VISCOELASTICITY IN GLASS, RUBBER AND MELT PHASE WHAT IS - - PowerPoint PPT Presentation

VISCOELASTICITY IN GLASS, RUBBER AND MELT PHASE

WHAT IS VISCOELASTICITY?

Elastic and viscous

 A viscoelastic material has at the same time both

elastic and viscous properties.

Cause of viscoelasticity

 Viscoelasticity is caused by entanglement of long

particles.

 Any material that consists of long flexible fibre like

particles is in nature viscoelastic.

 Polymers are always viscoelastic.

Some viscoelastic materials

 A pile of snakes.  Spaghetti.  Tobacco.  All fibre-like particles.

ABOUT POLYMER MOLECULES

Repeat unit (1)

 Polymer molecules are long chains built from many

small identical repeat units (or monomers).

 Polyvinylchloride (PVC) consists of many vinyl chloride

(-CH2-CHCl-) repeat units.

 Polyethylene (PE) consists of many ethylene

(-CH2-CH2-) repeat units.

 The number of repeat units in a macromolecule can

be very large: up to 10000 or more.

Repeat unit (2)

 The mutual direction between two neighbouring

repeat units is not fixed but can change due to thermal movements.

 Each repeat unit is hindered in its freedom by

neighbouring repeat units. Their possibility to change their direction is limited.

Kuhn segment (1)

 It takes several repeat units in a row in order to be

able to randomly take any direction.

 Such a group of repeat units is called a Kuhn

segment.

Repeat units Kuhn element

Kuhn segment (2)

 The number of repeat units in a Kuhn segment is a

fixed number for each polymer.

 It is called the characteristic ratio C∞.  Examples:  Number of Kuhn segments (NK) in a molecule with N

repeat units:

Characteristic ratio and Kuhn length for several polymers. PB PP PE PVC PMMA PS PC C∞ 5.5 6.0 8.3 6.8 8.2 9.5 1.3 lK (Ǻ) 10 11 15 26 15 18 2.9



 C N NK

Size of the macromolecule

 Each Kuhn segment can randomly take any direction

in space.

 The shape of the macromolecule in space therefor

follows a random path.

 Average size (r0) macromolecule:

K K

N l r 

r0 lK

Entanglements and blobs (1)

 Each macromolecule will be entangled with several

• ther macromolecules.

 At each entanglement the possible movements of

the Kuhn segments will be seriously limited.

 In between two entanglements the Kuhn segments

will follow a random path. This part of the macromolecule is called a blob.

blob

 If there are on average Ne Kuhn segments in a blob

then the average diameter of the blobs Dblob will be:

 A macromolecule contains NK/Ne blobs. The blobs

follow a random path in space.

 The start to end distance L of the macromolecule

will be:

Entanglements and blobs (2)

e K blob

N l r 

K K e K blob

N l N N r r  

POLYMER STRUCTURE

Network density (1)

 The polymer molecules form a disordered structure.  The molecules are entangled with many

neighbouring molecules. They form a network.

 The network density c is the number of

entanglements per volume:

e K c

N m   

Network density (2)

The network density influences:

 Strain hardening modulus (glass phase).  Rubber modulus (rubber and melt phase):  Stress crack resistance.

kT G

c rub

 

Free volume

 In between the molecules free volume is present.  The free volume is small. The molecules hinder

each other strongly in their movements.

 The free volume fraction free

is the relative difference between the amorphous and the crystalline volume:

  



     T T v v v

c a a c a free

  

T∞ Tg Tm

MOBILITY OF POLYMER MOLECULES

Movement possibilities of polymer molecules

Polymer molecules have two ways to move:

 Rotation of Kuhn segments.  Reptation of the entire molecule.

Rotation Reptation

Movement possibilities of polymer molecules

Polymer molecules have two ways to move:

 Rotation of Kuhn segments.  Reptation of the entire molecule.

Rotation Reptation

Movement possibilities of polymer molecules

Polymer molecules have two ways to move:

 Rotation of Kuhn segments.  Reptation of the entire molecule.

 Reptation is caused by the rotation of the Kuhn

segments in random directions.

Movement possibilities of polymer molecules

Polymer molecules have two ways to move:

 Rotation of Kuhn segments.  Reptation of the entire molecule.

Rotation

• Parts of the chain rotate; the molecule

itself is not displaced

• The rotation time rot is strongly

dependent on temperature.

• Rotation is important for the glass phase

properties:

• Glass transition temperature
• Yield stress
• Glass stress relaxation

Movement possibilities of polymer molecules

Polymer molecules have two ways to move:

 Rotation of Kuhn segments.  Reptation of the entire molecule.

Reptation

• The molecule moves into another position.
• The reptation time is proportional to the

rotation time (rep =  rot) with  = 104 – 108.

• Reptation is important for the fluid

properties:

• Viscosity
• Elasticity
• Rubber stress relaxation

Movement possibilities of polymer molecules

Polymer molecules have two ways to move:

 Rotation of Kuhn segments.  Reptation of the entire molecule.

Rotation Reptation

• Parts of the chain rotate; the molecule

itself is not displaced

• The rotation time rot is strongly

dependent on temperature.

• Rotation is important for the glass phase

properties:

• Glass transition temperature
• Yield stress
• Glass stress relaxation
• The molecule moves into another position.
• The reptation time is proportional to the

rotation time (rep =  rot) with  = 104 – 108.

• Reptation is important for the fluid

properties:

• Viscosity
• Elasticity
• Rubber stress relaxation

Rotation of Kuhn segments

 The polymer feels stiff when the rotation time is

much more than 1 second (glass phase).

 The polymer feels flexible when the rotation time is

much shorter than 1 second (rubber and melt phase).

 The glass transition temperature Tg is the

temperature at which the rotation time of the Kuhn segments is 1 second.

Rotation of Kuhn segments

 All molecules attract each other.

 Below the melting temperature they form a regular

crystalline structure.

Rotation of Kuhn segments

 The repeat units in a polymer also attract each

• ther.

 Below the melting temperature the formation of a

crystalline structure is difficult due to the limited mobility of the repeat units.

 They cluster together in

cooperatively rearranging regions (CRR’s).

 The seriously hinders the

rotation of the Kuhn segments.

 The rotation time rot of the Kuhn segments increases

strongly with reducing temperature.

       kT E z

rot rot , exp

 

3

2 3 3          

 

p p z

T Tm

p

/

2 



with and

Rotation of Kuhn segments

Rotation of Kuhn segments

 Above and below the glass transition temperature

Tg cooperative rotation of the Kuhn segments:

 The level of cooperativity z0 is only a function of

temperature.

 Above the glass transition temperature a dynamic

equilibrium is always reached.

 Below glass transition temperature Tg the Kuhn

rotation time is very long (>> 1 s).

 Reaching equilibrium takes time.  Time dependent properties of the polymer.

       kT E z

rot rot , exp

 

3

2 3 3          

 

p p z

T Tm

p

/

2 



Reptation of the macromolecule

 At times longer than the reptation time the polymer

behaves like a fluid.

 At times shorter than the reptation time the polymer

behaves like a rubber.

rubber fluid

Reptation of the macromolecule

 The reptation time is proportional to the rotation

time.

 The proportionality strongly depends on the number

• f Kuhn segments NK in the macromolecule:

 1 Kuhn segment: + or - give step –lK or +lK during rot.  2 Kuhn segments: ++ or -- give step step –lK or +lK

+- and -+ give no displacement  Step -lK or +lK takes 2rot.

 NK Kuhn segments: Step -lK or +lK takes NKrot.  Reptation over NK Kuhn segments takes NK

2 steps:

rot rot K K rep

K

N N N   

3 2

 

GLASS, RUBBER AND MELT PHASE

Glass phase (short term)

 Kuhn segments have a rotation time of (much) more

than 1 second.

 The plastic is rigid on a human time scale (observation

time is a few seconds).

 The polymer is difficult to deform:

 Chain segments can only bend a little bit. The

macromolecules are rigid.

 An applied force will only result in a small deformation

• f the plastic.

Glass phase (long term)

 Kuhn segments have a rotation time of (much) more

than 1 second.

 The plastic is rigid on a human time scale (observation

time is a few seconds).

 A force applied for a long time is still able to

deform the polymer in the glass phase.

 The time should be longer than the time that the Kuhn

segments need to rotate.

 This slow deformation is called creep.  The polymer now behaves like a rubber.

Rubber phase

 In the rubber phase the Kuhn segments rotate in a

time less than 1 second.

 The plastic is flexible.

 The reptation time of the macromolecules is much

higher than 1 s.

 The relative position of the macromolecules will not

change.

Melt phase

 In the melt phase the reptation time of the

macromolecules is less than 1 second.

 The macromolecules can change their relative position.

 In this condition the plastic can be shaped into

products by means of extrusion, injection moulding

• r blow moulding.

Glass, rubber and melt phase

Rotation time Reptation time Glass phase > 1 s >> 1 s Glass transition temperature 1 s >> 1 s Rubber phase < 1 s > 1 s Rubber – melt transition temperature << 1 s 1 s Melt phase << 1 s < 1 s

Glass, rubber and melt phase

 Relaxation of stress in the glass phase is caused by

rotation of the Kuhn segments.

 Relaxation of stress in the rubber and melt phase is

caused by reptation of the macromolecules

INFLUENCE OF STRESS ON RELAXATION TIME

Stress and relaxation time

 Relaxation of stresses in the glass phase is caused

by rotation of the Kuhn segments.

 Rotations that reduce the stress will speed up.  Rotations that increase the stress will slow down.

Stress and relaxation time

 Nett result: The relaxation time decreases

exponentially with the applied stress.

1000 s 100 s 10 s 1 s 10 MPa 30 MPa

• 30 MPa
• 10 MPa

stress relaxation time

Stress and relaxation time

 Relaxation of stresses in the glass phase is caused

by rotation of the Kuhn segments.

 Rotations that reduce the stress will speed up.  Rotations that increase the stress will slow down.  On average any stress will reduce the rotation time.

 

          kT V E T

gla rot rot rot rot rot

    exp ,

,

 

          kT V E T

gla rot rot rot rot rot

    exp ,

,

Stress and relaxation time

 Vrot is the activation volume.  Vrotgla is the energy that is consumed during

rotation of a Kuhn segment in a blob.

 If the deformation of the blob during Kuhn segment

rotation is ∆ and the stress gla is approximately constant then:

 c is the network density.

gla c gla c gla rot

d V       



  





1

Stress and relaxation time

 The average number of rotations will increase.  The average rotation time will decrease.  Since rotations can occur in any direction the

average must be determined by integration over all stresses from -gla to + gla:

 Net result: The glass stress relaxation time will

strongly decrease with increasing stress.

 

                              

  



kT V kT V kT E d V

gla rot gla rot rot rot rot gla rot av rot

gla gla

        

 

sinh exp 2 1 1

, 1 1

Stress and relaxation time

 The rotation time decreases with stress:  The reptation time is proportional to the rotation

time:

 Therefor the reptation time also decreases with

stress:

 Net result: The rubber stress relaxation time will

strongly decrease with increasing stress.

rot rep

K

N  

3



,0 exp

sinh

rep rub rep rub rot rep rep

V V E kT kT kT                 

               kT V kT V kT E

gla rot gla rot rot rot rot

    sinh exp

,

STRESS RELAXATION

Stress relaxation below glass transition temperature (1)

 Deformation of the polymer causes bending of the

chain segments.

 Rigid material; high glass stress.

 Rotation of the Kuhn segments reduces the bending.

Stress relaxation below glass transition temperature (2)

 Deformation of the polymer causes bending of the

chain segments.

 Rigid material; high glass stress.

 Rotation of the Kuhn segments reduces the bending.

 Deformation by bending is converted into deformation

by rotation:

 Glass stress changes with change in deformation by

bending.

 Rubber stress is 1000 x lower than glass stress.

 The typical relaxation time is the Kuhn segment

rotation time:

ben rot ben

d        

rot

d constant

rot gla

  

Stress relaxation below glass transition temperature (3)

 Differential equation for relaxation below the glass

transition temperature:

 The term dgla/dben relates the change in glass

stress to the change in deformation by bending.

 Shear deformation: dgla/dben is glass shear modulus.

rot gla ben gla gla

dt d d d dt d        

               kT V kT V kT E

gla rot gla rot rot rot rot

    sinh exp

,

Stress relaxation above glass transition temperature (1)

 Deformation of the polymer causes rotation of the

Kuhn segments.

 Macromolecules deformed; rubber stress.

 Reptation of the macromolecules into new positions

reduces deformation.

Stress relaxation above glass transition temperature (2)

 Deformation of the polymer causes rotation of the

Kuhn segments.

 Macromolecules deformed; rubber stress.

 Reptation of the macromolecules into new positions

reduces deformation.

 Elastic energy from deformation by rotation is

converted into heat.

 Rubber stress reduces to zero.

 The typical relaxation time is the reptation time:

rep melt

  

Stress relaxation above glass transition temperature (3)

 Differential equation for relaxation above the glass

transition temperature:

 The term drub/drot relates the change in rubber

stress to the change in deformation by rotation.

 Shear deformation: drub/drot is rubber shear modulus.

rep rub rot rub rub

dt d d d dt d        

,0 exp

sinh

rep rub rep rub rot rep rep

V V E kT kT kT                 

Stress relaxation all temperatures (1)

 Deformation of the polymer causes bending of the

chain segments  glass stress.

 Rotation of the Kuhn segments reduces the bending.

 Deformation by bending is converted into deformation

by rotation.

 Glass stress reduces to rubber stress.

 Differential equation for relaxation of the glass

stress:

rot gla ben gla gla

dt d d d dt d        

Stress relaxation all temperatures (2)

 Reptation of the macromolecules into new positions

reduce deformation by rotation to zero.

 Elastic energy from deformation by rotation is

converted into heat.

 Rubber stress reduces to zero.

 Differential equation for relaxation of the rubber

stress:

rep rub rot gla gla ben rot rub rub

d d d d dt d           

Modulus Deformation by rotation due to relaxation of deformation by bending

Stress relaxation all temperatures (3)

 Two relaxation times: rot and rep  Two coupled differential equations:

rep rub rot gla gla ben rot rub rub

d d d d dt d           

rot gla ben gla gla

dt d d d dt d        

rub gla

    

rot K rep

N  

3



Stress relaxation all temperatures (3)

 Two relaxation times: rot and rep  Two coupled differential equations:

rep rub rot gla gla ben rot rub rub

d d d d dt d           

rot gla ben gla gla

dt d d d dt d        

rub gla

    

Glass phase

rot K rep

N  

3



gla rub

  

Stress relaxation all temperatures (3)

 Two relaxation times: rot and rep  Two coupled differential equations:

rep rub rot gla gla ben rot rub rub

d d d d dt d           

rot gla ben gla gla

dt d d d dt d        

rub gla

    

rot K rep

N  

3



Rubber and melt phase

Stress relaxation all temperatures (3)

 Two relaxation times: rot and rep  Two coupled differential equations:

rep rub rot gla gla ben rot rub rub

d d d d dt d           

rot gla ben gla gla

dt d d d dt d        

rub gla

    

Rubber and melt phase

= 0

rep rub rot rub rub

dt d d d dt d        

rot K rep

N  

3



Stress relaxation small deformations

 In case of small deformations the moduli dgla/dben

and drub/drot are independent of strain.

 The differential equations then reduce to:

rep rot rot ben rot

dt d       

rot ben ben

dt d dt d      

Deformation by bending is converted into deformation by rotation.

YIELD STRESS

Yield stress

Yield stress

 In the glass phase the rotation time of the Kuhn

segments is very long.

 The rotation time strongly reduces with stress.  At a certain stress the rotation time has reduced to

a few seconds.

 The polymer starts to deform quickly.  The yield stress has been reached.

Yield stress

 The yield stress is determined in the glass phase.  Equations to use:  Uniaxial elongation:

rot gla ben gla gla

dt d d d dt d        

               kT V kT V kT E

gla rot gla rot rot rot rot

    sinh exp

,

gla ben gla

G d d 3   

rot gla gla gla

dt d G dt d       3

Yield stress

 At yield the stress is constant (gla = y):  Resulting yield stress:

rot y gla y

dt d G dt d        3 dt d G

rot gla y

   3 

               kT V kT V kT E

y rot y rot rot rot rot

    sinh exp

,

              



dt d kT E kT V G V kT

rot rot rot gla rot y

  

, 1

exp 3 sinh                  dt d V kT kT V G V kT V E

rot rot rot gla rot rot rot y

  

,

ln 6 ln

Yield stress

 Resulting yield stress:

              



dt d kT E kT V G V kT

rot rot rot gla rot y

  

, 1

exp 3 sinh                  dt d V kT kT V G V kT V E

rot rot rot gla rot rot rot y

  

,

ln 6 ln

Strain rate (s-1)  Yield stress (MPa) 

VISCOSITY

Viscosity of several polymers

100 1000 10000 100000 1 10 100 1000 10000 shear rate (s-1) viscosity (Pas) HDPE PP PB PVC

Viscosity of PVC

1,00E+00 1,00E+01 1,00E+02 1,00E+03 1,00E+04 1,00E+05 1,00E+06 0,00E+00 2,00E+05 4,00E+05 6,00E+05 8,00E+05 1,00E+06 1,20E+06 Shear stress (Pa) Viscosity (Pas) Data 180 C Data 190 C Data 200 C Calc 180 Calc 190 Calc 200

Reduction of viscosity

 With increasing stress the reptation time of the

polymer molecules reduces.

 The viscosity is the product of rubber shear modulus

and reptation time:  = Grubrep

 The viscosity will reduce with stress because the

reptation time reduces with stress.

 High stress = high shear rate:  The viscosity will reduce with shear rate.

Viscosity

 The viscosity is determined in the melt phase.  Equations to use:  Shear deformation:

rub rot rub

G d d   

rep rub rub rub

dt d G dt d      

rep rub rot rub rub

dt d d d dt d        

,0 exp

sinh

rep rub rep rub rot rep rep

V V E kT kT kT                 

Viscosity

 During shear rate d/dt stress is constant (rub = ):  Resulting viscosity:

rep rub dt

d G dt d        0 dt d G

rep rub

   

,0 exp

sinh

rep rub rep rub rot rep rep

V V E kT kT kT                 

rep rub

G                   kT V kT V kT E G

rep rep rot rep rub

    sinh exp

,

Viscosity

 Resulting viscosity:

rep rub

G                   kT V kT V kT E G

rep rep rot rep rub

    sinh exp

,

Shear rate (s-1)  Viscosity (Pas)  Stress (Pa)  Viscosity (Pas) 

ROD CLIMBING EFFECT

Rod climbing effect

Rod climbing effect

Rod climbing effect

 The rotating rod pulls at the entangled molecules.  The molecules move towards the rod.  The molecules near the rod are pushed upwards.

DIE SWELL

Die swell

Die swell

 The reduction of the cross-section creates a stress in

the polymer.

 At the exit the stress is released; the thickness

increases.

Large cross-section Temporary small cross-section The diameter increases The stress reduces due to relaxation