1. What happens to Tg with increasing pressure?

2. Bar = 1 atm = 100 kPa
Why?

3. A Demonstration of Polymer Viscoelasticity
Poly(ethylene oxide) in water

4. “Memory” of Previous State
Poly(styrene)
Tg ~ 100 °C

5. Chapter 5. Viscoelasticity
Is “silly putty” a solid or a liquid?
Why do some injection molded parts warp?
What is the source of the die swell phenomena that is often observed in extrusion processing?
Expansion of a jet
of an 8 wt% solution of
polyisobutylene in decalin
Under what circumstances am I justified in ignoring viscoelastic effects?

6. Rheology is the science of flow and deformation of matter
What is Rheology?
Rheology is the science of flow and deformation of matter
Rheology Concepts, Methods, & Applications, A.Y. Malkin and A.I. Isayev; ChemTec Publishing, 2006

8. Temperature & Strain Rate
The properties of polymers are strongly affected by temperature and strain rate

10. Time dependent processes: Viscoelasticity
The response of polymeric liquids, such as melts and solutions, to an imposed stress may resemble the behavior of a solid or a liquid, depending on the situation.

12. increasing loading rate
Stress
Strain
increasing loading rate

13. Network of Entanglements
There is a direct analogy between chemical crosslinks in rubbers and “physical” crosslinks that are created by the entanglements.
The physical entanglements can support stress (for short periods up to a time tT), creating a “transient” network.

14. Entanglement Molecular Weights, Me, for Various Polymers
Me (g/mole)
Poly(ethylene) 1,250
Poly(butadiene) 1,700
Poly(vinyl acetate) 6,900
Poly(dimethyl siloxane) 8,100
Poly(styrene) 19,000

15. Pitch drop experiment
Started in 1927 by University of Queensland Professor Thomas Parnell.
A drop of pitch falls every 9 years
Pitch drop experiment apparatus
Pitch can be shattered by a hammer

16. Viscoelasticity and Stress Relaxation
Whereas steady-shear measurements probe material responses under a steady-state condition, creep and stress relaxation monitor material responses as a function of time.
Stress relaxation studies the effect of a step-change in strain on stress. ?

17. Physical Meaning of the Relaxation Time
Constant strain applied
time s
Stress relaxes over time as molecules re-arrange
time
Stress relaxation:

18. Introduction to Viscoelasticity
All viscous liquids deform continuously under the influence of an applied stress – They exhibit viscous behavior.
Solids deform under an applied stress, but soon reach a position of equilibrium, in which further deformation ceases. If the stress is removed they recover their original shape – They exhibit elastic behavior.
Viscoelastic fluids can exhibit both viscosity and elasticity, depending on the conditions.
Viscous fluid
Viscoelastic fluid
Elastic solid
Polymers display VISCOELASTIC properties

19. Static Testing of Rubber Vulcanizates
Static tensile tests measure retractive stress at a constant elongation (strain) rate.
Both strain rate and temperature influence the result
Note that at common static test conditions, vulcanized elastomers store energy efficiently, with little loss of inputted energy.

20. Dynamic Testing of Rubber Vulcanizates: Resilience
Resilience tests reflect the ability of an elastomeric compound to store and return energy at a given frequency and temperature.
Change of rebound
resilience (h/ho) with
temperature T for:
1. cis-poly(isoprene);
2. poly(isobutylene);
3. poly(chloroprene);
4. poly(methyl methacrylate).

21. Hooke and Newton
It is difficult to predict the creep and stress relaxation for polymeric materials.
It is easier to predict the behaviour of polymeric materials with the assumption  it behaves as linear viscoelastic behaviour.
Deformation of polymeric materials can be divided to two components:
Elastic component – Hooke’s law
Viscous component – Newton’s law
Deformation of polymeric materials  combination of Hooke’s law and Newton’s law.

22. Hooke’s law & Newton’s Law
The behaviour of linear elastic were given by Hooke’s law: or
The behaviour of linear viscous were given by Newton’s Law:
E= Elastic modulus
s = Stress
e = strain
de/dt = strain rate
ds/dt = stress rate
= viscosity
** This equation only applicable at low strain

23. Viscoelasticity and Stress Relaxation
Stress relaxation can be measured by shearing the polymer melt in a viscometer (for example cone-and-plate or parallel plate). If the rotation is suddenly stopped, ie. g=0, the measured stress will not fall to zero instantaneously, but will decay in an exponential manner. .
Relaxation is slower for Polymer B than for Polymer A, as a result of greater elasticity.
These differences may arise from polymer microstructure (molecular weight, branching).

24. STRESS RELAXATION
CREEP
Constant strain is applied  the stress relaxes as function of time
Constant stress is applied  the strain relaxes as function of time

25. Time-dependent behavior of Polymers
The response of polymeric liquids, such as melts and solutions, to an imposed stress may under certain conditions resemble the behavior of a solid or a liquid, depending on the situation.
Reiner used the biblical expression that “mountains flowed in front of God” to define the DEBORAH number

26. metal
elastomer
Viscous liquid

27. Static Modulus of Amorphous PS
Glassy
Leathery
Rubbery
Viscous
Polystyrene
Stress applied at x
and removed at y

28. Stress Relaxation Test
Strain
Elastic
Stress
Stress
Viscoelastic
Stress
Viscous fluid
Viscous fluid
Viscous fluid
Time, t

29. Stress relaxation Go (or GNo) is the “plateau modulus”:
Stress relaxation after a step strain go is the fundamental way in which we define the relaxation modulus:
Go (or GNo) is the “plateau modulus”:
where Me is the average mol. weight between entanglements
G(t) is defined for shear flow. We can also define a relaxation modulus for extension:

30. Stress relaxation of an uncrosslinked melt
perse
Glassy behavior
Transition Zone
Terminal Zone
(flow region)
slope = -1
Plateau Zone
Mc: critical molecular weight above which entanglements exist
3.24

32. Network of Entanglements
There is a direct analogy between chemical crosslinks in rubbers and “physical” crosslinks that are created by the entanglements.
The physical entanglements can support stress (for short periods up to a time tT), creating a “transient” network.

33. Relaxation Modulus for Polymer Melts
Elastic
tT = terminal relaxation time tT
Viscous flow

34. Viscosity of Polymer Meltsho
Extrapolation to low shear rates gives us a value of the “zero-shear-rate viscosity”, ho.
Shear thinning behaviour
Poly(butylene terephthalate) at 285 ºC
For comparison: h for water is 10-3 Pa s at room temperature.

35. Rheology and Entanglements.
The elastic properties of linear thermo-plastic polymers are due to chain entanglements. Entanglements will only occur above a critical molecular weight.
When plotting melt viscosity o against molecular weight we see a change of slope from 1 to 3.45 at the critical entanglement molecular weight. o Mn
Slope = 1
Slope = 3.4
Entanglement
molecular weight

36. h ~ tTGP Scaling of Viscosity: ho ~ N3.4 ho ~ N3.4 N0 ~ N3.4 Why? 3.4
Data shifted for clarity!
Viscosity is shear-strain rate dependent. Usually measure in the limit of a low shear rate: ho
h ~ tTGP
3.4
ho ~ N3.4 N0 ~ N3.4
Universal behaviour for linear polymer melts
Applies for higher N: N>NC
Why?
G.Strobl, The Physics of Polymers, p. 221

38. Application of Theory: Electrophoresis
From Giant Molecules

40. Mechanical Model
Methods that used to predict the behaviour of visco-elasticity.
They consist of a combination of between elastic behaviour and viscous behaviour.
Two basic elements that been used in this model:
Elastic spring with modulus which follows Hooke’s law
Viscous dashpots with viscosity h which follows Newton’s law.
The models are used to explain the phenomena creep and stress relaxation of polymers involved with different combination of this two basic elements.

41. Dynamic Viscosity (dashpot)
Shear stress
Lack of slipperiness
Resistance to flow
Interlayer friction
SI Unit: Pascal-second
Shear rate
1 centi-Poise = milli Pascal-second

42. stress
stress input
Strain in dashpot
dashpot
27/06/46

43. Maxwell model In series Viscous strain remains after load removal.
stress input
Model
Strain Response
Maxwell model
27/06/46

44. Kelvin or Voigt model In parallel
Nonlinear increase in strain with time
Strain decreases with time after load removal because of the action of the spring (and dashpot).
stress input
Model
Strain Response
Voigt model
27/06/46

45. Typical Viscosities (Pa.s)
Asphalt Binder
Polymer Melt
Molasses
Liquid Honey
Glycerol
Olive Oil
Water
Acetic Acid
100,000
1,000
100 10 1
0.01
0.001
Courtesy: TA Instruments

46. Shear stress Non Newtonian Fluids Shear rate Casson Plastic
Bingham Plastic
Pseudoplastic (or Shear thinning)
Shear stress
Newtonian
Dilatant (or Shear thickening)
Non Newtonian Fluids
Shear rate

47. The Theory of Viscoelasticity
The liquid behavior can be simply represented by the Newtonian model. We can represent the Newtonian behavior by using a “dashpot” mechanical analog:
The simplest elastic solid model is the Hookean model, which we can represent by the “spring” mechanical analog.
The theory of linear viscoelasticity is phenomenological; no first principle prediction exists.
The aim of the theory is to link properties in one circumstance to the measurable in the other.
The basic components in the theory are the spring and the dashpot
stress
strain
viscosity
Gmodulus

48. Maxwell Model Let’s create a VISCOELASTIC material:
At least two components are needed, one to characterize elastic and the other viscous behavior. One such model is the Maxwell model:
stress
strain
viscosity
Gmodulus

49. Maxwell Model Let’s try to deform the Maxwell element stress
strain
viscosity
Gmodulus

50. Maxwell model too primitive
Maxwell: solid line
Experiment: circles
Maxwell model too primitive

51. Maxwell Model l is the relaxation time . Exponential decay in stresses
The deformation rate of the Maxwell model is equal to the sum of the individual deformation rates:
l is the relaxation time
If the mechanical model is suddenly extended to a position and held there (g=const., g=0): .
Exponential decay in stresses
stress
strain
viscosity
Gmodulus

52. Examples of Viscoelastic Materials
Mattress, Pillow
Tissue, skin
27/06/46

53. Elastic
Viscous
The common mechanical model that use to explain the viscoelastic phenomena are:
Maxwell
Spring and dashpot  align in series
Voigt
Spring and dashpot  align in parallel
Standard linear solid
One Maxwell model and one spring  align in parallel.

54. Measurements of Shear Viscosity
Melt Flow Index
Capillary Rheometer
Coaxial Cylinder Viscometer (Couette)
Cone and Plate Viscometer (Weissenberg rheogoniometer)
Disk-Plate (or parallel plate) viscometer

55. Weissenberg Effect

56. Dough Climbing: Weissenberg Effect
Other effects:
Barus
Kaye

57. Courtesy: Dr. Osvaldo Campanella

58. Dynamic Mechanical Testing Response for Classical Extremes
Purely Viscous Response
(Newtonian Liquid)
Purely Elastic Response
(Hookean Solid)
 = 90°
 = 0°
Stress
Stress
Strain
Strain
Courtesy: TA Instruments

59. Dynamic Mechanical Testing Viscoelastic Material Response
Phase angle 0° < d < 90°
Strain
Stress
Courtesy: TA Instruments

60. DMA Viscoelastic Parameters: The Complex, Elastic, & Viscous Stress
The stress in a dynamic experiment is referred to as the complex stress *
The complex stress can be separated into two components:
1) An elastic stress in phase with the strain. ' = *cos
' is the degree to which material behaves like an elastic solid.
2) A viscous stress in phase with the strain rate. " = *sin
" is the degree to which material behaves like an ideal liquid.
Phase angle d
Complex Stress, *
* = ' + i"
Courtesy: TA Instruments
Strain, 

61. 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)cos
The Viscous (loss) Modulus:
The ability of the material to dissipate energy. Energy lost as heat.
G" = (stress*/strain)sin
Tan Delta:
Measure of material damping - such as vibration or sound damping.
Tan = G"/G'
Courtesy: TA Instruments

62. DMA Viscoelastic Parameters: Damping, tan 
Dynamic measurement represented as a vector
It can be seen here that G* = (G’2 +G”2)1/2
G"
Phase angle  G'
The tangent of the phase angle is the ratio of the loss modulus to the storage modulus.
tan  = G"/G'
"TAN DELTA" (tan )is a measure of the damping ability of the material.
Courtesy: TA Instruments

63. Frequency Sweep: Material Response
Transition
Region
Rubbery Plateau
Region
Terminal Region
Glassy Region
log G'and G" 1 2
Storage Modulus (E' or G')
Loss Modulus (E" or G")
log Frequency (rad/s or Hz)
Courtesy: TA Instruments

64. Viscoelasticity in Uncrosslinked, Amorphous Polymers
Logarithmic plots of G’ and G” against angular frequency for uncrosslinked poly(n-octyl methacrylate) at 100°C (above Tg), molecular weight 3.6x106.

65. Dynamic Characteristics of Rubber Compounds
Why do E’ and E” vary with frequency and temperature?
The extent to which a polymer chains can store/dissipate energy depends on the rate at which the chain can alter its conformation and its entanglements relative to the frequency of the load.
Terminal Zone:
Period of oscillation is so long that chains can snake through their entanglement constraints and completely rearrange their conformations
Plateau Zone:
Strain is accommodated by entropic changes to polymer segments between entanglements, providing good elastic response
Transition Zone:
The period of oscillation is becoming too short to allow for complete rearrangement of chain conformation. Enough mobility is present for substantial friction between chain segments.
Glassy Zone:
No configurational rearrangements occur within the period of oscillation. Stress response to a given strain is high (glass-like solid) and tand is on the order of 0.1

66. Dynamic Temperature Ramp or Step and Hold: Material Response
Glassy Region
Transition
Region
Rubbery Plateau
Region
Terminal Region
Log G' and G" 1
Loss Modulus (E" or G")
Storage Modulus (E' or G') 2
Temperature
Courtesy: TA Instruments

67. One more time: Dynamic (Oscillatory) Testing
In the general case when the sample is deformed sinusoidally, as a response the stress will also oscillate sinusoidally at the same frequency, but in general will be shifted by a phase angle d with respect to the strain wave. The phase angle will depend on the nature of the material (viscous, elastic or viscoelastic)
Input
Response
where 0°<d<90°
stress
strain
viscosity
Gmodulus
3.29

68. One more time: Dynamic (Oscillatory) Testing
By using trigonometry:
(3-1)
In-phase component of the stress, representing solid-like behavior
Out-of-phase component of the stress, representing liquid-like behavior
Let’s define:
where:
3.30

69. Physical Meaning of G’, G”
Equation (3-1) becomes:
We can also define the loss tangent:
For solid-like response:
For liquid-like response:
G’storage modulus
G’’loss modulus

70. Typical Oscillatory DataG’
G’’
log G
log 
Rubber
G’storage modulus
G’’loss modulus
Rubbers – Viscoelastic solid response:
G’ > G” over the whole range of frequencies

71. Typical Oscillatory DataG’
G’’
log G
log 
Melt or solution G0
G’storage modulus
G’’loss modulus
Polymeric liquids (solutions or melts) Viscoelastic liquid response:
G” > G’ at low frequencies
Response becomes solid-like at high frequencies
G’ shows a plateau modulus and decreases with w-2 in the limit of low frequency (terminal region)
G” decreases with w-1 in the limit of low frequency

72. Typical Oscillatory Data
For Rubbers – Viscoelastic solid response:
G’ > G” over the whole range of frequencies
For polymeric liquids (solutions or melts) – Viscoelastic liquid response:
G”>G’ at low frequencies
Response becomes solid-like at high frequencies
G’ shows a plateau modulus and decreases with w-2 in the limit of low frequency (terminal region)
G” decreases with w-1 in the limit of low frequency

74. Sample is strained (pulled, ) rapidly to pre-determined strain ()
Stress required to maintain this strain over time is measured at constant T
Stress decreases with time due to molecular relaxation processes
Relaxation modulus defined as:
Er(t) also a function of temperature
Er(t) = (t)/e0