Transcript No Slide Title

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 Surf aces
e-mail: [email protected] u.edu
web site: http://www.chem.cm u.edu/berry
Carnegie Mellon
1

Introduction

3
(12)
Rheological methods
16
(19)

Linear elastic parameters
26
(5)

Linear viscoelastic f unctions
33
(12)

Several viscoelastic experiments
44
(16)

Relations among linear viscoelastic f unctions
62
(10)

Ex amples of linear viscoelastic functions
73
(9)

Time-temperature equivalence
83
(9)

The glass transition temperature
93
(13)

The viscosity
107
(26)

Ef f ects of polydispersity
134
(4)

Network f ormation
139
(13)

Isochronal Behavior
153
(6)

Ex amples f rom the literature
160
(45)
Branched and li near metallocene polyolefins
161
(10)
Colloidal dispersions
172
(9)
Wormlike Micelles
182
(4)
Def ormation of rigid materials
187
(4)
Nonlinear shear behavior
192
(16)
209
(6)
Carnegie Mellon Linear and nonlinear bulk properties
2
Carnegie Mellon

Introduction

Rheological methods

Linear elastic parameters

Linear viscoelastic f unctions

Several viscoelastic experiments

Relations among linear viscoelastic f unctions

Examples of linear viscoelastic functions

Time-temperature equivalence (Thermo-rheological simplicity)

The glass transition temperature

The viscosity

Ef f ects of polydispersity

Network f ormation

Isochronal Behavior

Examples f rom the literature
3
POLYM ERS
NATURAL
PROTEINS
POLY NUCLEOTIDES
POLY SACCHARIDES
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SYNTHETIC
GUMS
RESINS
THERMOPLASTIC
THERMOSETTING
ELASTOMERS
4
S ome Commo n Elastomers, Plastics and Fib ers
ELASTOME RS
PLASTICS
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 
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5
Fraction of Molecules With
Molecular Weight M
Mn
Mw
Mz
Molecular Weight M
A Schem atic Il l ustration of a Typical Distri bution
of Molecular W eigh ts, showi ng Mn, Mw, an d Mz
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6
A generalized Average of molecular weights:
wµ is the weight fraction of polymer w ith molecular weight Mµ:
M()
=


 wµM µ 
µ

Special Cases:
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 "Molecular Weight Distribution" Encyclopedia of Materials
Science and Engineering, ed. M. B. Bever, Pergamon Press, Oxford, 3759-68 (1986)
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7
Specific Volume
Tm
Temperature
A schematic v-T diagram for a typical non polymeric mate rial.
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Specific Volume
Tg
Tm
Te m pe r atur e
A schematic v-T diagram for a typical
s em i-crys tal li n e pol yme ric mate ri al.
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Specific Volume
Tg
Temperature
A schematic v-T diagram for a typical
noncrystal l i ne polyme ric m ate rial .
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10
Stress
Rigid Plastic
Flexible Plastic
Elastomer
Strain
Ty pical Stress-S train Behavior for Plastics and Elastome rs
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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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2-D projection of a random arrangement of a chain
with 1000 non-overlapping bonds, each s tep
otherwise randomly selected
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Mean chain dimensions:
For a linear chain with contour length L
(without excluded volu me e ffe cts):
Mean square-end-to-end dim ensio n:
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
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15
Carnegie Mellon

Introduction

Rheological methods

Linear elastic parameters

Linear viscoelastic f unctions

Several viscoelastic experiments

Relations among linear viscoelastic f unctions

Examples of linear viscoelastic functions

Time-temperature equivalence (Thermo-rheological simplicity)

The glass transition temperature

The viscosity

Ef f ects of polydispersity

Network f ormation

Isochronal Behavior

Examples f rom the literature
16
Schematic of Rheometer System
Computer System
for
Data Acquisition
and
Instrument Control
She ar Stre s s
vs
Tim e (Fre que ncy)
She ar Strain
vs
Tim e (Fre que ncy)
Norm al Force
vs
Tim e (Fre que ncy)
Te m pe rature
vs
Tim e
Carnegie Mellon Output
Interfaces
Torque
Trans duce r
Force
Trans duce r
Pos ition
Trans duce r
Shape
Trans duce r
Te m pe rature
Trans duce r
Rheometer
17
CONTROLLED STRESS
IN TENSION
"Frictionless"
Bearing
Position
Transducer
Sample
Removable
Weight
De vice
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Tare
O utpu t
Removable Weight
In pu t
Controlled weight
Posit ion T ransducer
Measure of shaft posit ion
Voltage (current )
Cont rolled force
18
CONTROLLED DEFORMATION
IN TENSION
Drive Screws
Crosshead
Position
Transducer
Sample
De vice
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O utpu t
Crosshead Drive
In pu t
Controlled Drive
Posit ion T ransducer
Measure of shaft posit ion
Voltage (current )
Cont rolled force
19
CONTROLLED STRESS RHEOMETER
Controlled
Torque
Drive
Angle
Position
Transducer
Shaft
"Frictionless mount"
Sample
Fixtures
Fixed Shaft
(Alternate: controlled rotation)
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Device
Input
Outpu t
Cont rolled Torque Drive
Cont rolled volt age
Cont rolled torque
Angle P osition T ransducer
Measure of shaft angle
Voltage (current )
20
CONTROLLED DEFORMATION RHEOMETER
Controlled
Rotation
Drive
Angle
Position
Transducer
Shaft
"Frictionless mount"
Sample
Fixtures
Torque Transducer
(Force Transducer)
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De vice
In pu t
O utpu t
Cont rolled Deformat ion Drive
Cont rolled voltage
Cont rolled shaft rotat ion
21
Electromagnetic Coi ls
I: A-F
II: a-f
d E
F
e
,
c
D
f
G
Iron Core
C
b
g
a
•
•
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B
A h
H
Aluminum Cyl inder
Attached to Rotor
Phasing of t he cu rrents in Coils I a nd II c an produce a timedependent t orque:
³
Constant torque amplitude
³
Sinusoidal torque amplitude
Torque amplitude may readily be var ied ove r a factor of 10 .
22

Parallel Plates
Sample
Fixtures
Height h
2R

Cone & Plate
Sample
Fixtures
Angle 
2R

Concentric Cylinders
Sample
Fixtures
h
2R
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R
23
Geometric Factors in Rheometry
Ge ometry
Me asured
Translational geom etries
Parallel P late
width,w; breadth b; separat ion h
Force:
C alcu lated
a
F
Stress:
 = F /wb
Displacement : D
Force:
F
Strain:
 = D/h
Displacement : D
Stress:
Strain:
 = F /2šRh
 = D/Rln(1 + /R)
Rotational ge ometries
Parallel P late
outer radius R; separat ion h
Torque:
M
Stress:
 = (2r/R)M /R
Rotat ion:

Strain:
Cone & P late
outer radius R; cone angle š - 
Torque:
Rotat ion:
M

Stress:
Strain:
 = (3/2)M /R
 = (1/) 
Concentric Cylinders
inner radius R; gap ; height h
Torque:
Rotat ion:
M

Stress:
Strain:
  (R/2h)M /R
(r)  (R/R) f(R,r)
Concentric Cylinders
inner radius R; gap ; height h
(r) = (r/h) 
2 1 + R
f(R,r) = (R/r) 1 + /2R
a
 and  are the shear st ress and st rain, respectively
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24
Functions and Parameters Used
Function /Paramete r
Symbol
Un its
Time
t

T
Frequency
Strain Component
ij
---
Elongational strain

---
Shear strain

---
Rate of shear
Stress Component
Ý, Ý
T-1
ML-1T-2
Shear stress

ML-1T-2
Modulus
Compliance
G, K, E
J, B, D

ML-1T-2
M-1LT2
Viscosity
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T-1
Sij
ML-1T-1
25
Carnegie Mellon

Introduction

Rheological methods

Linear elastic parameters

Linear viscoelastic f unctions

Several viscoelastic experiments

Relations among linear viscoelastic f unctions

Examples of linear viscoelastic functions

Time-temperature equivalence (Thermo-rheological simplicity)

The glass transition temperature

The viscosity

Ef f ects of polydispersity

Network f ormation

Isochronal Behavior

Examples f rom the literature
26
Linear elastic phenomenology
Shear stress 
Shear strain 
 = J = (1/G)
Elongational stress 
Elongational strain 
 = D = (1/E)
Pressure ² P
Volum e change ² V
² V/V = B² P = (1/K)² P
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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
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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  = S12
Shear strain  = 212
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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




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J = 1/G, B = 1/K, D = 1/E
30
1 Pa = 1.45·10 -4 psi
Graphite w hisker
12
Carbon f iber
KevlarTM f iber
PE
chain direction
Cellulose
chain direction
Log E/Pa
11
PVOH
Avg textile f iber
10
PE
Amorphous Glass
9
Nonpolar
Polar
Interchain
stretch f orces
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Rotation
Bending
Stretch
Intrachain forces
Deformation Modes
31
Carnegie Mellon

Introduction

Rheological methods

Linear elastic parameters

Linear viscoelastic f unctions

Several viscoelastic experiments

Relations among linear viscoelastic f unctions

Examples of linear viscoelastic functions

Time-temperature equivalence (Thermo-rheological simplicity)

The glass transition temperature

The viscosity

Ef f ects of polydispersity

Network f ormation

Isochronal Behavior

Examples f rom 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 Modulus a
1/E
ˆ (s) = 1/3Gˆ (s) + 1/9Kˆ (s)
a. The superscript "ˆ" denotes a Laplace transform.
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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)
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t1
t2
t
35
Linear viscoelastic phenomenology—
Strain Controlled

(t)
=
G(t – ti) i
i=
=
(t)
•0
d(u)G(t – u)
(t)
=
•
(u)
– u) u
(t)
=
Go(t) +
•0 du (t – u)
t
duG(t
-•
•
G(u)
u
(t)
(t)
(t)


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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  = S12
Shear strain  = 212
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37
Linear viscoelastic phenomenology

uj
1 ui
ij = 2 x + x ;
 j
i
2ij(t) =
Sij(s)
{J(t – s) s –
t
ds
-•
•
u is the displacement vector
1
3
S(s)
ij s 
S(s)
+ (2/9)ij B(t – s) s }
ij(s) 1 (s)
Sij(t) = •
ds{2G(t – s) s – 3 ij s 
-•
t
(s)
+ ijK(t – s) s }
In this notation,
Shear stress (t) = S12 (t)
Shear strain (t) = 212(t)
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38
Relation between G(t) and J(t)

1 t
t 0du
•
G(t – u) J(u)
= 1
2ˆ
ˆ
s G(s)
J(s)
= 1
with Laplace transform:
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39
Shear Compliance J(t) and
Recoverable Shear Co mpliance 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, with 1/ = 0 for a solid ,
Ge the equilib rium modulu s, with Ge = 0 for a fluid,
Go the "instantaneous" modulu s, 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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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 viscoelast ic solid:
1/ = 0,
J• = Je = 1/Ge,
0 < (t) < (t) Š 1

Linear viscoelast ic fluid:
1/ > 0,
J• = Js (= Joe),
0 < (t) < (t) Š 1
Bulk Compliance
B(t )
= Be – Be – Bo(t)
Bulk Modulus
K(t)
= Ke + Ko – Ke(t)
41
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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 mod el.
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43
Often used relations for (t) and (t)
A weight set of exponentials with N relaxation times:
N-1
 exp(–t/ ) = J
(t) = 
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 or 1 for a solid and fluid, resp.
m0 > 1 > 1 > … > i > i > i+1 > … > N-1 > N
(The contribution 0 is absen t for a flu id)
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44
Carnegie Mellon

Introduction

Rheological methods

Linear elastic parameters

Linear viscoelastic f unctions

Several viscoelastic experiments

Relations among linear viscoelastic f unctions

Examples of linear viscoelastic functions

Time-temperature equivalence (Thermo-rheological simplicity)

The glass transition temperature

The viscosity

Ef f ects of polydispersity

Network f ormation

Isochronal Behavior

Examples f rom 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
Time
Time
(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
Time
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Time
46
Creep and recovery with 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 = Te) - (t)
(t)
t = Te
0
t
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q
=t- Te
Time
47
The strain in creep for t Š Te:
(t)
=
t
o 0du
•
J(t – u) (u - 
= oJ(t) = oR(t) + t/
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48
The strain in creep for t Š Te:
(t)
t
= o•
du J(t – u) (u - 
0
= oJ(t) = oR(t) + t/
The strain for  = t – Te > 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/
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The strain in creep for t Š Te:
(t)
t
= o•
du J(t – u) (u - 
0
= oJ(t) = oR(t) + t/
The strain for  = t – Te > 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/
The recoverable strain R() = (Te) – (t) for  > 0:
R() = o{J(Te) – J( + Te) – J()}
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= o{R() + R(Te) – R( + Te)}
50
Stress relaxation after a step shear strain
Strain history:
(t) = 0
t<0
(t) = o
t •0
Stress
o
(t)
)
(•
Strain
0
(t) =
o
0
t
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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
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Recovery after a ramp shear s train
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
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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
0
(t) = 0 = Ý• du G(t – u) +
t
du
Te
•
(u)
G(t – u) u
For large Te and t, (full recoil after steady flow) it can be shown
that for a fluid this gives:
c Js =
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•
•
•0 ds s(s)/•0 ds (s)
54
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The strain response for t > 0:
t
(t) = o•
du J(t – u)cos(u)
0
In the steady-s tate l imit with large t :
(t) = o{J'()sin(t) – J''()cos(t)}
In-phase (or real or storage) dynamic compliance:
•
J'() = J• – J• – Jo•
ds(s)sin(s)
0
Out-of-phase (or imaginary or loss) dynamic compliance
•
ds(s)cos(s)
0
J"() = (1/) + J• – Jo•
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Alternatively
(t) = o |J*()|sin t – 
"Dynamic compliance":
|J*()|2 = J'()2 + J"()2
Phase angle ():
tan () = J"()/J'()
For small :
J'() - J•,
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J"() - 1/,
and
J"() – 1/ - 
57
Oscillation with a sinusoid shear 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-phase (or real or storage) dynamic compliance:
•
G'() = Ge + Go – Ge•0 ds(s)sins)
Out-of-phase (or imaginary or loss) dynamic compliance
•
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G''() = Go – Ge•0 ds(s)cos(s)
58
Alternatively
(t) = o |G*()|sin t + 
"Dynamic compliance":
|G*()| = G'() + G"()
2
2
2
Phase angle ():
tan () = G"()/G'()
For small :

•
ds
0
G'() - Ge +  Go – Ge•

s(s) fluid
 ( Js
•
fluid
G''() = Go – Ge•
ds(s)
 
0
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Exact relation s among the dynamic moduli and compliances:
|G*()||J*()| = 1
2
J'() = G'()/|G*()|
J"() = G"()/|G*()|2
2
G'() = J'()/|J*()|
G"() = J"()/|J*()|2
tan () = J"()/J'() = G"()/G'()
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The dynamic viscosity:
In-phase with the strain rate:
'() = G"()/
Out-of-phase with the strain rate:
"() = G'()/
For small :
•
fluid
'() = Go – Ge•
ds(s)
 
0
•

fluid
''() - Ge/ + Go – Ge•
ds
s(s)


Js
0
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Carnegie Mellon

Introduction

Rheological methods

Linear elastic parameters

Linear viscoelastic f unctions

Several viscoelastic experiments

Relations among linear viscoelastic f unctions

Examples of linear viscoelastic functions

Time-temperature equivalence (Thermo-rheological simplicity)

The glass transition temperature

The viscosity

Ef f ects of polydispersity

Network f ormation

Isochronal Behavior

Examples f rom 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 Modulus a
1/E
ˆ (s) = 1/3Gˆ (s) + 1/9Kˆ (s)
a. The superscript "ˆ" denotes a Laplace transform.
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63
Relation between G(t) and J(t)

1 t
t 0du
•
G(t – u) J(u)
= 1
2ˆ
ˆ
s G(s)
J(s)
= 1
with Laplace transform:
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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)
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65
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)
-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)
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66
-4
Log(Compliance/cgs)
G(t)
J’(w)
-6
6
-8
4
G’(w)
0
2
4
6
8
10
12
Log(Modulus/cgs)
8
R(t)
14
Log (Time/sec) & –Log(Frequency/sec-1)
Figure 14
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67
An often used relation between G(t) and J(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/)
• – Jo -•
1
•
•
dln H()exp(–t/)
o – Ge -•
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)
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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.:
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L()
-
lnL()/ln  (interative)


[J(t)/ln t –  J(t)/ln t) ]t = 2
69
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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.
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71
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72
Carnegie Mellon

Introduction

Rheological methods

Linear elastic parameters

Linear viscoelastic f unctions

Several viscoelastic experiments

Relations among linear viscoelastic f unctions

Examples of linear viscoelastic functions

Time-temperature equivalence (Thermo-rheological simplicity)

The glass transition temperature

The viscosity

Ef f ects of polydispersity

Network f ormation

Isochronal Behavior

Examples f rom the literature
73
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74
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75

H()
Go
'() = G''()
t/
G(t)
tan ()
L( )
J(t)
Jo
s lope = 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 ()
s lope = 1
J(t)
Jo
slope = 1/3
L( )
logor log 
log t
log 
Polymeric Fluid with M < M e
Go

G(t)
t/
s lope = -1/ 2
Jo
JN
'() = G''()
H()
GN
tan ()
s lope = 1
L( )
J(t)
slope = 1/3
log t
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logor log 
Polymeric Fluid with M >> M e
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)
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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 molecules;
the decrease of L() to zero being eviden t in examples of the
latter.
 The area under peak I provide s the contribution JA – Jo to the
total recoverable compliance Js.
 It seems lik ely that the mechanism giving rise to peak I may be
distinctly different from the largely entropic origin s of peaks II
and III described in the follo wing.
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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)
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79
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 
 Peak II is ascribed to Rouse-like modes of motion, either fluidlike for low molecular weight in the range for which the area
increases with M, or pseudo-solid like (on the relevan t time
scale) in the range of M after peak III develops.
 For low molecular weigh t, the Rouse model give s the area of
peak II as
Js – (JA + Jo) = (2M/5RT).
 For the pseudo-solid like behavior, obtain ing 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/RT).
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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)
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81
Peak III that develops 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 moves to larger  as MAX 
(M/Mc)3.4 for M > Mc
 The area under peak III, also invariant with M, ascribed to the
effects of chain entanglements 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+sM/kMc) 
2.2
2.0
2
1.8
1.6
S
Log (J ) + Cst.
Js – (JA + Jo)
1.4
1.2
1.0
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1.0
1.5
2.0•
~
Log (X)
2.5
3.0
3.5
82
Carnegie Mellon

Introduction

Rheological methods

Linear elastic parameters

Linear viscoelastic f unctions

Several viscoelastic experiments

Relations among linear viscoelastic f unctions

Examples of linear viscoelastic functions

Time-temperature equivalence (Thermo-rheological simplicity)

The glass transition temperature

The viscosity

Ef f ects of polydispersity

Network f ormation

Isochronal Behavior

Examples f rom the literature
83
Consider the following reduced expressions:
[J(t/c) – Jo]/Js
= [R(t/c) – Jo]/ Js + t/ Js
[J(t/c) – Jo]/ Js
= [R(t/c) – Jo]/ Js + t/c
c
= Js'(0) (=
Js)
The "time–temperature equiva lence" approxim ation :
[J(t/c) – Jo]/Js is a single-valu ed function o f t/c over a range of
temperature.
Although rarely, if ever, truly accurate for all t emperature, it is
never-the-less a useful and widely us ed approxim ation for use with
materials exhibit ing no phase transition over the temperature
range of interest.
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84
Since c may not be known over the range of 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
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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)
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87
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88
4
T1 (Highest)
2
Experim ental
Time Range
T2
log (J(t)/Pa)
0
T3
-2
T4
-4
T5
-6
T6
-8
T7 (Low est)
-10
-2
0
2
4
6
log (t/s)
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89
4
T1 (Highest)
2
Experim ental
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 (Low est)
-10
-4
-2
0
2
log (t/s) or log (a
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4
6
8
10
T-1t/s)
90
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92
Carnegie Mellon

Introduction

Rheological methods

Linear elastic parameters

Linear viscoelastic f unctions

Several viscoelastic experiments

Relations among linear viscoelastic f unctions

Examples of linear viscoelastic functions

Time-temperature equivalence (Thermo-rheological simplicity)

The glass transition temperature

The viscosity

Ef f ects of polydispersity

Network f ormation

Isochronal Behavior

Examples f rom the literature
93
Specific Volume
Tg
Temperature
A schematic v-T diagram for a typical
noncrystal l i ne polyme ric m ate rial .
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A Free Volume Model:
(vf)i = (v – v o)i at a certain position r i,
v = (specific) volume
vf = free vo lume
vo = occupied vo lume
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The g lass transition te mperature Tg
Tg depends on both intramolecular conformation and
intermolecular interactions.
Various Mod els/Treatments:
 Iso Free Volum e:
f(Tg) =
constant
 Iso Viscous:
(Tg) =
constant
 Iso Entropic:
² S(Tg) =
constant
None of these are fully s atisfactory are free of arbitrary
assumptions, and all contain parameters that can not be
independently evalu ated.
The free volum e and entropic models provid e similar expectations
re the dependence of Tg on chain l ength and dilu ent.
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120
PMMA
T g (°C)
100
80
60
40
0
0.2
0.4
0.6
0.8
1
Syndiotactic fraction
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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 are 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.
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2.4
2.2
boyer
kr evel en avg
kr evel en calc
Tm/T g
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
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Both free volu me 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 volum e model, KM and R arise from the
extra free volume provid ed by chain ends and diluen t, respectively :
typically, 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) ]
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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/M n
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Carnegie Mellon

Introduction

Rheological methods

Linear elastic parameters

Linear viscoelastic f unctions

Several viscoelastic experiments

Relations among linear viscoelastic f unctions

Examples of linear viscoelastic functions

Time-temperature equivalence (Thermo-rheological simplicity)

The glass transition temperature

The viscosity

Ef f ects of polydispersity

Network f ormation

Isochronal Behavior

Examples f rom 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 poly mers with 0 Š (T – Tg)/K < - 200:
LOC(T)  expC/(T – To)
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if T < (1.5-2)Tg
108
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The temperature dependence of the viscosity:
(T) - LOC(T) F (large scale structure, T)
- LOC(T) F (large scale structure)
For amorphous poly mers 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 constants, and ²
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REF
= TREF – To.
110
If TREF = Tg then


 K (T – Tg) 
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 interpreted in terms of the "free-volum e"
model
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Viscosity of Polymers and Their Solutions
M, c, T - LOC(T) F (M, c, T)
Dilut e solu tions
LOC(T) - Solvent(T)
F (M, c, T) - 1 + []c + …
[] = šN AKRG2 RH/M
G. C. Berry J. Rheology 40:1129-54 (1996)
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F (M, c, T) - 1 + []c + …
[] = šN AKRG2 RH/M
Spherical Particles
R = RH = (5/3)1/2RG;
K = 50/9
[]c = (5/2)
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F (M, c, T) - 1 + []c + …
[] = š NAKRG2 RH/M
Flexible Chain L inear Polymers
RG2 = (âL/3)2;  the chain expansion factor
â the persistence length
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F (M, c, T) - 1 + []c + …
[] = šN AKRG2 RH/M
Flexib le 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
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F (M, c, T) - 1 + []c + …
[] = šN AKRG2 RH/M
Flexib le 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
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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
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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
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Flexib le 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;
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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(R G/L)
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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) independent of c in t his
range)
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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 = šN A(â/3)Mc - 100
for many polym ers
~
Mc = Xc/šN A(â/3) - 100/šN A(â/3)
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The dependence of Tg on the diluent concentration must be
considered for poly mer solutions:


 K (T – Tg) 
LOC(T)/LOC(Tg) = exp–

 T – Tg + 
where  = Tg – To and K = C/².
For many polymers:
K = 2300 K and  = 57.5 K
² is approxim ately independent of the polym er concentration
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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)
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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
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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 exceptions to this known
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
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for many polym ers
129
6
Slope >3.4
Linear
Logc )
~~
4
Slope 3.4
2
Branched
0
Slope 1
-2
-2
0
2
Log (w)
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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(R G/L)
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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. Rh eology 40:1129-54 (1996)]
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Carnegie Mellon

Introduction

Rheological methods

Linear elastic parameters

Linear viscoelastic f unctions

Several viscoelastic experiments

Relations among linear viscoelastic f unctions

Examples of linear viscoelastic functions

Time-temperature equivalence (Thermo-rheological simplicity)

The glass transition temperature

The viscosity

Ef f ects of polydispersity

Network f ormation

Isochronal Behavior

Examples f rom the literature
134
Molecular Weight Polydispersity
 LOC(T) scales with Mn through Tg
 LOC(T) scales with Mw, except perhaps for unusual
distribution s
 Peak I in L() is essentially unaffected by molecular weigh t
dispersion
 Peak II in L() may comprise two piec es:
i) an area proportiona l to LMzMz+1/Mw, with the averages
calculated for chains with M < Me at volume fraction L, and
ii) an area proportiona l to (1 – L)Me for chains with M > Mc at
volu me fraction 1 – L
 Peak III in L() has an area proportional to
(1 – L)-2(Mz/Mw)2.5
 The maxima for peaks II and III separate in  as (1 – L)Mw
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Theoretical treatments are usually cast in terms of G(t), often in the
form:
 
G(t) = { wiGi(t) }
i
Gi(t) = shear modulu s for chains with Mi
at weigh t fraction wi
For example:
 = 1 in the "reptation model
 = 1/2 in the "double-reptation" model
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Theoretical treatments are usually cast in terms of G(t), often in the
form:
G(t) = { wiGi(t)}
i
Gi(t) = shear modulu s for chains with Mi
at weigh t fraction wi
For example:
 = 1 in the "reptation model
 = 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 Peak II
in L(). This is seen in L() for a polym er undergoing
crosslink ing to form a branched poly mer, leading to a network
poly mer
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138
Carnegie Mellon

Introduction

Rheological methods

Linear elastic parameters

Linear viscoelastic f unctions

Several viscoelastic experiments

Relations among linear viscoelastic f unctions

Examples of linear viscoelastic functions

Time-temperature equivalence (Thermo-rheological simplicity)

The glass transition temperature

The viscosity

Ef f ects of polydispersity

Network f ormation

Isochronal Behavior

Examples f rom the literature
139
log (L()/Pa)
-2
Crosslink prior t o
gelat ion
-4
-6
Init ial
-8
-4
-2
0
2
4
6
8
10
log (t/s) or log ( /s)
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Inc ipient gelat ion
log (L()/Pa)
-2
?
Crosslink prior t o
gelat ion
-4
-6
Init ial
-8
-4
-2
0
2
4
6
8
10
log (t/s) or log ( /s)
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Inc ipient gelat ion
log (L()/Pa)
-2
?
Crosslink prior t o
gelat ion
-4
-6
"W eak "
gel
"Gel”
Init ial
-8
-4
-2
0
2
4
6
8
10
log (t/s) or log ( /s)
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Power-law behavior
G(t) = [Go – Ge](t) + Ge
J(t) = Jo + (t) + t/
(t) = (Js – Jo)[1 – (t)]
Suppo se 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
•
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Eµ(x) =
 (nµxn+ 1) :
n=0
The Mittag-L effler function
146
For small µ,
G(t) - Go{1 + t/)µ}
For any µ, for l arge 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
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Bounded power-law behavior for (t) migh t be obtained in the 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/.
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An alternative relation that also exh ibits partial power-law behavior is given
by:
(t)
 
 





=   (/i) n/m exp(–t/i)/  (/i) n/m 
i = 
 i = 

where i = /i m; 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]} ()
where
= (1 + n)/m (µ =, for the Rouse model).
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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 )
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µ=1
m = 2; n = 1
-6
-8
0
2
4
6
log( c )
8
150
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152
Carnegie Mellon

Introduction

Rheological methods

Linear elastic parameters

Linear viscoelastic f unctions

Several viscoelastic experiments

Relations among linear viscoelastic f unctions

Examples of linear viscoelastic functions

Time-temperature equivalence (Thermo-rheological simplicity)

The glass transition temperature

The viscosity

Ef f ects of polydispersity

Network f ormation

Isochronal Behavior

Examples f rom the literature
153
ISOCHRONAL BEHAVIOR
•
In s ome cases, the te mperature is scanned while the dynam ic
properties are determined at fixed f requency; s uch experiments
might typ ically be reported as G'(;T) an d 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 s een to be mappings
in which c(T) increases with decreasing temperature with:




K
c(T)  expT - (Tg - )
•
For a reference temperature equal to th e glass temperature Tg,
so that a  = c(T)/c(Tg):


K  T - Tg 
ln a = ln –  1 + (T - Tg)


 k + k(T - Tg) +
…
with t he linear ap proximation valid for (T - Tg) << ; k = ln  and
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k= K /.
154
1
Log G' /Go
T- Tg = 0
0
Log G' /Go and Log tan 
Log tan 
-1
-2
-3
-4
-2
0
2
4
 log (a T )
1
=1s
0
-1
Log tan 
-1
Log G' /Go
-2
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-3
-10
10
0
T - Tg
20
155
Carnegie Mellon
156
"Iso-chronal" behavior for Poly(vinyl Chloride);
(No rotational is omers f or 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)
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157
Carnegie Mellon
158
Carnegie Mellon
159
Carnegie Mellon

Introduction

Rheological methods

Linear elastic parameters

Linear viscoelastic f unctions

Several viscoelastic experiments

Relations among linear viscoelastic f unctions

Examples of linear viscoelastic functions

Time-temperature equivalence (Thermo-rheological simplicity)

The glass transition temperature

The viscosity

Ef f ects of polydispersity

Network f ormation

Isochronal Behavior

Examples f rom the literature
160
Examples from the literature
Carnegie Mellon

Branched and li near metallocene polyolefins

Colloidal dispersions

Wormlike Micelles

Def ormation 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
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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. MŸnstedt Rheo. Acta, 38:393-403 (1999)
Carnegie Mellon
168
From creep/recovery
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169
log '()/'(0)
0
mLDPE-Linear
mLDPE-Branched
-1
LDPE-Branched
0
2
4
6
log '(0)
C. Gabriel and H. MŸnstedt Rheo. Acta, 38:393-403 (1999)
Carnegie Mellon
170
Carnegie Mellon
171
Examples from the literature
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
Branched and li near metallocene polyolefins

Colloidal dispersions

Wormlike Micelles

Def ormation 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
potentia l:
  LOC{1 + (5/2) + k'(5/2)  + …}
2
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
(5/2)
= volume fract ion = c/
= c
LOC
-
solv.
k'
-
1.0
2
173
Concentrated dispersion of hard-core spheres:
Empiric al relations:
 -
LOC{1
– /n –5n /2
 -
LOC{1
– (5/2)1 – /n –5n k'/2
1
1
2
2
designed to force agreement wit h the virial expansion at least to order  and ,
respectively,
n1
=
5/8 to give k' - 1.0
n1
=
max
- 0.64
Theoretical rela tions:

=
LOC{1 + (5/2) + k'1() + 2()(5/2)22}
1(): hydrodynamics
2(): thermodynamics
1() + 2() =1
U:
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1() - (4/5)(1 – /max)
2() - (1/5)(1 – /max)
(semi-empircial)
2
174
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175
Concentrated dispersion of hard-core spheres:
Linear Viscoelastic Response:

'() = '(0) for small , as expected, but also show a plateau '() - '(L) for a
regime at an int ermediate range of  - L, before decreasing to zero with increasing
.

'(L) is estimated with () = 0, reflecting the suppression of thermodynamic
interact ions 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 radial distribution at t he contact condit ion 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 range of  < L

J'EFF (L)
-
1/G'(L) - 1/G1 - R3/kT20()
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
178
Carnegie Mellon
179
Concentrated dispersion of interacting spheres:
 Van der Waals int eractio ns
 Electrostatic interactions a mong charged spheres
 Interactions among spheres and a dissolved polymer
 True or apparent yield behavior may obtain
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180
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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 dispersions

Wormlike Micelles

Def ormation of rigid materials

Nonlinear shear behavior

Linear and nonlinear bulk properties
182
Wormlike micelles
Certain amphillic molecules organize to form curvilinear cy linders, or wormlike
micelles. For example, in an aqueous medium, the amphiphile might organize
with its hy drophobic parts aggregated in the interior of the cy linder, and its
hydrophopic pieces arranged on the "surface" of the cylinder
The mic elle structure will exhibit a lifetime ruptu re f or 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 dy namics may dominate the observed
rheological behavior,
The chain may respond to a deformation by micellar dy namics similar to those
for a structure without rupture, abetted by the rupture process.
With one model, this approximates Maxwell behavior with a time constant
effe ctive - ruptu reruptu re
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
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 f or the Maxwell model, with a single
relaxation time effe ctiveso that
J(t) = Js
+ t/;
with Js = effe ctive
G(t) = (1/Js)exp(-t/effe ctive)
With this simple model,
J'() = Js
2
'() = (1/Js)/[1 + (effe ctive) ]
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
C alc ulated
or
)/ J
log J'(
These data reveal several deviations
from simple Maxwell behavior, including:
◊ 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 '/
2
4.0
10%
3.5
20%
-3
30
-4
-3
35
Tempe ratur e (°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 Jp
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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 dispersions

Wormlike Micelles

Def ormation 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 – Te > 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 observed nonlin ear behavio r
DR(t, o) = DA{1 + R(o)t1/3}
oR  106
(sec1/3)
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/cm2
Carnegie Mellon
190
Andrade Creep (with DNR(t) ° 0)
A nonrecoverable logarithmic creep is frequently observed 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
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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 dispersions

Wormlike Micelles

Def ormation 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 
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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
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theories, e.g., these give C2 > 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 approximation 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)
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 F1(
M1[(t,u)] =



n F1() 

= F1()1 +

n



197
Nonlinear Response in Simple Shear for a
Fluid
(In the approximation with t >> R)
First–Normal Stress Differe nce (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)] =

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

n F1() 

= F1()2 +

n



198
A Theoretical Expression for the Strain Function:
The theory due 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
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2
|
3

| / ''
4
199
Response to a Step Shear Strain
Strain Jump: (t)
=
(t) =
t = 0+
°
t
° G(t-s)(0) 

F1()1

S12 (t, )/ °
=
 n F1() 

+
 n   ds
°G(t) F1(°)
(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 +

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

·
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) = 
(
)
SS

t >> c
· = ()/2{
·
· 
N()
()}
SS
SS
· = Js
lim N()
=
· = Js S  /''
·
N()
c
N
•



 uG(u)M [ u]du
2
0
·
·
S c/''
=
N
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•

 uG(u)du
0
•
-2


·
0 G(u)M1[ u]du
 •

  G(u)du 
 0

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Steady-State Flow
Steady-State Recoverable Compliance
·
lim  R(t,) = R()
t; >>c
· =  ()/
· ()
·
R ()
SS
R
SS
· = J
lim RSS()
s
=
· = J S  /''
·
R ()
s R c
SS
·
SRc/''
= Result of an it erativ e calculation
involving G(t) and F1()
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Suppose
G(t) = Go•iexp(–t/i);
•i =
1
Then, with the approximate 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 both cases, the factors i i in the terms in the summation are
weigh ted by functions that decrease term–by–term with increasing ·or
.
Consequently, these expressions exhib it the Cox-Merz approxim ation :
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(·) - '·
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Narrow MW D
. log SN()
.
log H();
0
Broad MW D
-1
-2
-2
-1
0
log
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. 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. Polym. Sci: P olym. Phys. Ed. 22:1119-48 (1984)
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-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
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Examples from the literature
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
Branched and li near metallocene polyolefins

Colloidal dispersions

Wormlike Micelles

Def ormation of rigid materials

Nonlinear shear behavior

Linear and nonlinear bulk properties
209
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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()
t
(t ,s)P(s) ds

=

B[(t
–
s)

-•

V()
s
Frequently,
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B(t) =
BA{1 + (t/A)1/3}; t < 
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