ηLOC ≈ ηsolv.
k' ≈ 1.0
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Carnegie Mellon 174
Concentrated dispersion of hard-core spheres:Empirical relations:
η ≈ ηLOC{1 – ϕ/n1}–5n1/2
η ≈ ηLOC{1 – (5/2)ϕ[1 – ϕ/n2]–5n2k /' 2}
desi gnedto force agreemen twith t he viria l expansion at least to orde r ϕ and ϕ2,
respectivel ,y
n1 = 5/8 to gi vek' ≈ 1.0
n1 = ϕmax ≈ 0.64
Theoretica l relations:
η = ηLOC{1 + (5/2)ϕ + k'[ψ1(ϕ) + ψ2(ϕ)](5/2)2ϕ2}
ψ1(ϕ): hydrodynamics
ψ2(ϕ): thermodynamics
ψ1(0) + ψ2(0) =1
U: ψ1(ϕ) ≈ (4/5)(1 – ϕ/ϕmax)
ψ2(ϕ) ≈ (1/5)(1 – ϕ/ϕmax)2
(se -miempircial)
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Carnegie Mellon 175
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Carnegie Mellon 176
Concentrated dispersion of hard-core spheres:
Linear Viscoelastic Response:
• η'(ω) = η'(0) for small ω, as expected, but also show a plateau η'(ω) ≈ η'(ωL) for a
regime at an intermediate range of ω ≈ ωL, before decreasing to zero with increasing
ω.
• η'(ωL) is estimated with ψ2(ϕ) = 0, reflecting the suppression of thermodynamic
interactions at high ω
• G'(ωL) ≈ G1; G1R3/kTϕ2 ≈ ψ0(ϕ) for spheres of radius R
• ψ0(ϕ) ≈ 0.78(η'(ωL)/ηsolv)g(2, ϕ)
g(2, ϕ) is the radial distribution at the contact condition r/R = 2
Theory:
g(2, ϕ) = (1 – ϕ/2)2/(1 – ϕ)3 for ϕ < 0.5 and
g(2, ϕ) = (6/5)(1 – ϕ/ϕmax) for ϕ ≥ 0.5
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Carnegie Mellon 177
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Carnegie Mellon 178
Concentrated dispersion of hard-core spheres:
Linear Viscoelastic Response:
Theory:• η'(0) = ηLOC{ 1 + (5/2)ϕ + k'[ψ1(ϕ) + ψ2(ϕ)](5/2)2ϕ2}
• η'(ωL) = ηLOC{ 1 + (5/2)ϕ + k'[ψ1(ϕ)](5/2)2ϕ2}
• J'EFF(ω) ∝ ω-1/2 for a range of ω < ωL
• J'EFF(ωL) ≈ 1/G'(ωL) ≈ 1/G1 ≈ R3/kTϕ2ψ0(ϕ)
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Carnegie Mellon 179
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Carnegie Mellon 180
Concentrated dispersion of interacting spheres:
• Van der Waals interactions
• Electrostatic interactions among charged spheres
• Interactions among spheres and a dissolved polymer
• True or apparent yield behavior may obtain
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Carnegie Mellon 181170 nm beads (0.05 to 0.2 volume fraction), in 15% polystyrene solutionD. Meitz, L. Yen, G. C. Berry and H. Markovitz J. Rheol. 32:309-51 (1988)
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Carnegie Mellon 182
Examples from the literature
• Branched and linear metallocene polyolefins
• Colloidal dispersions
• Wormlike Micelles
• Deformation of rigid materials
• Nonlinear shear behavior
• Linear and nonlinear bulk properties
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Carnegie Mellon 183
Wormlike micelles
Certain amphillic molecules organize to form curvilinear cylinders, or wormlike
micelles. For example, in an aqueous medium, the amphiphile might organize
with its hydrophobic parts aggregated in the interior of the cylinder, and its
hydrophopic pieces arranged on the "surface" of the cylinder
The micelle structure will exhibit a lifetime τrupture for rupture of its components
If τruptur e is les s th an a longes t rheologica l tim e constan t τrheol the intac t wormlike
micelle would exhib ,it then the rupture dynamics may dominate the observed
rheological behavior,
The chain ma y respond to a deformation b y micellar dynamics similar t o those
for a structu re without rupture, abetted by the rupture process.
With one mode ,l this approximate s Maxwell behavio r wit h a tim e constant
τeffectiv e≈ (τruptureτrupture)1/2
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Carnegie Mellon 184
Cetyl triethylammonium tosylate
+hydrophobic
hydrophilic
+++
+++
++ + + + +
+ +++++++
+
-
-
micelles grow10 nm
micellar network
CTA+ -T
Schematic courtesy Dr. Lynn M. Walker
–
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Carnegie Mellon 185
In an extreme case, the system might approximate behavior for the Maxwell model, with a single
relaxation time τeffectiveso that
J (t) = J s + t/η; wit h J sη = τeffective
(G t) = (1/J s)ex (p -t/τeffective)
With this simple model,
J '(ω) = J s
η'(ω) = (1/J s)/[1 + (ωτeffective)2]
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Carnegie Mellon 186J. F. A. Soltero and J. E. Puig Langmuir 12: 141-8 (1996)
20%
10%These data reveal several deviations from simple Maxwell behavior, including:
◊ The rate of decrease of ⏐ '(⏐ ) with increasing ⏐ for larger ⏐ , to the extentof an increase in ⏐ '(⏐ ) with increasing⏐ for the data on the less concentratedSample
◊ The increase of J'(⏐ ) above the imputed Js for smaller ⏐ for the data on the more
concentrated sample
◊ It may be likely that these samples exhibitsolid-like behavior with a Je at smaller ⏐
than the experimental range, and that Jp
is truly Js
◊ The relatively constant J'(⏐ ) is expectedwith the Maxwell model, but this may be fortuitous
-3 -2 -1 0 1 2
-3
-2
-1
0
1
2
-3
-2
-1
0
30 35 40
10%
20%3.5
4.0
4.5
-3
-2
Temperature (°C)
log η/η
solvent
/log J Pas
log ω η Jpp
log η '(ω) / η (or log J' ω )/J
p
p
30
35
40
(° )T Cη ' (ω) / η (J' ω )/Jpp
3
-4
Calculated
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Carnegie Mellon 187
Examples from the literature
• Branched and linear metallocene polyolefins
• Colloidal dispersions
• Wormlike Micelles
• Deformation of rigid materials
• Nonlinear shear behavior
• Linear and nonlinear bulk properties
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Carnegie Mellon 188
Deformation of Rigid Materials
Creep and Recovery in Tension
Creep for 0 ≤ t ≤ Te
ε(t) = σo (D t) = σo[DR(t) + DNR(t)]
Recover y fo r ϑ = t– Te > 0
ε(ϑ, Te) = σo[DR(ϑ + Te) – DR(ϑ) + DNR(Te)]
εR(ϑ, Te) = ε(Te) – ε(ϑ, Te)
= σo{DR(Te) – DR(ϑ + Te) + DR(ϑ)}
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Carnegie Mellon 189
G. C. Berry J. Polym. Sci.: Polym. Phys. Ed. 14:451-78 (1976)
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Carnegie Mellon 190
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Carnegie Mellon 191
(t/sec)1/3 (ϑ/sec) or [ϑ + T )/sec] – (ϑ/sec) 1/31/3 1/3
0 5 10 15 20 250 5 10 15 200
1
2
3
D(t)/MPa
-1
(a) (b)
Andrade Creep (with DNR(t) ≠ 0)
A nonrecoverable logarithmic creep is frequently observed under
larger stress:
DNR(t) ≈ DL ln(1 + µt/DL) µt/DL <<1
⇒ µt
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Carnegie Mellon 192
Examples from the literature
• Branched and linear metallocene polyolefins
• Colloidal dispersions
• Wormlike Micelles
• Deformation of rigid materials
• Nonlinear shear behavior
• Linear and nonlinear bulk properties
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Carnegie Mellon 193
An "Incompressible" Isotropic Elastic Material
Suppose K >> G, then for infinitisimal strains
Sij = 2 G {εij – δij3 εαα} – δij P
Mor e generall ,y for finit e strain :s
Sij = W1 Bij + W2Bij-1 – δij P
Wi = Wi (I B;1, IB;2) – ∂W ∂IB;i
For simple extension:
f/ A ≈ 2(λ2 – λ-1)(W1 + W2/ λ)
For simple shea :r
S12 = 2(W1 + W2) γ = G γ
S11 – S33 = 2W1 γ2 ; S22 – S33 = – 2W2 γ2
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Carnegie Mellon 194
An expansion of the strain energy function gives theMooney–Rivlin Equation for small deformations:
W ≈ C1 (IB;1 – 3) + C2 (IB;2 – 3)
W1 = C1 and W2 = C2
For the original Kinetic Theory of Rubber Elasticity thecontributions to C1 are entropic in origin, and.:
2C1 = νEkT = ρR /T MXL
2C2 = 0
νE = Numb erof chains understress
MXL = Molecula r weight ofchains
betw eencrosslinks
The precedin g estimat esfor C1 an d C2 a renotaccura ,te an d hav ebeen modified in m oremoderntheori ,es e. .,g th esegive C2 > 0.
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Carnegie Mellon 195
An "Incompressible" Viscoelastic Material
Suppose K(t) >> G(t), then for infinitisimal strains
Sij(t) = 2⌡⎮⎮⌠
-∞
t
G(t – s)⎝⎜⎜⎛
⎠⎟⎟⎞∂εij( )s
∂s – δij3 ∂εαα( )s
∂s ds – δijP
Several relations are proposed for finit e strains,including that du e to Bernste ,in Kearsley an dZapas::
Sij(t) = 2⌡⎮⎮⌠
-∞
t
⎝⎜⎜⎛
⎠⎟⎟⎞∂U
∂IB;1 B(t)ij( )s –
∂U ∂IB;2
B(t)ij( )s-1 ds – δijP
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Carnegie Mellon 196
An "Incompressible" Viscoelastic Material
Suppose K(t) >> G(t), then for infinitisimal strains
Sij(t) = 2⌡⎮⎮⌠
-∞
t
G(t – s)⎝⎜⎜⎛
⎠⎟⎟⎞∂εij( )s
∂s – δij3 ∂εαα( )s
∂s ds – δijP
Several relations are proposed for finit e strains,including that du e to Bernste ,in Kearsley an dZapas::
Sij(t) = 2⌡⎮⎮⌠
-∞
t
⎝⎜⎜⎛
⎠⎟⎟⎞∂U
∂IB;1 B(t)ij( )s –
∂U ∂IB;2
B(t)ij( )s-1 ds – δijP
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Carnegie Mellon 197
Nonlinear Response in Simple Shear for aFluid
(In the approximation with t >> τR)
Sh ear Stress σ(t) = S12(t):
σ(t) = – ⌡⎮⌠
0
∞
[Δγ(t,u)] F1[Δγ(t,u)] ∂G(u)
∂u du
σ(t) = ⌡⎮⌠
-∞
t
G(t – u) ∂γ(u) ∂u M1[Δγ(t,u)] du
Δγ(t,u) = γ(t) – γ(u)
M1[Δγ(t,u)] = ∂γ F1(γ)
∂γ = F1(γ)⎩⎨⎧
⎭⎬⎫
1 + ∂ l n F1(γ)
∂ l n γ
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Carnegie Mellon 198
Nonlinear Response in Simple Shear for aFluid
(In the approximation with t >> τR)
First–Normal Stress Difference ν (1)(t) = σ11(t) – σ22(t) :
ν(1)(t) = – ⌡⎮⌠
0
∞
[Δγ(t,u)]2 F1[Δγ(t,u)] ∂G(u)
∂u du
ν(1)(t) = ⌡⎮⌠
-∞
t
G(t – u) ∂γ(u) ∂u M2[Δγ(t,u)] du
M2[Δγ(t,u)] = ∂ γ2 F1(γ)
∂γ = γF1(γ)⎩⎨⎧
⎭⎬⎫
2 + ∂ l n F1(γ)
∂ l n γ
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Carnegie Mellon 199
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Carnegie Mellon 200
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Carnegie Mellon 201
Response to a Ramp Deformation
γ(t) = ·γ t t > 0
Stress Growth:
σ(t) = ·γ ∫t
0 G(s) ×
F1(·γ s)
⎩⎪⎨⎪⎧
⎭⎪⎬⎪⎫
1 + ∂ l n F1(
·γ s)
∂ l n ·γ s ds
ν(1)(t) = ·γ2 ∫t0 sG(s) ×
F1(·γ s)
⎩⎪⎨⎪⎧
⎭⎪⎬⎪⎫
2 + ∂ l n F1(
·γ s)
∂ l n ·γ s ds
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Carnegie Mellon 202
Steady-State Flow
Viscosity
limt >> τc
σ( )t =σSS(·γ)
η(·γ) = σSS(·γ)/·γ
limγ=0
η(·γ) = η(0) = η
η(·γ) = η(0) H(τc·γ /γ'')
H(τc·γ /γ'') =
⌡⎮⎮⌠
0
∞
(G u)M1[·γ ]u du
⌡⌠
0
∞
(G u)du
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Carnegie Mellon 203
Steady-State Flow
First-Normal Stress Difference
limt >> τc
ν(1)(t) = ν(1)SS(·γ)
N(1)(·γ) = ν(1)SS(·γ) /2{σ
SS(·γ)}2
limγ=0
N(1)(·γ) = Js
N(1)(·γ) = Js SN(τc
·γ/γ'')
SN(τc
·γ/γ'') =
⌡⎮⎮⌠
0
∞
uG(u)M2[·γ u]du
⌡⌠
0
∞
uG(u)du
⎝⎜⎜⎜⎜⎛
⎠⎟⎟⎟⎟⎞
⌡⎮⎮⌠
0
∞
G(u)M1[·γ u]du
⌡⌠
0
∞
G(u)du
-2
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Carnegie Mellon 204
Steady-State Flow
Steady-State Recoverable Compliance
limt;θ >>τc
γR( ,tθ) = γ
R(·γ)
RSS(·γ) = γ
R(·γ)/σ
SS(·γ)
limγ=0
RSS(·γ) = Js
RSS(·γ) = Js SR(τc
·γ/γ'')
SR(τc·γ/γ'') = Result of a n iterativ e calculation
involving (G t) andF1(γ)
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Carnegie Mellon 205
Suppose
G(t) = Go∑ϕ iexp(–t/τi); ∑ϕ i = 1
The , n with t he approxim ateF1(γ) given above
η(γ · ) =Go∑ϕ i τi (H γ ·τi/γ )''
(H γ ·τi/γ )'' ≈1
[1 + (βγ·τi/γ ' )' ε]2/ε ; ε ≈ 6/5, β ≈ 1
By comparison,
η (' ω) =Go∑ϕ i τi 1
[1 + (ωτi)2]
I n both cases, the factors ϕ i τi in the term s in t he summation are
weighted by functions tha t decreas eter – –m by term with increasing γ · orω.
Consequent , ly thes eexpression sexhibit the Cox-Merz approximation:
η(γ ·) ≈η'(ω = γ ·)
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Carnegie Mellon 206
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Carnegie Mellon 207
-3 -2 -1 0 1 2 3
0
-1
0
-1
log( τ γ )
c
·
[log η(γ )/ η(0 )]·
[ log S (γ )/J ]
·
(1)
s
[log J (γ )/J ]
·
s
s
0
-1
PolyethyleneK. Nakamura, C.-P. Wong and G. C. Berry J. Polym. Sci: Polym. Phys. Ed. 22:1119-48 (1984)
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Carnegie Mellon 208
-2 -1 0 1 2 3
0
-1
log( τ γ )
c
·
[log η (' ω )/ η(0 )]
[ (log J' ω )/J ]
s
[ log S (γ )/J ]
·
(1)
s
0
-1
[log η ( γ )/ η(0 )]·
-1
0
-2
-1
0
-2
Linear and nonlinear behavior for a polymer with a relatively narrow MWD
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Carnegie Mellon 209
Examples from the literature
• Branched and linear metallocene polyolefins
• Colloidal dispersions
• Wormlike Micelles
• Deformation of rigid materials
• Nonlinear shear behavior
• Linear and nonlinear bulk properties
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Carnegie Mellon 210
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Carnegie Mellon 211
D. J. Plazek and G. C. Berry , in Glass: Science and Technology, Vol. 3 Viscosity and Relaxation, ed. D. R. Uhlmann and N.J. Kreidl, Academic Press (1986).
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Carnegie Mellon 212
An Inherent Nonlinearity in Response
B(t) = B(0) + ΔB β(t)
β(t) = β(t/ τκ)
But τκ = τκ(V,T)
An attempt to accoun t for this effec t m akes use of anmaterial t imeconstant averaged over t he timeinterval o f interes :t
⟨τκ−1(t2 ,t1)⟩ =1
(t2 - t1) ∫
t2t1 τκ−1(u) du
(V t) – (V 0)(V 0) = ∫t
-∞ [(B t – s) ⟨τκ−1(t ,s)⟩]∂P(s) ∂s ds
Frequently,
B(t) = BA{1 + (t/τA)1/3}; t < το
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Carnegie Mellon 213
D. J. Plazek and G. C. Berry , in Glass: Science and Technology, Vol. 3 Viscosity and Relaxation, ed. D. R. Uhlmann and N.J. Kreidl, Academic Press (1986).
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Carnegie Mellon 214D. J. Plazek and G. C. Berry , in Glass: Science and Technology, Vol. 3 Viscosity and Relaxation, ed. D. R. Uhlmann and N.J. Kreidl, Academic Press (1986).
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Carnegie Mellon 215