R h e ol o g i ca l
Behav i o r
and
P o l ym er
Pr op er t i es
G. C. Berry
Department of Chemistry
Carnegie Mellon University
Colloids, Polymers and Surfaces
e-mail: gcberry@andrew.cmu.edu
web site: http://www.chem.cmu.edu/be rry
Carnegie Mellon
1

Int roduction

3
(12)
Rheological methods
16
(19)

Linear elastic parameters
26
(5)

Linear visc oelastic functions
33
(12)

Several viscoelastic experiments
44
(16)

Relations among linear visc oelastic functions
62
(10)

Examples of linear visc oelastic functions
73
(9)

Time-temperature equivalence
83
(9)

The glass transition temperature
93
(13)

The visc osity
107
(26)

Effects of polydispersity
134
(4)

Network formation
139
(13)

Isochronal B ehavior
153
(6)

Examples from the literature
160
(45)
Branched and li near metallocene polyolefins
161
(10)
Colloidal d ispersions
172
(9)
Wormlike Micelles
182
(4)
Deformation of rigid materials
187
(4)
Nonlinear shear behavior
192
(16)
209
(6)
Carnegie Mellon Linear and nonlinear bulk properties
2
Carnegie Mellon

Int roduction

Rheological methods

Linear elastic parameters

Linear visc oelastic functions

Several viscoelastic experiments

Relations among linear visc oelastic functions

Examples of linear visc oelastic functions

Time-temperature equivalence (Thermo-rheological simplicity)

The glass transition temperature

The visc osity

Effects of polydispersity

Network formation

Isochronal B ehavior

Examples from the literature
3
POLYMERS
NATURAL
PROTEINS
POLYNUCLEOTIDES
POLYSACCHARIDES
Carnegie Mellon
SYNTHETIC
GUMS
RESINS
THERMOPLASTIC
THERMOSETTING
ELASTOMERS
4
Some Common Elastomers, Plastics and Fibers
ELAS TOME RS
PLAS TI CS
Polyisoprene
polyethylene
polyisobutylene
polytetrafluoroethylene
poybutadiene
polystyrene
FIBERS
poly(methyl methacrylate)
Phenol-formaldehyde
Urea-formaldehyde
Melami ne-formaldehyde
 Poly(vinyl chloride) 
 Polyurethanes
 Polysiloxanes 
 Polyami de
 Polyester 
 Polypropylene 
Carnegie Mellon
5
Fraction of Molecules With
Molecular Weight M
Mn
Mw
Mz
Molecular Weight M
A Schematic Illustrat ion of a Typical Distribution
of Molecular We ights, showing Mn, Mw, and Mz
Carnegie Mellon
6
A generalized Average of molecular weights:
wµ is the weight f raction of polymer with molecular w eight Mµ:
M()
=


 wµM µ 
µ

Special C ases:
Number average:
=



M()/M = 1/ wµM µ 
µ

Mw
=
M()/M()
Mz
=
M/M()
Mn
Weight average:
=
 wµMµ
µ
z-average:



=  wµMµ wµMµ
µ
 µ
G. C. Berry "Molec ular Weight Distribution" Encyclopedia of Materials
Science and Engineering, ed. M. B. Bever, Pergamon Press, Oxford, 3759-68 (1986)
Carnegie Mellon
7
Specific Volume
Tm
Temperature
A schematic v-T diagram fo r a typical nonpolymeric material.
Carnegie Mellon
8
Specific Volume
Tg
Tm
Temperature
A schematic v-T diagram for a typical
semi-crystalline polymeric material.
Carnegie Mellon
9
Specific Volume
Tg
Temperature
A schematic v-T diagram fo r a typical
noncrys talline polymeric material.
Carnegie Mellon
10
Stress
Rigid Plastic
Flexible Plastic
Elastomer
Strain
Typical Stress-Strain Behavior for Plastics and Elastomers
Carnegie Mellon
11
F. W. Billme yer Jr. (1976):
J. Po lym. Sc i.: Symp. (1976) 55: 1-10
"…characterization of polymers is inherently more
difficult than that of other materials. Polymers are
roughly equivalent in comple xity to, if not more compl ex
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 n ever be economically feasible."
Carnegie Mellon
12
Carnegie Mellon
13
2-D projection of a random arrangement of a chain
with 1000 non-overlapping bonds, each step
otherwise randomly selected
Carnegie Mellon
14
Mean chain dimensions:
For a linear chain with contour length L
(without excluded volume effects):
Mean square-end-to-end dimension:
RL2 = 2âL
â is the persistence length (2â is the Kuhn length)
for a flexible chain, â << L.
Mean square-radius of gyration:
RG2 = RL2 /6 = âL/3
Carnegie Mellon
15
Carnegie Mellon

Int roduction

Rheological methods

Linear elastic parameters

Linear visc oelastic functions

Several viscoelastic experiments

Relations among linear visc oelastic functions

Examples of linear visc oelastic functions

Time-temperature equivalence (Thermo-rheological simplicity)

The glass transition temperature

The visc osity

Effects of polydispersity

Network formation

Isochronal B ehavior

Examples from the literature
16
Schematic of Rheometer System
Computer System
for
Data Acquisition
and
Instrument Control
Shear Stress
vs
Time (Frequency)
Shear Strain
vs
Time (Frequency)
Normal Force
vs
Time (Frequency)
Temperature
vs
Time
Carnegie Mellon Output
Interfaces
Torque
Transducer
Force
Transducer
Position
Transducer
Shape
Transducer
Temperature
Transducer
Rheometer
17
CONTROLLED STRESS
IN TENSION
"Frictionless"
Bearing
Position
Transducer
Sample
Removable
Weight
Device
Carnegie Mellon
Tare
Output
Remov able Weigh t
Input
Controlled weight
Positi on Transduc er
Measure of sha ft positi on
Volt age (current)
Controll ed force
18
CONTROLLED DEFORMATION
IN TENSION
Drive Screws
Crosshead
Position
Transducer
Sample
Device
Carnegie Mellon
Output
Crosshead D rive
Input
Controlled Drive
Positi on Transduc er
Measure of sha ft positi on
Volt age (current)
Controll ed force
19
CONTROLLED STRESS RHEOMETER
Controlled
Torque
Drive
Angle
Position
Transducer
Shaft
"Frictionless mount"
Sample
Fixtures
Fixed Shaft
(Alternate: controlled rotation)
Carnegie Mellon
Device
Input
Output
Controlle d To rque Drive
Controlle d vol tage
Controlle d torque
Ang le Posit ion Transdu cer
Measure of shaft angl e
Volt age (current)
20
CONTROLLED DEFORMATION RHEOMETER
Controlled
Rotation
Drive
Angle
Position
T ransducer
Shaft
"Frictionless mount"
Sample
Fixtures
T orque T ransducer
(Force T ransducer)
Carnegie Mellon
Device
Input
Output
Controll ed Deformation D rive
Controll ed vo lt age
Controll ed shaft rotation
21
Electromagnetic Coils
I: A-F
II: a-f
d E
F
e
,
c
D
f
G
Iron Core
C
b
g
a
•
•
Carnegie Mellon
B
A h
H
Aluminum Cylinder
Attached to Rotor
Phasing of the currents in Coils I and II can produce a timedependent torque:
³
Constant torque amplitude
³
Sinusoidal torque amplitude
Torque amplitude may readily be varied over a factor of 10.
22

Parallel Plates
Sample
Fixtures
Height h
2R

Cone & Plate
Sample
Fixtures
Angle 
2R

Concentric Cylinders
Sample
Fixtures
h
2R
Carnegie Mellon
R
23
Geometric Factors in Rheometry
a
Geometry
Measured
Translational geome tries
Parall el Plate
Force:
F
Stress:
 = F /wb
Displa cement:
Force:
D
F
Strain:
 = D/h
Displa cement:
D
Stress:
Strain:
 = F /2š Rh
 = D/Rln(1 + /R)
Rotational geometries
Parall el Plate
outer radius R; separation h
Torque :
M
Stress:
 = (2r/R)M /R
Rotation:

Strain:
Cone & Plate
outer radius R; cone ang le š - 
Torque :
Rotation:
M

Stress:
Strain:
 = (3/2)M /R
 = (1/) 
Concentric Cyli nde rs
inne r radius R; gap ; heigh t h
Torque :
Rotation:
M

Stress:
Strain:
  (R/2h)M /R
(r)  (R/R) f(R,r)
width,w; breadth b; separation h
Concentric Cyli nde rs
inne r radius R; gap ; heigh t h
Calculated
(r) = (r/h) 
2 1 + R
f(R,r) = (R/r) 1 + /2R
a
 and  are the shea r stress and s train, respectively
Carnegie Mellon
24
Functions and Parameters Used
Function/Para meter
Symbol
Units
Time
t

T
Frequency
Strain Co mponen t
ij
---
Elong ation al s train

---
Shea r strain

---
Rate of shea r
Ý, Ý
T-1
Stress Componen t
Sij
Shea r stress

ML-1T-2
Modulus
G, K, E
ML-1T-2
Compli anc e
J, B, D

M-1LT2
Viscosit y
Carnegie Mellon
T-1
ML-1T-2
ML-1T-1
25
Carnegie Mellon

Int roduction

Rheological methods

Linear elastic parameters

Linear visc oelastic functions

Several viscoelastic experiments

Relations among linear visc oelastic functions

Examples of linear visc oelastic functions

Time-temperature equivalence (Thermo-rheological simplicity)

The glass transition temperature

The visc osity

Effects of polydispersity

Network formation

Isochronal B ehavior

Examples from the literature
26
Linear elastic phenomenology
Shear stress 
Shear strain 
 = J = (1/G)
Elongational s tress 
Elongational s train 
 = D = (1/E)
Pressure ² P
Volume change ² V
² V/V = B² P = (1/K)² P
Carnegie Mellon
27
Linear Elastic Functions
Shear Compliance
J
Shear Modulus
G
Bulk Compliance
B
Bulk Modulus
K
Tensile Compliance
D = J/3 + B/9
Tensile Modulus
1/E = 1/3G + 1/9K
Carnegie Mellon
28
Linear elastic phenomenology
ij

uj
1 ui
= 2 x + x ;
 j
i
u is the displacement
vector
2ij = J [Sij –
1
3
Sij = 2G [ij –
ij S] + (2/9)ij B S
1
3 ij 
] + ij K 
ij = 1 if i = j, and ij = 1 if i ° j
In this notation,
Shear stress  = S 12
Shear strain  = 212
Carnegie Mellon
29
Relations Among Linear Elastic Constants
K, G
E, G
K, E
K, 
E, 
G, 
K
K
EG
33G – E
K
K
E
31 – 2
2G1 + 
31 – 2
E
9KG
3K + G
E
E
3K(1 – 2)
E
2G(1 + )
G
G
G
3KE
9K – E
3K1 – 2
21 + 
E
21 + 
G
3K – 2G
6K + 2G
E
2G – 1
3K – E
6K




Carnegie Mellon
J = 1/G, B = 1/ K, D = 1/E
30
1 Pa = 1.45·10-4 psi
Graphite whisker
12
Carbon fiber
KevlarTM fiber
PE
chain direction
Cellulose
chain direction
Log E/Pa
11
PVOH
Avg textile fiber
10
PE
Amorphous Glass
9
Nonpolar
Polar
Interchain
stretch forces
Carnegie Mellon
Rotation
Bending
Stretch
Intrachain forces
Deformation Modes
31
Carnegie Mellon

Int roduction

Rheological methods

Linear elastic parameters

Linear visc oelastic functions

Several viscoelastic experiments

Relations among linear visc oelastic functions

Examples of linear visc oelastic functions

Time-temperature equivalence (Thermo-rheological simplicity)

The glass transition temperature

The visc osity

Effects of polydispersity

Network formation

Isochronal B ehavior

Examples from the literature
32
Linear Elastic Functions
Shear Compliance
J
Shear Modulus
G
Bulk Compliance
B
Bulk Modulus
K
Tensile Compliance
D = J/3 + B/9
Tensile Modulus
1/E = 1/3G + 1/9K
Carnegie Mellon
33
Linear Viscoelastic Functions
Shear Compliance
J(t)
Shear Modulus
G(t)
Bulk Compliance
B(t)
Bulk Modulus
K(t)
Tensile Compliance
D(t) = J(t)/3 + B(t)/9
Tensile Modulusa
1/Ê(s) = 1/3Ĝ(s) + 1/9K̂(s)
a. The superscript "ˆ" denotes a Laplace transform.
Carnegie Mellon
34
Linear viscoelastic phenomenology—
Stress Controlled

(t)
J(t – ti) i
=
=
i=
t
(t)
•0
d(u) J(t – u)
(u)
u
(t)
=
(t)
J(u)
•
= Jo(t) + •
du
(t
–
u)
u
0
(t)
•
duJ(t – u)
-•
 2
 1
(t)
(t)
Carnegie Mellon
t1
t2
t
35
Linear viscoelastic phenomenology—
Strain Controlled

(t)
=
G(t – ti) i
i=
=
(t)
•0
(t)
=
(u)
•
duG(t – u) u
-•
(t)
=
Go(t) +
d(u)G(t – u)
t
•
•0 du (t – u)
G(u)
u
(t)
(t)
(t)


Carnegie Mellon
t1
t2
t
36
Linear elastic phenomenology
ij

uj
1 ui
= 2 x + x ;
 j
i
u is the displacement
vector
2ij = J [Sij –
1
3
Sij = 2G [ij –
ij S] + (2/9)ij B S
1
3 ij 
] + ij K 
ij = 1 if i = j, and ij = 1 if i ° j
In this notation,
Shear stress  = S 12
Shear strain  = 212
Carnegie Mellon
37
Linear viscoelastic phenomenology

uj
1 ui
ij = 2 x + x ;
 j
i
u is the disp lacement vector
Sij(s)
2ij(t) = •
ds{J(t – s) s –
-•
t
1
3
S(s)
ij s 
S(s)
+ (2/9)ij B(t – s) s }
ij(s)
Sij(t) = •
ds{2G(t – s)  s –
-•
t
(s)
s 
1
3 ij
(s)
+ ijK(t – s) s }
In this notation,
Shear stress (t) = S12 (t)
Shear strain (t) = 212(t)
Carnegie Mellon
38
Relation between G(t) and J(t)

1 t
t 0du
•
G(t – u) J(u)
= 1
s2Ĝ(s)Ĵ(s)
= 1
with Laplace transform:
Carnegie Mellon
39
Shear Compliance J(t) and
Recoverable Shear Compliance R(t)
R(t) = J(t) – t/ = J• – J• – Jo(t)
(t): Retardation Function
Shear Modulus G(t)
G(t) = Ge + Go – Ge(t)
(t): Relaxation Function

the (linear) viscosity, w ith 1/ = 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
Carnegie Mellon
40
Creep Shear Compliance J(t)
R(t) = J(t) – t/
= J• – J• – Jo(t)
Shear Modulus G(t)
G(t)
= Ge + Go – Ge(t)

Linear elastic solid:
1/ = 0,
J• = Je = 1/Ge,
(t) = (t) = 

Linear viscous fluid:
1/ > 0,
Go = 0,
(t) = (t) = (t)

Linear viscoelastic soli d:
1/ = 0,
J• = Je = 1/Ge,
0 < (t) < (t) Š 1

Linear viscoelastic fluid:
1/ > 0,
J• = Js (= J oe),
0 < (t) < (t) Š 1
Bulk Compli ance
B(t)
= Be – Be – Bo(t)
Bulk Modulus
K(t)
= Ke + Ko – Ke(t)
41
Carnegie Mellon
42
Simple example of the relation between G(t) and J(t)
Maxwell fluid:
G(t) = Goexp(- t/);
J(t) = Js + t/;
 = /Go
Js = Jo = 1/Go
R(t) = Js
Note: (t) = exp(-t/) and (t) = 0 for this model.
Carnegie Mellon
43
Often used relations for (t) and (t)
A weight set of exponentials with N relaxation times:
N-1
(t) = 
 exp(–t/ ) = J
i
i
m
N
(t)
=  exp(–t/ ) =
G
i
1
i
1
•
•
dln L()exp(–t/)
-•
–
J
•
o
1
•
•
dln H()exp(–t/)
o – Ge -•
Notes: i = i = 1, and
m is equal to 0 o r 1 for a solid and flu id, resp.
m0 > 1 > 1 > … > i > i > i+1 > … > N-1 > N
(The contribution 0 is absent for a flu id)
Carnegie Mellon
44
Carnegie Mellon

Int roduction

Rheological methods

Linear elastic parameters

Linear visc oelastic functions

Several viscoelastic experiments

Relations among linear visc oelastic functions

Examples of linear visc oelastic functions

Time-temperature equivalence (Thermo-rheological simplicity)

The glass transition temperature

The visc osity

Effects of polydispersity

Network formation

Isochronal B ehavior

Examples from the literature
45
(b) Stress Relaxation
(t) =  o
(t)
0
(•
)
0
(t) = a + bt
R()
Strain
Strain
o
Stress
Stress
(a) Creep & Recovery
(t) =  o
(t)
t = Te
0
0
 = t - Te
t
t
Tim e
Tim e
(c) Ramp Deformation & Recovery
o
Stress
Stress
(t = Te )
(d) Sinusoid Deformation
(t)
0
(t)
P = 1/2  
 ( )
R
o
Strain
Strain
0
(t)
0
.
(t) =    t
t = Te

0
 = t - Te
t
Tim e
Carnegie Mellon
Tim e
46
Creep and recovery w ith a step shear stress
Stress
Stress history:
(t) = 0
t <0
(t) = o
0 Š t Š Te
(t) = 0
t > Te
(t) = o
0
Strain
(t) = a + bt
R() = (t = T e) - (t)
(t)
t = Te
0
t
Carnegie Mellon
q
= t - Te
Time
47
The strain in creep for t Š T e:
(t)
t
= o•
du J(t – u) (u - 
0
= oJ(t) = oR(t) + t/
Carnegie Mellon
48
The strain in creep for t Š T e:
(t)
t
= o•
du J(t – u) (u - 
0
= oJ(t) = oR(t) + t/
The strain for  = t – T e > 0 in recovery:
Te
t
(t) = o•0 du J(t – u) (u - 0) – o•
du J(t – u) (u - Te)
T
e
() = oJ( + Te) – J() = oR( + Te) – R() + Te/
Carnegie Mellon
49
The strain in creep for t Š T e:
(t)
t
=  o•
du J(t – u) (u - 
0
= oJ(t) = oR(t) + t/
The strain for  = t – T e > 0 in recovery:
T
t
(t) = o•0edu J(t – u) (u - 0) – o•
du J(t – u) (u - Te)
T
e
() = oJ( + Te) – J() = oR( + Te) – R() + Te/
The recoverable strain R() = (Te) – (t) for  > 0:
R() = o{J(Te) – J( + Te) – J()}
Carnegie Mellon
= o{R() + R(Te) – R( + Te)}
50
Stress relaxation after a step shear strain
Strain h istory:
(t) = 0
t<0
(t) = o
t •0
Stress
o
(t)
)
(•
Strain
0
(t) =
o
0
t
Carnegie Mellon
Time
51
The stress response for t > 0:
t
(t) = o•
du G(t – u) (u - 0) = oG(t)
0
= o{Ge + (Go – Ge)(t)}
(•) = oGe
Carnegie Mellon
52
Recovery after a ramp shear strain
Strain history:
Stress history:
(t) = 0
t <0
(t) = Ýt
0 Š t Š Te
(t) = 0
t > Te
Stress
(t = T e )
(t)
0
Strain
R(q) =
(t = T
e)
- (t)
.
(t) =  t
t = Te
0
=t-T
t
e
Time
Carnegie Mellon
53
The stress response for t Š Te :
t
t
(t) = Ý•
du G(t – u) = ÝGet + (Go – Ge)•
ds (s)
0
0
For a fluid in steady-state deformation,  = Ý
, or
•
 = (•)/Ý
 = Go •
ds (s)
0
The strain for t > Te:
Te
(t) = 0 = Ý•0
(u)
du G(t – u) + •
du G(t – u) u
Te
t
For large Te and t, (full recoil after steady flow) it can be shown
that for a fluid this give s:
c Js =
Carnegie Mellon
•
•
•0 ds s(s)/•0 ds (s)
54
Carnegie Mellon
55
The strain response for t > 0:
t
(t) = o•
du J(t – u)cos(u)
0
In the steady-state limit with large t :
(t) = o{J'()sin(t) – J''()cos(t)}
In-ph ase (or real or storage) dyna mic compliance:
•
J'() = J• – J• – J o•
ds(s)sin(s)
0
Out-of-phase (or imaginary or loss) dyn amic compliance
•
J"() = (1/) + J• – J o•
ds(s)cos(s)
0
Carnegie Mellon
56
Alternatively
(t) = o |J*()|sin t – 
"Dynam ic compliance":
2
2
2
|J*()| = J'() + J"()
Phase angle ():
tan () = J"()/J'()
For small :
J'() - J•,
Carnegie Mellon
J"() - 1/,
and
J"() – 1/ - 
57
Oscillation with a si nusoid s hear strain
Strain his tory: (t) = 0
t<0
(t) = osin(t)
t •0
The stress response for t > 0 is giv en by
t
(t) = o •
du G(t – u)cos(u)
0
In the steady- state limit with large t,
(t) = o{G'()sin(t) + G''()cos(t)}
In-ph ase (or real or storage) dyna mic compliance:
•
G'() = Ge + Go – Ge•0 ds(s)sins)
Out-of-phase (or imaginary or loss) dyn amic compliance
•
Carnegie Mellon
G''() = Go – Ge•0 ds(s)cos(s)
58
Alternatively
(t) = o |G*()|sin t + 
"Dynam ic compliance":
|G*()| = G'() + G"()
2
2
2
Phase angle ():
tan () = G"()/G'()
For small :

•
ds
0
G'() - Ge +  Go – G e•

s(s) fluid
 ( Js
•
flu id
G''() = Go – Ge•
ds(s)
 
0
Carnegie Mellon
59
Exact relations among the dynam ic modul i and comp liances:
|G*()||J*()| = 1
2
J'() = G'()/|G*()|
2
J"() = G"()/|G*()|
2
G'() = J'()/|J*()|
2
G"() = J"()/|J*()|
tan () = J"()/J'() = G"()/G'()
Carnegie Mellon
60
The dynamic viscosity:
In-pha se with the s train rate:
'() = G"()/
Out-of-pha se with the s train rate:
"() = G'()/
For small :
•
'() = Go – Ge•
ds(s)
0
flu id
 
•

fluid
''() - Ge/ + Go – G e•
ds
s(s)


Js
0
Carnegie Mellon
61
Carnegie Mellon

Int roduction

Rheological methods

Linear elastic parameters

Linear visc oelastic functions

Several viscoelastic experiments

Relations among linear visc oelastic functions

Examples of linear visc oelastic functions

Time-temperature equivalence (Thermo-rheological simplicity)

The glass transition temperature

The visc osity

Effects of polydispersity

Network formation

Isochronal B ehavior

Examples from the literature
62
Linear Viscoelastic Functions
Shear Compliance
J(t)
Shear Modulus
G(t)
Bulk Compliance
B(t)
Bulk Modulus
K(t)
Tensile Compliance
D(t) = J(t)/3 + B(t)/9
Tensile Modulusa
1/Ê(s) = 1/3Ĝ(s) + 1/9K̂(s)
a. The superscript "ˆ" denotes a Laplace transform.
Carnegie Mellon
63
Relation between G(t) and J(t)

1 t
t 0du
•
G(t – u) J(u)
= 1
s2Ĝ(s)Ĵ(s)
= 1
with Laplace transform:
Carnegie Mellon
64
J(t)
8
Log(Compliance/cgs)
R(t)
-6
6
1/G(t)
G(t)
-8
0
2
4
6
8
10
4
12
Log(Modulus/cgs)
-4
14
Log (Time/sec)
Carnegie Mellon
65
J(t)
8
Log(Compliance/cgs)
R(t)
-6
6
1/G(t)
G(t)
-8
2
0
4
6
8
10
4
12
Log(Modulus/cgs)
-4
14
Log (Time/sec)
-4
Log(Compliance/cgs)
-6
J”(w)
6
-8
G”(w)
4
G’(w)
Log(Modulus/cgs)
8
J’(w)
-10
-14
-12
-10
-8
-6
-4
-2
0
Log(Frequency/sec -1)
Carnegie Mellon
66
-4
Log(Compliance/cgs)
G(t)
J’(w)
-6
6
-8
4
G’(w)
0
2
4
6
8
10
Log (Time/sec) & –Log(Frequency/sec
12
Log(Modulus/cgs)
8
R(t)
14
-1)
Figure 14
Carnegie Mellon
67
An often used relation between G(t) and J(t)
A weight set of exponentials with N relaxation times:
N-1
(t) =
1
•

 i exp(–t/i) =
•
dln L()exp(–t/)
J• – Jo -•
m
N
(t) =
i exp(–t/i) = Go 1– Ge •-•• dln H()exp(–t/)
1
Notes: i = i = 1, and
m is equal to 0 or 1 for a solid and fluid
0 > 1 > 1 > … > i > i > i+1 > … > N-1 > N
(0 absent for a fluid)
Carnegie Mellon
68
Determination of L() (or the i-i set) from J(t)
(Similar considerations apply to the determination of
H() (or the i-i set) from G(t))
Derivative methods for L():
1st Approx.:
L()
-
M(m) [R(t)/ln t]t = 
M(m) =
2nd Approx.:
Carnegie Mellon
L()
-
lnL()/ln (interative)
[J(t)/ln t – J(t)/ln t)]t = 2
69
Carnegie Mellon
70
Determination of L() (or the i-i set) from J(t)
(Similar considerations apply to the determination of
H() (or the i-i set) from G(t))
Inverse transform methods for i-i:
The inverse transform is "ill-posed", and a stable
solutions requires constraints (e.g., i • 0)
In an often used strategy, a set of logarithmically spaced
i are chosen such that the span in 1/I does not exceed
the time span in the experimental data. A constrained
nonlinear least squares analysis is then used to
determine the i. Commercial packages are available for
this transform.
Carnegie Mellon
71
Carnegie Mellon
72
Carnegie Mellon

Int roduction

Rheological methods

Linear elastic parameters

Linear visc oelastic functions

Several viscoelastic experiments

Relations among linear visc oelastic functions

Examples of linear visc oelastic functions

Time-temperature equivalence (Thermo-rheological simplicity)

The glass transition temperature

The visc osity

Effects of polydispersity

Network formation

Isochronal B ehavior

Examples from the literature
73
Carnegie Mellon
74
Carnegie Mellon
75

H( )
Go
'() = G''()
t/
G(t)
tan ()
L( )
J(t)
Jo
slope = 1/3
slope = 1
logor log 
log t
log 
Low Molecular Weight Glass Former
Go
G(t)
t/

H( )
'() = G''()
slope = -1/2
tan ()
slope = 1
J(t)
Jo
slope = 1/3
L( )
logor log 
log t
log 
Polymeric Fluid with M < Me
Go

G(t)
t/
slope = -1/2
Jo
JN
'() = G''()
H( )
GN
tan ()
slope = 1
L( )
J(t)
slope = 1/3
log t
Carnegie Mellon
logor log 
Polymeric Fluid with M >> Me
log 
76
log (L( )/Pa)
log (R(t)/Pa)
-4
Js
-6
Slope = 1/3
Narrow MWD
-8
-4
Slope = 1/3
-6
3
2
1
-8
-4
-2
0
Narrow MWD
2
4
6
8
10
log (t/s) or log (/s)
Carnegie Mellon
77
Peak I with L() linear in 1/3 before the peak decreases sharply to
zero.
 The behavior ascribed to peak I, first reported by Andrade, is
seen in a variety of materials, such as metals, ceramics,
crystalline and glassy polym ers and small organic mole cules;
the decrease of L() to zero being eviden t in examples of the
latter.
 The area under peak I provides the contribution JA – Jo to the
total recoverable compliance J s.
 It seems likely that th e mechan ism giving rise to peak I may be
distinctly different from the largely entropic origins of peaks II
and III described in the following.
Carnegie Mellon
78
log (L( )/Pa)
log (R(t)/Pa)
-4
Js
-6
Slope = 1/3
Narrow MWD
-8
-4
Slope = 1/3
-6
3
2
1
-8
-4
-2
0
Narrow MWD
2
4
6
8
10
log (t/s) or log (/s)
Carnegie Mellon
79
Peak II that inc reases in peak area with increasing M until reaching
a certain level, beyond which the peak is inva riant with increasing
M, both in area and position in 
 Peak II is ascribed to Rouse-like modes of motion, ei ther fluidlike 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 o f M after peak III develops.
 For low molecular weight, the Rou se model give s the area of
peak II as
Js – (JA + Jo) = (2M/5RT).
 For the pseudo-solid like behav ior, obtaining when peak III has
developed, reflecting the e ffects of intermolecular
entangl ement, the a rea of peak II becomes inva riant with M and
given by
JN – (JA + Jo) = (Me/RT).
Carnegie Mellon
80
log (L( )/Pa)
log (R(t)/Pa)
-4
Js
-6
Slope = 1/3
Narrow MWD
-8
-4
Slope = 1/3
-6
3
2
1
-8
-4
-2
0
Narrow MWD
2
4
6
8
10
log (t/s) or log (/s)
Carnegie Mellon
81
Peak III that deve lops as peak II area ceases to increase with
increasing M, with peak III developing an a rea invariant with
M, and a maximum at MAX that mov es to larger  as MAX 
(M/Mc)3.4 for M > Mc
 The area under peak III, a lso invariant with M, ascribed to the
effects of chain entang lements is given by
2+s
Js – (JN + JA + Jo) = (kMe/ RT),
where k is in the range 6-8 in most cases, and
s - 2( – 1)/(3 – 2) - 0 to 1/4 with  = ln RG2 /ln M
 Overall,
= (2M/5RT)1 + (
1+s
 
M/kMc) 
2.2
2.0
2
1.8
1.6
S
Log (J ) + Cst.
Js – (JA + Jo)
1.4
1.2
1.0
Carnegie Mellon
1.0
1.5
2.0•
~
Log (X)
2.5
3.0
3.5
82
Carnegie Mellon

Int roduction

Rheological methods

Linear elastic parameters

Linear visc oelastic functions

Several viscoelastic experiments

Relations among linear visc oelastic functions

Examples of linear visc oelastic functions

Time-temperature equivalence (Thermo-rheological simplicity)

The glass transition temperature

The visc osity

Effects of polydispersity

Network formation

Isochronal B ehavior

Examples from the literature
83
Consider the following r educed exp ressions:
[J(t/c) – J o]/Js
= [R(t/c) – J o]/ Js + t/ Js
[J(t/c) – J o]/ Js
= [R(t/c) – J o]/ Js + t/c
c
= Js'(0) (=
Js)
The "time–temperature equivalence" approximation:
[J(t/c) – J o]/Js is a singl e-valued function of t/c over a range o f
temperature.
Although rarely, i f ever, truly accurate for all temperature, it is
never-the-less a useful and widely us ed app roximation for use with
materials exhibit ing no phase transition over the temperature
range of interest.
Carnegie Mellon
84
Since c may not be kno wn over the range o f temperature of
interest, it is 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) + t/bTJs(TREF)
[J(t) – Jo]/bTJs(TREF) = [R(t) – Jo]/bTJs(TREF) + t/hTbTc(TREF)
[J(t/aT) – Jo]/bT = [R(t/aT) – Jo]/bT + t/hTbT(TREF)
[J(t/aT) – Jo]/bT = [R(t/aT) – Jo]/bT + t/aT(TREF)
bT
= b(T, TREF ) = Js(T)/Js(TREF )
hT
= h(T, TREF ) = '(0)[T]/'(0)[TREF ] {=(T)/(TREF )}
aT
= bT hT
Carnegie Mellon
85
Carnegie Mellon
86
bT = b(T, TREF ) = Js(T)/Js(TREF )
hT = h(T, TREF ) = '(0)[T]/'(0)[TREF ] (= (T)/(TREF ))
aT = bT hT
log R(t)/bT
0
T3 < T2 < T1
-2
-4
T3
T2
T1
-6
0
2
4
6
8
log t
H. Markovitz J. Polym. Sci. Symp. No. 50: 431-56 (1975)
Carnegie Mellon
87
Carnegie Mellon
88
4
T1 (Highest)
2
Experimental
Time Range
T2
log (J(t)/Pa)
0
T3
-2
T4
-4
T5
-6
T6
-8
T7 (Lowest)
-10
-2
0
2
4
6
log (t/s)
Carnegie Mellon
89
4
T1 (Highest)
2
Experimental
Time Range
T2
log (J(t)/Pa)
0
T3
-2
T4
-4
T5
-6
T6
Hypothetical example of
time-temperature superposition
(bT = 1).
-8
T7 (Lowest)
-10
-4
-2
0
2
4
6
8
10
log (t/s) or log (aT-1t/s)
Carnegie Mellon
90
Carnegie Mellon
91
Carnegie Mellon
92
Carnegie Mellon

Int roduction

Rheological methods

Linear elastic parameters

Linear visc oelastic functions

Several viscoelastic experiments

Relations among linear visc oelastic functions

Examples of linear visc oelastic functions

Time-temperature equivalence (Thermo-rheological simplicity)

The glass transition temperature

The visc osity

Effects of polydispersity

Network formation

Isochronal B ehavior

Examples from the literature
93
Specific Volume
Tg
Temperature
A schematic v-T diagram fo r a typical
noncrys talline polymeric material.
Carnegie Mellon
94
A Free Volume Model:
(vf)i = (v – v o)i at a certain position r i,
v = (specific) volume
vf = free volume
vo = occupied volume
Carnegie Mellon
95
Carnegie Mellon
96
The glass transition temperature Tg
Tg depends on both intramolecular conformation and
intermolecular interactions.
Various Mod els/Treatments:
 Iso Free Volume:
f(Tg) =
constant
 Iso Viscous:
(Tg) =
constant
 Iso Entropic:
² S(Tg) =
constant
None of these are fully s atisfactory are free of arbitrary
assumption s, and all contain pa rameters that can no t be
independ ently evaluated.
The free volume and entropic mod els provide similar expectations
re the dependen ce of Tg on chain l ength and dilu ent.
Carnegie Mellon
97
120
PMMA
T g (°C)
100
80
60
40
0
0.2
0.4
0.6
0.8
1
Syndiotactic fraction
Carnegie Mellon
98
Estimation of Tg and Tm via Group Contributions
Tg
-
M-1Yg,i
Tm
-
M-1Ym,i

The Y x,i represent molar group contributions to the relevant property

Higher order approximations a re available for both cases
D. W. van Krevelen, Properties of polymers : their correlation with chemical structure, their
numerical estimation and prediction from additive group contributions, 3rd Ed., Elsevier;
Amsterdam ; New York, 1990.
Carnegie Mellon
99
Carnegie Mellon
100
Carnegie Mellon
101
Carnegie Mellon
102
2.4
2.2
boyer
kre ve len avg
kre ve len cal c
Tm /Tg
2.0
1.8
1.6
1.4
1.2
200
300
400
500
600
700
Tm /K
D.W. Van Krevelen, op cit
R. F. Boyer, Rubber Reviews 36:1303-421
Carnegie Mellon
103
Both free volume and entropic models give results that may be cast
in the forms:
Tg(M) -
Tg (•
) {1 + kM/Mn}
 w
1
1 - w 
 w + R1(1 - w) 
= 
+ R
Tg(w)
Tg;DIL 

Tg(Mn)


Both KM and R are model specific parameters, best evaluated
experimentally.
For example, in the free volume model, KM and R arise from the
extra free volume provided by chain ends and diluent, respectively:
typ ically, R is in the range 0.5 to 1.5.
Note, that if Tg;DIL > Tg(Mn), then Tg(w, Mn) is increased by the
diluent.
[G. C. Berry J. Phys. Chem. 70:1194-8 (1966) ]
Carnegie Mellon
104
120
100
T g(°C)
Free Radical
p(Syndio) ~ 0.76
80
p(Syndio) ~ 0.50
60
0
1
2
3
4
5
10 4/Mn
Carnegie Mellon
105
Carnegie Mellon
106
Carnegie Mellon

Int roduction

Rheological methods

Linear elastic parameters

Linear visc oelastic functions

Several viscoelastic experiments

Relations among linear visc oelastic functions

Examples of linear visc oelastic functions

Time-temperature equivalence (Thermo-rheological simplicity)

The glass transition temperature

The visc osity

Effects of polydispersity

Network formation

Isochronal B ehavior

Examples from the literature
107
(T) - LOC(T) F (large scale structure, T)
- LOC(T) F (large scale structure)
"Arrheniu s" form:
LOC(T)  expW/T
if T > (1.5-2)Tg
For melts of crystalline polym ers, Tm > (1.5-2)Tg, permitting use
of this simple form.
"Vogel-Fulcher" form:
For amorphous polymers with 0 Š (T – Tg)/K < - 200:
LOC(T)  expC/(T – To)
Carnegie Mellon
if T < (1.5-2)Tg
108
Carnegie Mellon
109
The temperature dependence of the viscosity:
(T) - LOC(T) F (large scale structure, T)
- LOC(T) F (large scale structure)
For amorphous polymers with 0 Š (T – Tg)/K < - 200:
LOC(T)  expC/(T – To)
if T < (1.5-2)Tg
"WLF form":
LOC(T)/LOC(TREF ) = expC/(T – To) – C/(TREF – To)




C(T – TREF )
= exp –

 REF (T – TREF + REF )
with C and To being con stants, and ²
Carnegie Mellon
REF
= TREF – To.
110
If TREF = Tg then


 K (T – T g) 
LOC(T)/LOC(Tg) = exp– T – T + 


g
where  = Tg – To and K = C/.
For many polym ers:
K = 2300 K and  = 57.5 K
These parameters may be in terpreted in t erms of the "free-volume"
model
Carnegie Mellon
111
Carnegie Mellon
112
Viscosity of Polymers and Their Solutions
M, c, T - LOC(T) F (M, c, T)
Dilute solutions
LOC(T) - Solvent(T)
F (M, c, T) - 1 + []c + …
[] = šN AKRG2 RH/M
G. C. Berry J. Rheology 40:1129-54 (1996)
Carnegie Mellon
113
F (M, c, T) - 1 + []c + …
[] = šN AKRG2 RH/M
Spherical Particles
R = RH = (5/3)1/2RG;
K = 50/9
[]c = (5/2)
Carnegie Mellon
114
F (M, c, T) - 1 + []c + …
[] = š NAKRG2 RH/M
Flexible Chain L inear Polyme rs
RG2 = (âL/3)2;  the cha in expansion factor
â the pe rsistence length
Carnegie Mellon
115
F (M, c, T) - 1 + []c + …
[] = šN AKRG2 RH/M
Flexible Chain Linear Polyme rs
RG2 = (âL/3)2;  the chain exp ansion factor
â the persistence length
High M:
3RH/2 - RG L1/2;
K - 10/3
ML[] = šN A(20/9)(â/3)3/23L1/2 = '(â/3)3/23L1/2
Carnegie Mellon
116
F (M, c, T) - 1 + []c + …
[] = šN AKRG2 RH/M
Flexible Chain Linear Polyme rs
RG2 = (âL/3)2;  the chain exp ansion factor
â the persistence length
High M:
3RH/2 - RG L1/2;
K - 10/3
ML[] = šN A(20/9)(â/3)3/23L1/2 = '(â/3)3/23L1/2
Low M:
RH - L;
ML[] = šN A(â/3)L
Carnegie Mellon
K - 1
(Debye)
117
Flexible Chain B ranched Polymers
ML[] = š NAKRG2 RH/L
g = RG2 /(RG2 )LIN; calculated  = 1
High M:
h = RH/(RH)LIN;
h - g1/2
K - KLINf(g, shape)
[] = f(g, shape)g3/2[]LIN
Carnegie Mellon
118
Flexible Chain B ranched Polymers
ML[] = š NAKRG2 RH/L
g = RG2 /(RG2 )LIN; calculated  = 1
High M:
h = RH/(RH)LIN;
h - g1/2
K - KLINf(g, shape)
[] = f(g, shape)g3/2[]LIN
Star:
[] = g1/2[]LIN
Comb:
[] = g3/2[]LIN
Rando m:
[] = g[]LIN
Carnegie Mellon
119
Flexible Chain Branched Polymers
ML[] = šN AKRG2 RH/L
g = RG2 /(RG2 )LIN; calculated  = 1
High M:
h = RH/(RH)LIN;
h - g1/2
K - KLINf(g, shape)
[] = f(g, shape)g3/2[]LIN
Star:
[] = g1/2[]LIN
Comb:
[] = g3/2[]LIN
Random:
[] = g[]LIN
Low M:
[] = šN AKRG2 RH/LML
RH - L;
Carnegie Mellon
K - 1
120
[] = g[]LIN

(c)
2
1
Entanglement
Interactions
log([](c)/[])
1
0.8
Scaled screening
of
Intramolecular
Interactions
0.6
0.4
0.2
Virial
Expansion
0
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
log(RG/L)
Carnegie Mellon
121
Viscosity of Polymers and Their Solutions
M, c, T - LOC(T) F(M, c, T)
Concentrated solutions and undiluted linear flexible
chain polymers
LOC(T) - LOC(Tg)exp{–K(T – Tg)/(T – Tg +² )}
F(M, c, T) - 1 + [](c)c
Low M (Rouse behavior;  = 1):
~
~
F(M, c, T) - 1 + X - X
~
X = [](c)c;
ML[](c)
=
a modified Fox parameter
š NA(â/3)L;
([](c) ind ependen t of c in this
range )
Carnegie Mellon
122
Carnegie Mellon
123
High M (Entang lement regime)
~ ~ ~
~ ~ ~
F (M, c, T) - 1 + XE (X/Xc ) - XE (X/Xc )
~ ~
~ ~
E (X/Xc ) = {1 + (X/Xc )4.8}1/2
~
Xc = šN A(â/3)Mc - 100
for many polym ers
~
Mc = Xc/šN A(â/3) - 100/šN A(â/3)
Carnegie Mellon
124
Carnegie Mellon
125
The dependenc e of Tg on the diluent conc entration must be
considered for polymer solutions:


 K (T – Tg) 
LOC(T)/LOC(Tg) = exp –

 T – Tg + 
where  = Tg – To and K = C/².
For many polyme rs:
K = 2300 K and  = 57.5 K
² is approximately independen t of the polym er concent ration
Carnegie Mellon
126
400
Polystyrene/Dibenzyl ether
Temperature/K
300
Tg
200
To
100
Tg – To
0
0
0.2
0.4
0.6
0.8
1
Volume Fraction Polymer
G. C. Berry and T. G Fox Adv. Polym. Sci. 5:261-357 (1968)
Carnegie Mellon
127
1.0
5
0.75
4
0.50
log( /Pa·s)
3
2
0.25
1
0.125
0
-1
-2
3
4
5
6
log( M w)
Carnegie Mellon
128
Viscosity of Polymers and Their Solutions
M, c, T - LOC(T) F(M, c, T)
Branched Chain Polymers (Concentrated or undiluted)
LOC(T) - [LOC(T)]LIN; Rare excep tions to this kno wn
F(M, c, T) - 1 + [](c)c
ML[](c)
=
š NA(â/3) gL
~ ~ ~
F(M, c, T) - 1 + XE(X/Xc );
~
X = [](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 = š NA(â/3)Mc - 100 for many polym ers
Carnegie Mellon
129
6
Slope >3.4
Linear
Logc )
~~
4
Slope 3.4
2
Branched
0
Slope 1
-2
-2
0
2
Log (w)
Carnegie Mellon
130
Carnegie Mellon
131
Moderately Concentrated Solutions

(c)
2
1
Entanglement
Interactions
log([](c)/[])
1
0.8
Scaled screening
of
Intramolecular
Interactions
0.6
0.4
0.2
Virial
Expansion
0
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
log(RG/L)
Carnegie Mellon
132
Viscosity of Polymers and Their Solutions
M, c, T - LOC(T) F(M, c, T)
Moderately Concentrated Solutions
LOC(T) - [LOC(T)]1c -=µ0LOC(T)]µc = ; µ -  = c/
F(M, c, T) - 1 + [](c)c
ML[](c)
=
š NA(â/3)(c)2(RH(c)/L)L
~ ~ ~
F(M, c, T) - 1 + H(c)XE(X/Xc );
~
X = [](c)c
~ ~
~ ~
E(X/Xc ) = {1 + (X/Xc )4.8}1/2
~
Xc = š NA(â/3)Mc - 100 for many polym ers
[G. C. Berry J. Rheology 40:1129-54 (1996)]
Carnegie Mellon
133
Carnegie Mellon

Int roduction

Rheological methods

Linear elastic parameters

Linear visc oelastic functions

Several viscoelastic experiments

Relations among linear visc oelastic functions

Examples of linear visc oelastic functions

Time-temperature equivalence (Thermo-rheological simplicity)

The glass transition temperature

The visc osity

Effects of polydispersity

Network formation

Isochronal B ehavior

Examples from the literature
134
Molecular Weight Polydispersity
 LOC(T) scales with Mn through Tg
 LOC(T) scales with Mw, except pe rhaps for unusual
distributions
 Peak I in L() is essentially unaffected by molecular weight
dispersion
 Peak II in L() may comp rise two pieces:
i) an area proportional to LMzMz+1 /Mw, with the averages
calculated for chains with M < Me at volume fraction L, and
ii) an area proportional to (1 – L)Me for chains with M > Mc at
volume fraction 1 – L
 Peak III in L() has an a rea proportional to
(1 – L)-2(Mz/Mw)2.5
 The maxima for peak s II and III separate in  as (1 – L)Mw
Carnegie Mellon
135
Theoretical treatments are usually ca st in terms of G(t), often in the
form:
 
G(t) = { wiGi(t) }
i
Gi(t) = shear modulus for chains with Mi
at weight fraction wi
For example:
 = 1 in the "reptation mode l
 = 1/2 in the "double-reptation" model
Carnegie Mellon
136
Carnegie Mellon
137
Theoretical treatments are usually ca st in terms of G(t), often in the
form:
 
G(t) = { wiGi(t) }
i
Gi(t) = shear modulus for chains with Mi
at weight fraction wi
For example:
 = 1 in the "reptation mode l
 = 1/2 in the "double-reptation" model
The effects of increased dispersity of molecular species is usually
most prominent in Peak III in L(), followed by effects in P eak II
in L(). This is seen in L() for a polym er undergoing
crosslinking to form a branched polymer, leading to a network
polymer
Carnegie Mellon
138
Carnegie Mellon

Int roduction

Rheological methods

Linear elastic parameters

Linear visc oelastic functions

Several viscoelastic experiments

Relations among linear visc oelastic functions

Examples of linear visc oelastic functions

Time-temperature equivalence (Thermo-rheological simplicity)

The glass transition temperature

The visc osity

Effects of polydispersity

Network formation

Isochronal B ehavior

Examples from the literature
139
log (L()/Pa)
-2
Cross link prior to
gelation
-4
-6
Initial
-8
-4
-2
0
2
4
6
8
10
log (t/s) or log (/s)
Carnegie Mellon
140
Incipient gelation
log (L()/Pa)
-2
?
Cross link prior to
gelation
-4
-6
Initial
-8
-4
-2
0
2
4
6
8
10
log (t/s) or log (/s)
Carnegie Mellon
141
Incipient gelation
log (L()/Pa)
-2
?
Cross link prior to
gelation
-4
-6
"Weak"
gel
"Gel”
Initial
-8
-4
-2
0
2
4
6
8
10
log (t/s) or log (/s)
Carnegie Mellon
142
Carnegie Mellon
143
Carnegie Mellon
144
Carnegie Mellon
145
Power-law behavior
G(t) = [Go – Ge](t) + Ge
J(t)
= Jo + (t) + t/
(t) = (Js – Jo)[1 – (t)]
Suppose that for all t (note, this involves permissible, but peculiar
behavior for large t):
(t) = (t/)
With this expression, and 1/ = 0:
[J'() – Jo]/Jo = µ(µ)cos(µš /2) ( )-µ
J"()/Jo = µ(µ)sin(µ š /2) ( )-µ
Use of the convolution integral relating J(t) and G(t) gives
(t) = Eµ(-kµt/)µ)
with Ge = 0 and 1/ = 0, where kµ = µ(µ) and
•
Carnegie Mellon
Eµ(x) =
 (nµx + 1) :
n=0
n
The Mittag-Leffler function
146
For small µ,
G(t) - Go{1 + t/)µ}
For any µ, for large t/
G(t)  Gosin(µ š) /µš t/)µ
G(t)J(t)  sin(µ š) /µš < 1
(G'() – Ge)/(Go – Ge)  (-) sin[(-)/2] ( )
G"()/(Go – Ge)  (-) cos[(-)/2] ( )
[J'() – Jo]/Jo = µ(µ)cos(µš /2) ( )-µ
J"()/Jo = µ(µ)sin(µ š /2) ( )-µ
Carnegie Mellon
147
Bounded power-law behavior for (t) migh t be obtained in th e form
(t) = 1;
for t Š 
= (/t); for  < t Š , with 0 < µ < 1
= (q/t)m; for t > ,
with m > 1
where q = (/)m.Then,
G'() – Ge   and G"()   for  << 1/;
G'() = Go and G"() = 0 for >> 1/;
(G'() – Ge)/(Go – Ge)  (-) sin[(-)/2] ()
G"()/(Go – Ge)  (-) cos[(-)/2] ()
for the interval 1/  <  < 1/.
Carnegie Mellon
148
An alternativ e relation th at also exhibits partial power-law behavior is given
by:
(t)
 
 





n/m
n/m


=
(/i)
exp(–t/i)/
(/i)

i = 
 i = 

where i = /im; m = 2 and n = 0 in the Rouse model.
For the intermediate int erval 1/ <  < 1/,
(G'() – Ge)/(Go – Ge)  {/2m sin[(-)/2]} ()
G"()/(Go – Ge)  {/2m sin[(-)/2]} ()
CarnegieµMellon
where
= (1 + n)/m (µ =, for the Rouse model).
149
0
0
-2
-2
N = 1000
µ = n/(1 + m) = 1/2
N = 300
µ = n/(1 + m)
-4
µ = 1/3
m = 6; n = 1
-4
-6
0
log G'() or log G''()
-6
m = 4; n = 1
0
-2
-2
-4
-4
-6
0
µ = 1/2
m = 4; n = 1
m = 3; n = 0.5
0
-2
-2
-4
-4
m= 2; n = 0
-6
0
m = 1; n = 0.5
-2
0
-2
µ = 2/3
m =3; n = 1
-4
-4
0
2
4
6
log( c )
Carnegie Mellon
8
µ =1
m = 2; n = 1
-6
-8
0
2
4
6
log( c )
8
150
Carnegie Mellon
151
Carnegie Mellon
152
Carnegie Mellon

Int roduction

Rheological methods

Linear elastic parameters

Linear visc oelastic functions

Several viscoelastic experiments

Relations among linear visc oelastic functions

Examples of linear visc oelastic functions

Time-temperature equivalence (Thermo-rheological simplicity)

The glass transition temperature

The visc osity

Effects of polydispersity

Network formation

Isochronal B ehavior

Examples from the literature
153
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'(;T) and tan (; T) or '(;T)
versus T, depending on the application.
•
Insofar as G'(c(T)) and G"(c(T)) as functions of c(T) are
independent of T, the isochronal plots are seen to be mappings
in which c(T) increases with decreasing temperature with:




K
c(T)  expT - (Tg - )
•
For a reference temperature equal to the glass temperature Tg,
so that a  = c(T)/c(Tg):


K  T - Tg 
ln a = ln –  1 + (T - Tg)


 k + k(T - Tg) +
…
with the linear approximation valid for (T - Tg) << ; k = ln  and
Carnegie Mellon
k= K /.
154
1
Log G' /Go
T - Tg = 0
0
Log tan 
Log G' /G o and Log tan 
-1
-2
-3
-4
-2
0
2
4
log (a T )
1
=1s
0
-1
Log tan 
-1
Log G' /Go
-2
-3
Carnegie Mellon
-10
10
0
T - Tg
20
155
Carnegie Mellon
156
"Iso-chronal" behavior for Poly(v inyl C hloride);
(No rotational is omers for the side group)
10
Poly(vinyl chloride)
log G'/Pa
(Fixed )
8
tan  = E"/E'
Tg
Main-chain rotation
0.1


0.01
-200
-100
0
100
Temperature (°C)
Carnegie Mellon
157
Carnegie Mellon
158
Carnegie Mellon
159
Carnegie Mellon

Int roduction

Rheological methods

Linear elastic parameters

Linear visc oelastic functions

Several viscoelastic experiments

Relations among linear visc oelastic functions

Examples of linear visc oelastic functions

Time-temperature equivalence (Thermo-rheological simplicity)

The glass transition temperature

The visc osity

Effects of polydispersity

Network formation

Isochronal B ehavior

Examples from the literature
160
Examples from the literature
Carnegie Mellon

Branched and li near metallocene polyolefins

Colloidal d ispersions

Wormlike Micelles

Deformation of rigid materials

Nonlinear shear behavior

Linear and nonlinear bulk properties
161
5
4
log G''()
3
2
log G'()
1
Unmodified Linear
0
-1
0
1
2
log 
Metallocene polyethylenes
Claus Gabriel and Helmut Münstedt Rheol. Acta 38: 393-403 (1999)
Carnegie Mellon
162
Carnegie Mellon
Metallocene polyethylenes
Claus Gabriel and Helmut Münstedt Rheol. Acta 38: 393-403 (1999)
163
Carnegie Mellon
Metallocene polyethylenes
Claus Gabriel and Helmut Münstedt Rheol. Acta 38: 393-403 (1999)
164
Carnegie Mellon
Metallocene polyethylenes
Claus Gabriel and Helmut Münstedt Rheol. Acta 38: 393-403 (1999)
165
Carnegie Mellon
Metallocene polyethylenes
Claus Gabriel and Helmut Münstedt Rheol. Acta 38: 393-403 (1999)
166
0
log '( '(
-1
-4
log J'()/b
Unmodified Linear
-5
Modified Branched 1
Modified Branched 2
2
3
4
5
6
log  '(0) b
U
Carnegie Mellon
M1
M2
log '(0)
3.28 3.68 4.00
log b
-0.7 0
0
167
log J(t) or log R(t). Pa-1
0
mLDPE-Linear
J(t)
mLDPE-Branched
LDPE-Branched
-2
R(t)
-4
-6
-2
0
2
4
log t/s
C. Gabriel and H. Münstedt Rheo. Acta, 38:393-403 (1 999)
Carnegie Mellon
168
From creep/recovery
Carnegie Mellon
169
log '()/'(0)
0
mLDPE-Linear
mLDPE-Branched
-1
LDPE-Branched
0
2
4
6
log  '(0)
C. Gabriel and H. Münstedt Rheo. Acta, 38:393-403 (1 999)
Carnegie Mellon
170
Carnegie Mellon
171
Examples from the literature
Carnegie Mellon

Branched and li near metallocene polyolefins

Colloidal d ispersions

Wormlike Micelles

Deformation of rigid materials

Nonlinear shear behavior

Linear and nonlinear bulk properties
172
Colloidal dispersions:
Linear and nonlinear
viscoelastic behavior.
Dilute dispersion of spheres interacting via a hard-core
potential:
  LOC{1 + (5/2) + k'(5/2)  + …}
2

Carnegie Mellon
2
= volume fraction = c/
(5/2)
= c
LOC
-
solv.
k'
-
1.0
173
Concentrated dispersion of hard-core spheres:
Empirical relations:
 -
LOC{1
– /n –5n /2
 -
LOC{1
– (5/2)1 – /n –5n k'/2
1
1
2
2
designed to force agreement with the v iri al expans ion at least to order  and ,
respectively,
n1
=
5/8 to give k' - 1.0
n1
=
max
- 0.64
Theoretical relations:

=
LOC{1 + (5/2) + k'1() + 2()(5/2)22}
1(): hyd rodyna mi cs
2(): thermodyna mi cs
1() + 2() =1
U:
Carnegie Mellon
1() - (4/5)(1 – /max)
2() - (1/5)(1 – /max)
(semi- empircial)
2
174
Carnegie Mellon
175
Concentrated dispersion of hard-core spheres:
Linear Viscoelastic Response:

'() = '(0) for small , as exp ected, but als o show a plateau '() - '(L) for a
regim e at an in termediate range of  - L, before decreasing to zero wit h increasing
.

'(L) is estima ted wit h () = 0, reflecting the suppression o f thermodyna mi c
interactions at high 

G'(L) - G1;
G1R3/kT2 - 0() for spheres of radius R

0() - 0.78('(L)/solv)g(2 , )
g(2 , ) is the radia l distribution at the contact cond iti on r/R = 2
Theory :
g(2, ) = (1 – /2)2/(1 – )3 for  < 0.5 and
g(2, ) = (6/5)(1 – /max) for  •0.5
Carnegie Mellon
176
Carnegie Mellon
177
Concentrated dispersion of hard-core spheres:
Linear Viscoelastic Response:
Theory:

'()
=
LOC{1 + (5/2) + k'1() + 2()(5/2)22}

'(L)
=
LOC{1 + (5/2) + k'1()(5/2)22}

J'EFF()

-1/2 for a rang e of  < L

J'EFF(L)
-
1/G'(L) - 1/G1 - R3/kT20()
Carnegie Mellon

178
Carnegie Mellon
179
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
Carnegie Mellon
180
Carnegie Mellon
170 nm beads (0.05 to 0.2 volume fraction), in 15% polystyrene solution
D. Meitz, L. Yen, G. C. Berry and H. Markovitz J. Rheol. 32:309-51 (1988)
181
Examples from the literature
Carnegie Mellon

Branched and li near metallocene polyolefins

Colloidal d ispersions

Wormlike Micelles

Deformation of rigid materials

Nonlinear shear behavior

Linear and nonlinear bulk properties
182
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 ruptu re for rupture of its components
If ruptu re is less than a longest rheological time constant rheol the intact wormlike
micelle would exhibit, then the rupture dynamics may dominate the observed
rheological behavior,
The chain may respond to a deformation by micellar dynamics similar to those
for a structure without rupture, abetted by the rupture process.
With one model, this approximates Maxwell behavior with a time constant
eff ective -  ruptu reruptu re
Carnegie Mellon

183
Cetyl triethylammonium tosylate
-T
CTA+
hydrophobic
–
+
hydrophilic
+
+
+ + +
++
-
-
+ + +
+
+
+
+
+
+ +
+ + + +
micelles grow
10 nm
micellar
network
Schematic courtesy Dr. Lynn M. Walker
Carnegie Mellon
184
In an extreme case, the system might approximate behavior for the Maxwell model, with a single
relaxation time eff ectiveso that
J(t) = Js
+ t/;
with Js = eff ective
G(t) = (1/Js)exp(-t/eff ective)
With this simple model,
J'() = Js
2
'() = (1/Js)/[1 + (eff ective) ]
Carnegie Mellon
185
0
10%
-1
'/p J'()/Jp T (°C)
30
35
-2
◊ The rate of decrease of '() with
increasing  for larger , to the extent
of an increase in '() with increasing
 for the data on the less concentrated
Sample
40
p
-3
3
Calculated
2
◊ The increase of J'() above the imputed
Js for smaller  for the data on the more
concentrated sample
p
1
-1
-2
20%
s
log J /Pa
0
-3
solvent
4. 5
-2
log 
log '/ or log J'()/J
These data reveal several deviations
from simple Maxwell behavior, including:
4. 0
10%
3. 5
20%
-3
30
-4
-3
35
T emperature (°C)
-2
40
0
-1
1
2
◊ It may be likely that these samples exhibit
solid-like behavior with a Je at smaller 
than the experimental range, and that Jp
is truly Js
◊ The relatively constant J'() is expected
with the Maxwell model, but this may be
fortuitous
log p J p
Carnegie Mellon
J. F. A. Soltero and J. E. Puig Langmuir 12: 141-8 (1996)
186
Examples from the literature
Carnegie Mellon

Branched and li near metallocene polyolefins

Colloidal d ispersions

Wormlike Micelles

Deformation of rigid materials

Nonlinear shear behavior

Linear and nonlinear bulk properties
187
Deformation of Rigid Materials
Creep and Recovery in Tension
Creep for 0 Š t Š Te
(t) = oD(t) = o[DR(t) + DNR(t)]
Recovery for  = t – T e > 0
(, Te) = o[DR( + Te) – DR() + DNR(Te)]
R(, Te) = (Te) – (, Te)
= o{DR(Te) – DR( + Te) + DR()}
Carnegie Mellon
188
G. C. Berry J. Polym. Sci.: Polym. Phys. Ed. 14:451-78 (1976)
Carnegie Mellon
189
Andrade Creep (with DNR(t) = 0)
A frequently ob served nonlin ear behavio r
DR(t, o) = DA{1 + R(o)t1/3}
oR  106
(sec 1/3)
sinh(o/A)
R(o) - R()
; A a constant
o/A
30
299°C
20
231
50
10
34.5
0
0
20
40
60
80
100
 /Mdyn/cm 2
Carnegie Mellon
190
Andrade Creep (with DNR(t) ° 0)
A nonrecoverable logarithmic creep is frequently obs erved under
larger stress:
DNR(t) - DL ln(1 + µt/DL)
(a)

µt/DL <<1
µt
(b)
D(t)/MPa
-1
3
2
1
0
0
5
10
(t/sec)1/3
Carnegie Mellon
15
20
0
5
10
15
20
25
(/sec)1/3 or [  + T )/sec] 1/3– ( /sec)1/3
191
Examples from the literature
Carnegie Mellon

Branched and li near metallocene polyolefins

Colloidal d ispersions

Wormlike Micelles

Deformation of rigid materials

Nonlinear shear behavior

Linear and nonlinear bulk properties
192
An "Incompressible" Isotropic Elastic Material
Suppose K >> G, then for infinitisimal strains

Sij = 2 G {ij – 3ij } – ij P
More generally, for finite strains:
-1
Sij = W1 Bij + W2Bij – ij P
Wi = Wi (I B;1, IB;2) –
ŽW
ŽIB;i
For simple extension:
f/A - 2(2 – -1)(W1 + W2/)
For simple shear:
S12 = 2(W1 + W2) G 
Carnegie Mellon
S11 – S33 = 2W1 2 ;
S22 – S33 = – 2W2 2
193
An expansion of the strain energy function gives the
Mooney–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 the
contributions to C1 are entropic in origin, and.:
2C1 = EkT = RT/MXL
2C2 = 0
stress
chains
E = Number of chains under
MXL = Molecular weight of
between crosslinks
The preceding estimates for C1 and C2 are not
accurate, and have been modified in more modern
Carnegie Mellon
theories, e.g., these give C 2 > 0.
194
An "Incompressible" Viscoelastic Material
Suppose K(t) >> G(t), then for infinitisimal strains
t
Sij(t) =


2 G(t
–
-•

ij(s)
s) s
–
ij

(s) 

s  ds – ijP
Several relations are proposed for finite strains,
including that due to Bernstein, Kearsley and Zapas::
t
Sij(t) =

 U
2  I
 B;1
-•
Carnegie Mellon

U
-1
B(t)ij(s) – I B(t)ij(s)  ds – ijP

B;2
195
An "Incompressible" Viscoelastic Material
Suppose K(t) >> G(t), then for infinitisimal strains
t
Sij(t) =


2 G(t
–
-•

ij(s)
s) s
–
ij

(s) 

s  ds – ijP
Several relations are proposed for finite strains,
including that due to Bernstein, Kearsley and Zapas::
t
Sij(t) =

 U
2  I
 B;1
-•
Carnegie Mellon

U
-1
B(t)ij(s) – I B(t)ij(s)  ds – ijP

B;2
196
Nonlinear Response in Simple Shear for a
Fluid
(In the appro ximation with t >> R)
Shear Stress (t) = S12(t):
•

G(u)

(t) = –  [(t,u)] F1[(t,u)] u du

t

(u)

(t) =  G(t – u) u M1[(t,u)] du
-•
(t,u) = (t) – (u)
Carnegie Mellon
 F1(
M1[(t,u)] =



n F 1() 

= F1()1 +

n



197
Nonlinear Response in Simple Shear for a
Fluid
(In the appro ximation with t >> R)
First–Normal Stress Difference (1)(t) = 11(t) – 22(t) :
•

G(u)

(t) = –  [(t,u)] F1[(t,u)] u du

t

(u)


 (t) =  G(t – u) u M2[(t,u)] du
-•
  F1(
M2[(t,u)] =

Carnegie Mellon


n F 1() 

= F1()2 +

n



198
A Theoretical Expression for the Strain Function:
The theory du e to Doi and Edwards
F1() = [1 + (||/'')];
'' - 2.13
An Approximate form of the Strain Function:
F1 () = 1
for || Š '
F1 () = exp[ – (||– ')/'']
for || > '
log F1 ( )
0
'/ ''
[1 + (||/'')]
-1
-1
exp(-| - '|/")
-2
1
Carnegie Mellon
2
|
3

| / ''
4
199
Response to a Step Shear Strain
Strain Jump: (t)
t = 0+
 °
=
t
° G(t-s)(0) 
(t) =

F1()1

°G(t) F1(°)
=
S12 (t, )/ °
 n F1() 

+
 n   ds
(t)
0
-1
0.01
2.5
5
R
-2
c
-3
Carnegie Mellon
-4
-2
0
log t/sec
2
200
Response to a Ramp Deformation
(t)
=
· t
t>
Stress Growth:
(t) =
(t) =
·  G(s) 


 n F1(· s) 
 ds
F1(· s)1 +

 n · s 
t
·2  sG(s) 
t

F1(· s)2 +

Carnegie Mellon


·
n F1( s)
 ds
 n · s 

201
Steady-State Flow
Viscosity
·
lim  (t) = SS(

t >> c
· = SS(/
· ·
(
· = (0) = 
lim (

=
· = (0) H · /''
(
c
•



 G(u)M [ u]du
1
0
·
Hc· /'' =
Carnegie Mellon
•

 G(u)du
0
202
Steady-State Flow
First-Normal Stress Difference
·
lim  (t) = ()

t >> c
SS
· = ()/2{
·
·
N()
()}

SS
SS
· = Js
lim  N()

=
· = Js S  /''
·
N()
c
N
•



 uG(u)M [ u]du
2
0
·
·
S c/''
=
N
Carnegie Mellon
•

 uG(u)du
0
•
-2


·
0 G(u)M1[ u]du
 •

  G(u)du 
 0

203
Steady-State Flow
Steady-State Recoverable Compliance
·
lim   (t,) =  ()
R
t; >>c
R
· =  ()/
· ()
·
R ()
SS
R
SS
· = J
lim  R ()
s
SS
=

· = J S  /''
·
R ()
s
R
c
SS
·
SRc/''
= Result of an iterative calculation
involving G(t) and F1()
Carnegie Mellon
204
Suppose
G(t) = Go•iexp(–t/i);
•i =
1
Then, with the app roximate F1() given above
(· ) = Go•i i H(·i/'')
H(·i/'') -
1
;
·
[1 + (i/ '')]
 - 6/5,  - 1
By comparison,
1
'() =Go•i i [1 + (i)]
In bo th cases, the factors i i in the terms in the su mmation are
weighted by functions that decrease term–by–t erm with increasing ·or
.
Consequently, th ese expressions exhib it the Cox-Merz approximation:
Carnegie Mellon
(·) - '·
205
Narrow MWD
Broad MWD
-1
.
.
log H(); log SN()
0
-2
-2
-1
0
log
Carnegie Mellon
.

1
2
3
c
206
-1
s
·
0
s
log[J )/J ]
·
·
s
-1
0
(1)
log[ )/ )]
log[S )/J ]
0
-1
-3
-2
-1
0
1
2
3
·
log( )
c
Polyethylene
K. Nakamura, C.-P. Wong and G. C. Berry J. Poly m. Sci: Polym. Phys. Ed . 22:1119 -48 (1984)
Carnegie Mellon
207
-1
-1
s
·
0
(1)
0
-2
-1
-2
log[S )/J ]
s
log[J'( )/J ]
-2
0
·
log[ ()/ )]
log[ '( )/ )]
0
-1
-1
0
1
2
3
·
log( )
c
Linear and nonlinear behavior for a polymer with a relatively narrow MWD
Carnegie Mellon
208
Examples from the literature
Carnegie Mellon

Branched and li near metallocene polyolefins

Colloidal d ispersions

Wormlike Micelles

Deformation of rigid materials

Nonlinear shear behavior

Linear and nonlinear bulk properties
209
Carnegie Mellon
210
Carnegie Mellon
211
An Inherent Nonlinearity in Response
B(t) = B() + B(t)
^
(t) = (t/)
But
 = (V,T)
An attempt to account for this effect makes use of an
material time constant averaged over the time
interval of interest:
1
(t ,t) = (t - t )
 
t2
t1
 (u) du
V(t) – V()
P(s)
t


=

B[(t
–
s)

(t
,s)
-•

V()
s ds
Frequently,
Carnegie Mellon
B(t) =
BA{1 + (t/A)1/3}; t < 
212
Carnegie Mellon
213
Carnegie Mellon
214
Carnegie Mellon
215