E 1 Physics of Solid Polymers –Pt. 8: ViscoelasticitySeoul National University, WCU Program C

Physics of Solid Polymers – Pt. 8: Viscoelasticity

Seoul National University, WCU Program C

₂

E

₂

1

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

) relationship

Physics of Solid Polymers – Pt. 8 Viscoelasticity

2

Temperature Dependence for Polymers

Glassy

Leathery

rubbery rubbery flow

Viscous

Temperature dependence of elastic modulus Stree 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 t

J =

linear for strains below ~0.005 (0.5%) J_U: unrelaxed compliance J_R: relaxed compliance

ε Deformation (strain): ε or γ

(inverse modulus)

Physics of Solid Polymers – Pt. 8: Viscoelasticity

Seoul National University, WCU Program C

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₂

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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

G_U: Unrelaxed modulus G_U = J_U^-1 G_R: Relaxed modulus

G_R = J_R^-1

10

Dynamic Mechanical Analysis (DMA, DMTA)

Oscillatory sinusoidal strain of angular frequency ω

ω

t

γ γ

= ₀sin

For a linearviscoelastic material the stress is also sinusoidal

( ω δ )

σ

σ =

₀

sin

t

+

11

ωt γ γ= ₀sin

(

ω δ

)

σ σ= ₀sin t+

(

σ δ

)

ω^t

(

σ δ

)

ω^t

σ= 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

Physics of Solid Polymers – Pt. 8: Viscoelasticity

Seoul National University, WCU Program C

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E

₂

3

13

polyisobutylene

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

15

Linear and branched polyethylene

16

Measurement of T_gfrom V-T Curve

T V V d

d

= 1 α

T_gis 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 T_g

• Heat Capacity

• Dilatometry (volume)

• DMA

•Dielectric Spectroscopy

•NMR

•Gas permeability

18

Free Volume Theory of Glass Transition The free volume, V_f, 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 = V_o+ V_f V: total volume

V_o: volume actually occupied by molecules V_f: free volume

The free volume V_fis a measure of the space available for the polymer to undergo rotation and translation.

Physics of Solid Polymers – Pt. 8: Viscoelasticity

Seoul National University, WCU Program C

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E

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19

Temperature Dependence of the Volume Occupied volume V_o

• Linear function of temperature (thermal vibration)

• Irrespective of whether the polymer is glassy or rubbery Free volume V_f

• Linear (vs T) in the rubbery state (above T_g)

• Free volume contracts with decreasing T, and reaches a critical value at T_gthat there is insufficient free space for large scale chain movement

• V_fis essentially constant below T_gbecause molecular chains are immobilized.

20

Mechanical Models of Viscoelasticity:

Spring and Dashpot

G

σ

J

γ

=

σ

=

t d d

γ η σ

=

21

Maxwell and Voigt-Kelvin Models

η G

J η

22

η G

Maxwell Model: Creep

Creep:

σ

=

σ

0is constant Strain:

γ

=

γ

₁+

γ

₂ Stress:

σ

=

σ

₁=

σ

₂ Dashpot: d

γ

₁/dt =

σ

/

η

Spring:

γ

₂=

σ

/G

γ

1=

σ

0/

η

t

γ

2=

σ

0/G

γ

=

σ

₀/

η

t+

σ

₀/G

23

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

24

Zener Model (Standard Linear Solid)