1 t
σ
Stress vs.time t
ε
ideal elastic solids t
ε
viscous liquid
• Perfectly elastic solids
metals and ceramics at low strains Hooke’s law: σ = E ε
• Viscous liquid
Newton’s law: σ = η dε/dt Stress-strain (
σ
-ε
) relationshipPhysics of Solid Polymers – Pt. 8 Viscoelasticity
2
Temperature Dependence for Polymers
Glassy
Leathery
rubbery rubbery flow
Viscous
Temperature dependence of elastic modulus Strain vs time
3
Viscoelastic Behaviour of Polymers Deformation vs. Time
Polymers are viscoelastic
- behaviour both viscous and elastic.
t ε
glassy state t
ε
leathery state t
ε
ideal rubber
t ε
vulcanized rubber t
ε
unvulcanized rubber t
ε
viscous state
4
Creep
– strain vs. time
a, initial elastic response; b, creep; c, irrecoverable viscous flow.
5
Creep = progressive increase in strain over time at constant stress.
Creep
ε1(t) ε2(t)
6
Linear Viscoelastic Creep
Creep compliance J(t)
( ) ( )
2 2 1
1 σ
γ σ γ t = t
( ) ( )
σ γ
t tJ =
linear for strains below ~0.005 (0.5%) JU: unrelaxed compliance JR: relaxed compliance
ε Deformation (strain): ε or γ
(inverse modulus)
7
Linear and Non-linear Viscoelasticity
8
Stress-Relaxation constant strain (γ) is applied at t = 0
Æ measure stress σ (t) required to maintain constant γ
9
Linear Viscoelastic Stress-Relaxation
( ) ( )
2 2 1 1
γ σ γ σ t = t
( ) ( )
γ σt t
G =
Stress-relaxation modulus
GU: Unrelaxed modulus GU = JU-1 GR: Relaxed modulus
GR = JR-1
10
Dynamic Mechanical Analysis (DMA, DMTA)
Oscillatory sinusoidal strain of angular frequency ω
ω
tγ γ
= 0sinFor a linearviscoelastic material the stress is also sinusoidal
( ω δ )
σ
σ =
0sin
t+
11
ωt γ γ= 0sin
(
ω δ)
σ σ= 0sin t+
(
σ δ)
ωt(
σ δ)
ωtσ= 0cos sin + 0sin cos
[
G' ωt G'' ωt]
γ
σ= 0 sin + cos γ δ σ cos
0
= 0
G'
γ δ σ sin
0
= 0
G'' Storage modulus
Loss modulus
Loss tangent tan δ = G’’G’
Stress-Strain Relationship for a Dynamic Analysis
σ= G* γ G* =complex modulus
12
Torsion Pendulum for Dynamic Mechanical Analysis
1 n
ln n +
=
Λ A
A = π tanδ Logarithmic decrement:
Storage modulus: G’ = KMω2
polyisobutylene 13
Forced Oscillation Technique
• a reliable technique for high values of δ
• very easy to change frequency.
An oscillatory force is applied and the phase angle δ can be directly determined.
Effect of timescale on glass-rubber relaxation t ~ 102s, T ~ -66 ºC t ~ 10-2s, T ~ -10 ºC
14
Dynamic Mechanical Properties of PMMA
* CH2C *
C CH3
O O CH3
n
Constant frequency
~ 1Hz
tanδ
15
Linear and branched polyethylene
16
Measurement of Tgfrom V-T Curve
T V V d
d
= 1 α
Tgis defined as the point where the thermal expan- sion coefficient:
undergoes a discontinuity.
Effect of equilibrating time Curve 1: 0.02 hours Curve 2: 100 hours
17
Measurment of Tg
• Heat Capacity
• Dilatometry (volume)
• DMA
•Dielectric Spectroscopy
•NMR
•Gas permeability
18
Glass transition temperature of some materials
1175 Fused quartz
520-600 Soda-lime glass
245 Chalcogenide (AsGeSeTe)
145 Polycarbonate (PC)
Polymethyl methacrylate (PMMA) 105 (atactic)
15 Poly(3-hydroxybutyrate) (PHB)
95 Polystyrene (PS)
80 Poly(vinyl chloride) (PVC)
85 Poly(vinyl alcohol) (PVA)
70 Poly(ethylene terephthalate) (PET)
28 Poly(vinyl acetate) (PVAc)
−20 polypropylene (atactic)
Tyre Rubber −72
−105 or −30 also cited Polyethylene LDPE
Tg(°C) Polymer
19
Free Volume Theory of Glass Transition The free volume, Vf, is defined as the unoccupied space in a sample, arising from the inefficient packing of disordered chains in the amorphous regions of a polymer sample.
V = Vo+ Vf V: total volume
Vo: volume actually occupied by molecules Vf: free volume
The free volume Vfis a measure of the space available for the polymer to undergo rotation and translation.
20
Temperature Dependence of the Volume Occupied volume Vo
• Linear function of temperature (thermal vibration)
• Irrespective of whether the polymer is glassy or rubbery Free volume Vf
• Linear (vs T) in the rubbery state (above Tg)
• Free volume contracts with decreasing T, and reaches a critical value at Tgthat there is insufficient free space for large scale chain movement
• Vfis essentially constant below Tgbecause molecular chains are immobilized.
21
Mechanical Models of Viscoelasticity:
Spring and Dashpot
G
σ
Jγ
=σ
=dt d
γ η σ
=22
Maxwell and Voigt-Kelvin Models
η G
J η
23
η G
Maxwell Model: Creep
Creep:
σ
=σ
0is constant Strain:γ
=γ
1+γ
2 Stress:σ
=σ
1=σ
2 Dashpot: dγ
1/dt =σ
/η
Spring:γ
2=σ
/Gγ
1=σ
0/η
tγ
2=σ
0/Gγ
=σ
0/η
t+σ
0/G24
σ0J
Stress relaxation behaviour of Maxwell model
(
τ)
η σ σ
σ 0exp Gt 0exp t/
−
⎟⎟=
⎠
⎜⎜ ⎞
⎝
⎛−
=
Creep Stress-relaxation
Creep behaviour of Voigt-Kelvin model )]
/ exp(
1 [ )]
exp(
1
[ 0 R
0 σ τ
σ η
γ J t
J
J − − t = − −
=
σ0/G σ0/η
Gγ0
25
Zener Model (Standard Linear Solid)