VOIGT MODELMaxwell mdel essentially assumes a uniform distribution Of stress.Now assume uniform distribution of strain - VOIGT MODEL

Picture representation

Equation

__dε(t)dt

σ(t) = Eε(t) + η

(Strain in both elements of themodel is the same and the totalstress is the sum of the twocontributions)

VOIGT MODEL - creep and stress relaxationGives a retarded elastic response but does not allow for “ideal” stress relaxation,in that the model cannot be “instantaneously” deformed to a given strain.But in CREEP σ = constant, σ

0

σ(t) = σ 0__dε(t)dt

= Eε(t) + η

=

σΕ

0_ε(t) = [1- exp (-t/τ ‘ )]t

Strain

tt1

t2

__dε(t)dt

__ ε(t) τ ‘

ση

0_+t

τ ‘t - retardation time (η/E)

SUMMARY

Voigtelement

MaxwellelementDashpotSpring

E

η

E

E η

η

Voigtmodel

Maxwellmodel

DashpotSpringStrain

tt1

t1

t1

t1t

2t2

t2 t

2

PROBLEMS WITH SIMPLE MODELS

•The maxwell model cannot account for a retarded elastic response

•The voigt model does not describe stress relaxation

•Both models are characterized by single relaxation times - a spectrum of relaxation times would provide a better description

NEXT - CONSIDER THE FIRST TWO PROBLEMSTHEN -THE PROBLEM OF A SPECTRUM OF RELAXATION TIMES

ELASTIC + VISCOUS FLOW + RETARDED ELASTIC

FOUR - PARAMETER MODEL

EM

EV η

V

ηM

ε [1- exp (-t/τ )]t

σΕ

0_M

__ση M

0t= +σΕ

0_M

+

eg CREEP

applied removed

Elasticresponse

Retardedelastic

response

Permanentdeformation

Time (t)t1

t2

Strain

DISTRIBUTIONS OF RELAXATION AND RETARDATION TIMES The Maxwell - Wiechert Model

E1 E

2E

3

η3

η2

η1

=_dεdt

_ση

_dσdt

_1Ε +1

1 11

= _ση

_dσdt

_1Ε +2

2 22

= _ση

_dσdt

_1Ε +3

3 33

Consider stress relaxation _dεdt = 0

0σ = σ exp[-t/τ ]t11

0σ = σ exp[-t/τ ]t22

0σ = σ exp[-t/τ ]t33

DISTRIBUTIONS OF RELAXATION AND RETARDATION TIMES

E1 E

2E

3

η3

η2

η1

Stress relaxation modulus

E(t) = σ(t)/ε 0

σ(t) = σ + σ + σ1 2 3

E(t) = exp (-t/τ )σ__ε

010

01 + exp (-t/τ )σ__ε

020

02 + exp (-t/τ )σ__ε

030

03

Or, in generalσ__ε

0n0

E(t) = Σ E exp (-t/τ )tnn E =nwhere

SIMILARLY, FOR CREEP COMPLIANCE COMBINE VOIGT ELEMENTS TO OBTAIN

D(t) =ΣD [1- exp (-t/τ ‘ )]tnn

4

6

8

10

- 10

Glassyregion

Rubberyplateau

Transition

Lowmolecular

weight

- 8 - 4- 6 0- 2 + 2

Log time (sec)

Log

E(t )

(dy

nes

c m- 2

)

Highmolecular

weight

43210-1-20

2

4

6

8

10

Log time (min)

Log

E(t

) (

Pa)

DISTRIBUTIONS OF RELAXATION AND RETARDATION TIMES Example - The Maxwell - Wiechert Model with n = 2

E(t) = Σ E exp (-t/τ )tnn

n = 2

TIME - TEMPERATURE SUPERPOSITION PRINCIPLE

Recall that we have seen that there is a time - temperatureequivalence in behaviour

4

3

5

6

7

8

10

9

Glassyregion

Rubberyplateau

Cross-linkedelastomers

MeltLow

molecularweight

Log E(Pa)

Temperature

This can be expressed formally in terms of a suprposition principle

4

6

8

10

- 10

Glassyregion

Rubberyplateau

Transition

Lowmolecular

weight

- 8 - 4- 6 0- 2 + 2

Log time (sec)

Log

E(t )

(dy

nes

c m- 2

)High

molecularweight

TIME TEMPERATURE SUPERPOSITIONPRINCIPLE - creep

Log t

ε(t) Tσ

0

T3 > T

2 > T

1

T3

T2

T1

T0

T

log aT

0

ε(t) Tσ0

log t - log aT

TIME TEMPERATURE SUPERPOSITIONPRINCIPLE - stress relaxation

SIGNIFICANCE OF SHIFT FACTORWhat is the significance of the log scale for a ,and what does this tell us about the temperature dependence of relaxation behaviour in amorphous polymers ?Consider stress relaxation:

T

E(t) = Σ E exp (-t/τ )tn

n

a =__τt1

τt0T

Let a particular mode of relaxation have a characteristic time τ at T , and a characteristic time τ at T . Then DEFINE0 1t1t0

__τt1 τt0aT

t t___=So that the exponential term can be written

Hence, taking logs

log (t/τ ) = log (t/τ ) + log at1 t0 T

SIGNIFICANCE OF SHIFT FACTOR

log (t/τ ) = log (t/τ ) + log at1 t0 T

•ie relaxation behaviour at one temperature can be superimposed on that at another by shifting an amount a along a log scale.

•BUT ,real behaviour is characterized by a distribution of relaxation times and relaxation mechanisms vary and have different length scales as a function of temperature

•This implies that all the relaxation processes involved have (more or less) the same temperature dependence

T

RELAXATION PROCESSES ABOVE Tg - THE WLF EQUATION

Log a = _________-C (T - T )

1

s

C + (T - T )s2T

From empirical observation

For Tg > T < ~Tg + 100 C 0

Originally thought that C and C were universal constants,= 17.44 and 51.6, respectively,when T = Tg .Now known that these vary from polymer to polymer.Homework problem - show how the WLF equation can beobtained from the relationship of viscosity to freevolume as expressed in the Doolittle equation

s

2

1

DYNAMICS OF POLYMER CHAINS

obstacles

chain

An advanced topic that we will not discuss in detail

Rouse - Bouche modelA chain as a string ofBeads linked by springs

Reptation,scaling conceptsAnd other advanced theories