Chapter 9: Rheological and Mechanical
Properties of Polymers
AE 447: Polymer Technology, Dr. Cattaleeya Pattamaprom
Adapted from the workshop on
Fundamentals of Rheology Seminar
by
Dr. Abel Gaspar-Rosas
TA Instruments – Waters, Inc.
April, 2004
Thailand
Outline
1. Rheology – Introduction
2. Elasticity, Viscosity and Viscoelasticity
3. Rheology tests
•
•
•
•
Steady-state flow
Stress Relaxation
Creep Recovery
Oscillatory Tests
– Dynamic Frequency sweep
– Dynamic Temperature ramp
4. Molecular Properties vs. Rheology of Polymers
5. Mechanical Testing
Definition of Rheology
• Rheology is the science of
deformation and flow of matter.
รีโอโลยี (Rheology): ศาสตร์ ของรี โอโลยี เป็ นศาสตร์ ที่กล่าวถึงสมบัติการไหลหรื อ
การเสี ยรู ป รี โอโลยีของพอลิเมอร์ ส่วนใหญ่มกั กล่าวถึงค่าโมดูลสั และค่าความหนืด
Simple Shear Deformation and Shear Flow
Shear Deformation
x(t)
F
=
t
A
V
y0 q
y
A
z
x
.
g
=
Dg
Dt
x(t)
Strain, g = y
0
1V d x(t)
.
Strain Rate, g = yy d t
00
t
Viscosity, h = .
g
t
Shear Modulus, G =
g
Range of Rheological Material Behavior
 Rheology:
The study of deformation and
flow of matter.
Range of material behavior
Elastic solid
Solid Like ---------Liquid Like
Ideal Solid-------------Ideal Fluid
Viscoelastic
Classical Extremes
Viscous liquid
Classical Extremes: Elastic Solid
1678: Robert Hooks develops his
“True Theory of Elasticity”
 “The power of any spring is in the same
proportion with the tension thereof.”
 Hooke’s Law: t
=Gg
or
where G is the RIGIDITY MODULUS
.
G is constant
(Stress = G x Strain)
Classical Extremes: Viscosity
 1687: Isaac Newton addresses liquids and steady simple
shearing flow in his “Principia”
 “The resistance which arises from the lack of
slipperiness of the parts of the liquid, other things being
equal, is proportional to the velocity with which the parts
of the liquid are separated from one another.”
Newton’s Law: t = hg
h is constant
wherehis the Coefficient of Viscosity
Response for Classical Extremes
Spring
Purely Elastic
Response
Hookean Solid
t = Gg
Dashpot
Purely Viscous
Response
Newtonian
Liquid
t = hg
For the classical extremes cases, what only matters are
the values of stress, strain and strain rate.
The response is independent of the loading.
Overview
about the different rheological material behavior
ilustrated by the use of…
Ideal viscous
Newton‘s law
viscoelastic
fluid / solid
Ideal elastic
Hooke‘s law
shear viscosity h [Pas]
shear stress t [Pa]
shear rate g [1/s]
Oil, Solvent
shear modulus G [Pa]
Apparent Yield Point,
Thixotropy, …
Coatings, Food, Cosmetics,
Polymers
Glass, Steel
Physica Messtechnik Gmbh Vor dem lauch 6, 70567 Stuttgart, Germany Internet:http://www.physica .de
Viscoelasticity Defined
Range of Material Behavior
Solid Like ---------- Liquid Like
Ideal Solid ----- Most Materials ----- Ideal Fluid
Purely Elastic ----- Viscoelastic ----- Purely Viscous
Viscoelasticity: Having both viscous
and elastic properties
Realistic polymer material
- Materials are in between Elastic solid and
Newtonian liquid called “viscoelastic”
- Some energy stored, some dissipated.
- Some time independent, some time dependent.
A typical viscoelastic effect is tackiness (with long strings)
which is not occuring with purely viscous liquid (example: water),
or purely elastic solid (example: stone)
Physica Messtechnik Gmbh Vor dem lauch 6, 70567 Stuttgart, Germany Internet:http://www.physica .de
Viscoelastic Fluids
Viscoelastic behavior, illustrated by use of a dashpot and a spring
Polymers
at rest
Macromolecules are entangled
and have spherical shapes
Macromolecules are deformed
and disentangled
low viscosity
under shear
high viscosity
Physica Messtechnik Gmbh Vor dem lauch 6, 70567 Stuttgart, Germany Internet:http://www.physica .de
Newtonian and Non-Newtonian
Behavior of Fluids
Non-Newtonian
Region
1.000E5
Newtonian Region
h Independent of g
1.000E5
h = f(g)
10000
Zero-shear viscosity (ho)
h
t (Pa)
h(Pa.s)
1000
100.0
Shear thinning
Pseudo-plastic
t
10000
1.000E-5
1.000E-4
1.000E-3
10.00
0.01000
shear rate (1/s)
g
0.1000
1.000
1.000
Elastic Modulus vs. Temp.
4
E
brittle
3
leathery
2
rubbery
1
viscous
Temp
Area 1: Viscous state:
Area 2: Rubbery state
Area 3: Leathery state
Area 4: Glassy state
 E 
 = stress = F/A
 = strain = dL/L
E = elastic modulus
Linear and Non-Linear Stress-Strain
Behavior of Solids
Linear Region
G is constant
Non-Linear Region
100.0
G = f(g)
osc. stress (Pa)
1000
100.0
G
10.00

1.000
0.010000
0.10000
1.0000
10.000
% strain
100.00
0.01000
1000.0
Viscosity and Viscoelasticity
Viscosity: Definition
Viscosity is . . . .
synonymous with internal friction.
 resistance to flow.
Variables that Affect Viscosity –for polymer
•
•
•
•
•
Shear Rate - Molecular Weight
Time of Shearing
Temperature
Pressure
Surface Tension
Characteristic Diagrams for Newtonian Fluids
t, Pa
hPa.s
Ideal Yield Stress
(Bingham Yield)
gs
gs or tPa
Viscosity: Temperature dependence
v is c o s ity (P a .s )
10000
1000
100.0
10.00
20.0
30.0
40.0
50.0
60.0
temperature (Deg C)
70.0
80.0
90.0
100.0
Summary of Types of Flow
Shear Stress, t
Bingham (Newtonian w/yield stress)
Bingham Plastic
(shear-thinning w/yield stress)
Shear Thinning (Pseudoplastic)
ty
Newtonian
Shear Thickening (Dilatent)

Shear Rate, g
Viscosity vs. Shear Rate
Result: Rheological material behaviour
Shear stress & viscosity as a function of the shear rate
• Newtonian
Ideal viscous (e.g.: oil, solvent)
• Shear thinning
(e.g.: typical food, cosmetics, coatings, polymers, …)
• Shear Thickening
Shear thickening (e.g.: PVC plastisol & paper coatings under high
shear cond.)
Newtonian and Non-Newtonian
Behavior of Fluids
Non-Newtonian
Region
1.000E5
Newtonian Region
h Independent of g
1.000E5
h = f(g)
10000
h
t (Pa)
h(Pa.s)
1000
100.0
t
10000
1.000E-5
1.000E-4
1.000E-3
10.00
0.01000
shear rate (1/s)
g
0.1000
1.000
1.000
Idealized Flow Curve - Polymers
Power Law Region
First Newtonian Plateau
h0 = Zero Shear Viscosity
h0 = K x MW3.4
log h
Extend Range
with TimeTemperature
Superposition (TTS)
& Cox-Merz
Measure in Flow Mode on
AR1000/AR500
Extend Range
with Oscillation
& Cox-Merz
Molecular Structure
1.00E-5
1.00E-4
1.00E-3
Compression Molding
0.0100
0.100
1.00
Extrusion
10.00
shear rate (1/s)
100.00
Second Newtonian
Plateau
Blow and Injection Molding
1000.00
1.00E4
1.00E5
Comparison of Newtonian & Shear Thinning
Samples
viscosity (Pa.s)
10000
Standard oil A
1000
100.0
10.00
Xanthan/Gellan
Fructose Soln.
1.000
0.1000
Standard oil B
0.01000
1.000E-3
1.000E-6
1.000E-4
0.01000
1.000
shear rate (1/s)
10.00
100.0
1000
Non-Newtonian, Time Dependent Fluids
 Thixotropy
 A decrease in apparent viscosity with time under
constant shear rate or shear stress, followed by a
gradual recovery, when the stress or shear rate is
removed.
 Rheopexy
 An increase in apparent viscosity with time under
constant shear rate or shear stress, followed by a
gradual recovery when the stress or shear rate is
removed. Also called Anti-thixotropy or negative
thixotropy.
Reference:Barnes, H.A., Hutton, J.F., and Walters, K., An Introduction to Rheology,
Elsevier Science B.V., 1989. ISBN 0-444-87469-0
Non-Newtonian, Time Dependent Fluids
Viscosity
Rheopectic
Shear Rate = Constant
Thixotropic
time
Time-Dependent Viscoelastic Behavior:
Solid and Liquid Properties of "Silly Putty"
T is short [< 1s]
T is long [24 hours]
Deborah Number [De] = t/ 
Storage and Loss of Viscoelastic Material
SUPER BALL
LOSS
X
TENNIS
BALL
STORAGE
Rheology Tests
Some Types of Rheometers
• Typical Rheometer:
Low shear rate/Oscillatory
• Capillary Rheometer:
High shear rate
Rheology:Choosing Tests and Conditions
Geometries: Typical Rheometer
Max. 3°
Max. 3°

Delta < 1.2
Delta < 1.2
0.25mm < gap < 2mm
Ref.: Physica
1. Stepped Ramp or Steady-Shear Flow
A series of logarithmic stress steps allowed to reach steady
Viscosity
state, each one giving a single viscosity data point:
Shear
Thinning
Region
Shear Rate
Time
hg
(d dt)
Shear Rate, 1/s
Typical steady shear flow experiment
Polyethylene Rheology @ 150 C
1000000
HDPE
100000
viscosity (Pa.s)
LLDPE
10000
LDPE
1000
1.000E-4
1.00E-3
0.01000
0.1000
shear rate (1/s)
1.000
10.00
Idealized Full Flow Curve - Polymers
Power Law Region
First Newtonian Plateau
h0 = Zero Shear Viscosity
h0 = K x MWc 3.4
log h
Extend Range
with TimeTemperature
Superposition (TTS)
& Cox-Merz
Measure in Flow Mode on
AR1000/AR500
Extend Range
with Oscillation
& Cox-Merz
Molecular Structure
1.00E-5
1.00E-4
1.00E-3
Compression Molding
0.0100
0.100
1.00
Extrusion
10.00
shear rate (1/s)
100.00
Second Newtonian
Plateau
Blow and Injection Molding
1000.00
1.00E4
1.00E5
At very high shear rate  Melt fracture
Higher Shear Rate
2. Stress Relaxation Experiment
Strain is applied to sample instantaneously
Strain
(in principle) and held constant with time.
Stress is monitored as a function of time (t).
0
time
Stress Relaxation Experiment
0
time
Response of Classical Extremes
Hookean Solid
Newtonian Fluid
stress for t>0 is 0
stress for t>0
is constant
0
time
0
time
Stress Relaxation Experiment (cont’d)
Response of
Material
Stress decreases with time
starting at some high value
and decreasing to zero.
0
time
For small deformations (strains within the linear region)
the ratio of stress to strain is a function of time only.
This function is a material property known as the
STRESS RELAXATION MODULUS, G(t)
G(t) = (t)/g
3. Creep Recovery Experiment
Stress is applied to sample instantaneously, t1, and
Stress
held constant for a specific period of time. The strain is
monitored as a function of time g(t).
The stress is reduced to zero, t2, and the strain is
monitored as a function of timegt.
t1
time
t2
Creep Recovery Experiment
t1 time t2
Response of Classical Extremes
Stain for t>t1 is constant
– Strain for t >t2 is 0
–
t1
time t2
Stain rate for t>t1 is constant
– Strain for t>t1 increase with time
– Strain rate for t >t2 is 0
–
t1 time t2
Creep Recovery Experiment:
Response of Viscoelastic Material
Creep >0
Recovery  = 0 (after steady state)
/h
Recoverable
Strain
t
1
Strain rate decreases
with time in the creep
zone, until finally
reaching a steady state.
t2
time
In the recovery zone, the viscoelastic
fluid recoils, eventually reaching a
equilibrium at some small total strain
relative to the strain at unloading.
Reference: Mark, J., et.al., Physical Properties of Polymers ,American Chemical Society, 1984, p. 102.
4.Oscillatory Testing of
Polymers
(Dynamic Mechanical Tests)
Dynamic Mechanical Testing
 An oscillatory (sinusoidal)
Deformation
deformation (stress or strain)
is applied to a sample.
The material response
Response
(strain or stress) is measured.
The phase angle d, or phase
shift, between the deformation
and response is measured.
Phase angle
d
Dynamic Mechanical Testing Viscoelastic
Material Response
Phase angle
Stress
Strain
0° < d < 90°
DMA Viscoelastic Parameters:
The Complex, Elastic, & Viscous Stress
The stress in a dynamic experiment is referred to as the
complex stress t*
Phase angle d
Strain, g
Complex Stress, t*
t* = t' + it"
DMA Viscoelastic Parameters
The Complex Modulus: Measure of
materials overall resistance to
deformation.
G* = Stress*/Strain
G* = G′ + iG″
The Elastic (Storage) Modulus:
Measure of elasticity of material. The
ability of the material to store energy.
G' = (stress*/strain)cosd
The Viscous (loss) Modulus:
The ability of the material to dissipate
energy. Energy lost as heat.
G" = (stress*/strain)sind
Tan Delta:
Measure of material damping - such
as vibration or sound damping.
Tan d =
G"/G'
Storage and Loss of Viscoelastic Material
SUPER BALL
LOSS
X
TENNIS
BALL
STORAGE
4.1 Frequency Sweep
Deformation
Time
USES
The material response to
increasing frequency
(rate of deformation) is
monitored at a constant
amplitude (stress or
strain) and temperature.
Information - Zero Shear h, shear thinning
Elasticity (reversible deformation) in materials
 MW & MWD differences Polymer Melts and Polymer solutions.
Finding Yield in gelled dispersions
High and Low Rate (short and long time) modulus properties.
Extend time or frequency range with TTS
Viscosity
Frequency Sweep: Material Response
Terminal
Region
Rubbery
Plateau
Region
Transition
Region
Glassy Region
1
2
Storage Modulus (E' or G')
Loss Modulus (E" or G")
log Frequency (rad/s or Hz)
Typical Oscillatory Data
PDMS
1.000E6
1.000E5
1.000E6
1.000E5
1.000E5
10000
10000
10.00
1000
|n*| (Pa.s)
100.0
1000
G”
G'' (Pa)
G' (Pa)
10000
1000
100.0
G’
10.00
1.000
1.000
0.1000
0.1000
PDMS Extended frequency sweep-0001o, Frequency sweep step
0.01000
100.0
1.000E-5
1.000E-4
1.000E-3
0.010000.1000 1.000
frequency (Hz)
10.00
0.01000
100.0
Time-Dependent Viscoelastic Behavior:
Solid and Liquid Properties of "Silly Putty"
T is short [< 1s]
T is long [24 hours]
Deborah Number [De] = t/ 
Dynamic Moduli of a Polymer Melt vs. Frequency
1000000
1000000
1.00E7
PDMS at 20°C
100000
100000
10000
1000
1000
100.0
100.0
10.00
10.00
h*
100000
10000
G"
1000
1.000
1.000
G'
0.1000 0.1000
1.000E-3
1.000E-4
0.01000
0.1000
1.000
10.00
ang. frequency (rad/sec)
100.0
100.0
1000
h* (Pa.s)
10000
G'' (Pa)
G' (Pa)
1000000
Typical steady shear flow experiment
Polyethylene Rheology @ 150 C
1000000
HDPE
100000
viscosity (Pa.s)
LLDPE
10000
LDPE
1000
1.000E-4
1.00E-3
0.01000
0.1000
shear rate (1/s)
1.000
10.00
Cox-Merz Rule
v is c o s ity (P a .s )
100000
10000
Polydimethylsiloxane
Creep or Steady shear Flow
Dynamic
Frequency
Sweep
.
hg
h*w
PDMS.05F-Flow step
PDMS.08F-Flow step
1000
Dynamic data gives high
shear rates unattainable in flow
100.0
1.000E-5
1.000E-3
0.1000
shear rate (1/s)
10.00
1000
4.2 Dynamic Temperature Ramp:
Material Response
Glassy Region
Transition
Region
Rubbery Plateau
Region
Storage Modulus (E' or G')
Loss Modulus (E" or G")
Temperature
Terminal Region
Time – Temp Superposition:
Time – Temp Superposition: can be used for shifting
“moduluse” in creep test (E(t)), relaxation test (G(t)), dynamic oscillatory test
( G’, G”)
t
 1/f
“ Time scale (or frequency) of stress applies and temperature has a similar
influent on mechanical properties.! “
short time
( high f )
 = h/G long time
( low f )
low temp.
high temp.
works because t and T affect the relaxation time of polymer in same way
< Relaxation time determines mechanical properties
Relationship between
Molecular Properties and
Rheology of Polymers
Effect of Molecular Properties on Rheology
Molecular Structure
•MW & MWD
•Chain Branching and Cross-linking
•Interaction of Fillers with Matrix Polymer
•Single or Multi-Phase Structure
Viscoelastic Properties
As a function of::
•Strain Rate(frequency)
•Strain Amplitude
•Temperature
Processability & Product Performance
Molecular Structure - Effect of Molecular Weight
Glassy Region
Transition
Region
Rubbery
Plateau
Region
MW has practically
no effect on the
modulus below Tg
Low
MW
Temperature
High
Med. MW
MW
Effect of Molecular Weight on
Viscosity and Elasticity
MWc = critical MW
3.4
MWc
log MW
log Joe
log ho
MWc’
log MW
Ref. Graessley, Physical Properties of Polymers,
ACS, c 1984.
Zero Shear Viscosity ho
Sensitive to molecular weight, MW
For low MW (no entanglements) ho
is proportional to MW
For MW > Critical MW, ho is
proportional to MW3.4.
Effect of Molecular Weight
h0 =K x MW3.4
h0
Increasing MW
log g
Effect of Molecular Weight Distribution on h0
A Polymer with a broad MWD exhibits nonNewtonian flow at a lower rate of shear than a
polymer with the same h0 but has a narrow MWD
Narrow MWD
Broad MWD
Log Shear Rate (1/s)
IV.8 : Mechanical Tests
Tensile – Compression Test
< stress – strain behavior >
Normal Stress
Stress 
L(t)
F(t)
Force F / A
( normal stress)
F(t)
A0
=
(A
= cross-section area )
if A = A 0
engineering stress
A = A (t)
true stress
Strain 
=
L  L0
L0
(engineering )
 true
= ln ( 1 +  eng )
( true strain)
Derive จากการใช้ L(t) (True Length ) แทน L0
Shear Stress
DUxx
A
DUyy
F
Shear stress =
Shear strain =
y
Force F
surface area
DUxx
Dy
uyx = ระยะทางที่เคลือ่ นที่ไปในแนวแกน X
x
Stress
(psi * 103)
true
eng
fracture
Compressive strength
fracture
compression
Tensile strength
tension
30
Strain ( % )
Stress – strain curve for a normally brittle polymer under tension and compression
Elastic modulus : slope of stress – strain curve at origin
( indicator of stiffness )
Strength
: stress at fracture
(normally strength tension < strength compression)
Elongation at break : strain at fracture
ถ้ ามีคา่ มาก
ductile
น้ อย
brittle
Toughness : area under stress–strain curve
[ = ] energy/unit volume
desirable material
stiff, strong, ductile, and tough
Note Test can be performed in tension, compression, flexure, or torsion.
Elongation at yield
Ultimate
strength
stress
Plastic deformation
Yield stress
strain
Physical Testing