Rheological behaviour of molten polymer, models and experiments - - PowerPoint PPT Presentation

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Rheological behaviour of molten polymer, models and experiments

Rudy VALETTE Cemef – Mines Paristech

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Cemef – Mines Paristech

Center for Material Forming Polymer, pastes, composites, metals Industrial partnership Polymer processing

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Examples of polymers

Polystyrene Polypropylene Polyethylene

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Phenomenology

Density : ~water « Melting » temperature : ~150°C (PE, PP, PS) Viscosity : ~106 water « Internal » structure : remains/relaxes over differents timescales → → → → memory, elasticity « High » temperature : viscous + elastic → → → → visco-elastic fluid

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« Internal » structure ?

Macromolecules, ex PE : -[CH2]-n … 10 000 CH2 groups ; 1018 molecules per cm3 ; very flexible Polydispersity : etc …

molecular mass distribution

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Research topics

Link between structure and rheology Viscoelastic flows modelling → Polymer processing flows :

• large deformations
• large deformation rates
• complex geometries

How pertinent are viscoelastic constitutive equations ?

Polymer processing flows: extrusion

flow instabilites interfacial instabilities

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Which material ?

macromolecule : ... close view ... 0.2 nm

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Which material ?

macromolecule : ... further away. 20 nm

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Which material ?

macromolecule : ... further away : free segments 20 nm

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Flexible chain

Degrees of freedom « Free » equivalent segment Conformation = { }= "random flight" of N segments of length b φ θ

dof not dof

=

b r r

i i

=

• ,

i

r

• ↔

↔ ↔ ↔

∑

=

i

r R

Entropic elasticity

Energy kT Relaxation: diffusion in the space of R

∑

=

i

r R

• ( )

0 R

R R kT R F

• ≅

→

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Viscoelasticity

Relaxation of entropic elastic forces Relaxation processes involve:

• which object (chain, subchain) ?
• which time ?
• which distance/which topology ?

Role of the flow ? Overview:

• Scalings
• Linear viscoelasticity/elastic dumbell
• Topological constraints, tube model
• Treatment of non-linear deformation (flow)
• Some examples

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Entropic elasticity

Statistical conformation = = end-to-end vector Mean value : Mean square : Gaussian density distribution : Statistical entropy : Thermodynamic force :

∑

=

i

r R

• =

R

• (

)( )

2 2 2

R Nb r r R

j i

= = =

∑ ∑

• ( )

2 3

2 2

2 3 2 3 2         −

        =

R R

e R R π ψ

• ( )

( )

R k S

tot

• ψ

Ω = ln

( )

R R kT R d dS T R F

• 2

3 = − =

Doi-Edwards 1986

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Elastic dumbbell

Force: Spring constant: Einstein relation: Relaxation time:

( )

R R kT R F

• 2

3 =

2 2

3 3 Nb kT R kT =

b

N kT kT D ζ ζ = =

2 2 2 2

θ ζ θ N kT b N D R

b =

= =

Doi-Edwards 1986

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Macroscopic scale

Force: chains per unit volume chains per unit surface Stress: Elastic modulus: Relaxation time:

( )

R R kT R F

• 2

3 =

2

N ∝ θ ν R ν

2 2 2 2

3 3 R R kT R R kT

• ν

ν = kT G ν =

Doi-Edwards 1986

Relaxation function : experiment 1 on viscoelasticity

Response to a sudden macroscopic deformation

macro

γ

micro

γ

Sample at rest

(the arrows are proportional to the chains oriented and stretched in their direction)

Deformation a t=0 : stress = G0 x strain Stress decreases with time because microscopic strain γmicro decreases Whatever γmacro : G(t) is constant (same ratio stress/strain)

Relaxation function : experiment 1 on viscoelasticity

Stress relaxation is faster when far from equilibrium

micro micro

dt d γ θ γ 1 − = τ θ τ 1 − = dt d

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Linear (separable) response of a simple viscoelastic fluid

Step strain :

t=0 rest, t>0 constant shear log time log shear modulus

macro

γ

θ

γ τ

/

) ( /

t

e G t G

−

= = G θ G

s

2

10−

fraction of unrelaxed segments

Viscosity function : experiment 2 on viscoelasticity

Response to a continuous macroscopic deformation

macro

γ

• micro

γ

Stress = ,

τ = G0 γmacroθ γmicro = γmacroθ

Viscosity function : experiment 2 on viscoelasticity

Response to a continuous macroscopic deformation

drag

dt dt d

micro macro micro

γ θ γ γ 1 − =

macro micro

γ θ γ

• =

relax Stationary : Stress :

macro micro

G G γ θ γ τ

• =

=

Viscosity :

θ η G =

Response in steady shear

Steady shear:

t=0 rest, t>0 onstant shear rate, log time ~ deformation log shear stress

t V γ

• =

elastic: slope G Viscous: viscosity Gθ

) 1 ( /

/θ

θ γ τ

t

e G

−

− =

• G

θ G

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Steady shear

Shear rate: Drag (affine deformation)/relaxation balance → stress: Viscosity: Weissenberg effect:

γ

• θ

γ τ

• G

=

b

N Nb G ζ ν θ η

2

= =

Long chains are entangled

Entanglements : topological constrains between chains

Courtesy K. Kremer, from Everaers et al., Science, 2004

a

Long chains are entangled

Entanglements: topological constrains between chains

Courtesy K. Kremer, from Everaers et al., Science, 2004

Long chains are entangled

Entanglements: topological constrains between chains

Courtesy K. Kremer, from Everaers et al., Science, 2004

a

a kT R R R kT F ≅ ≅

Long chains are entangled

Entanglements: topological constrains between chains Relaxation: diffusion out of a « tube », reptation dynamics convenient mean field model (to make self-similar !)

From T.C.B. McLeish group, Leeds University

Long chains are entangled

Entanglements: topological constrains between chains Relaxation: diffusion out of a « tube », reptation dynamics Tube survival probability

Doi-Edwards 1986

Long chains are entangled

Entanglements: topological constrains between chains Tube length L = Za, scales as N Disengagement time:

a

3 2 2

θ ζ θ N kT N N D L

b d

∝ ∝ = a=10−9m Z = 5− 50 N / Z = 25 θZ=1 ≈ 0.1s

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Macroscopic scale

Elastic modulus: Relaxation time: Tension on each segment:

3

N

d ∝

θ a kT 3 kT G

e N

ν =

a

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Linear response of an entangled polymer melt

Step strain :

t=0 rest, t>0 constant strain log time log shear modulus

macro

γ

Pa

9

10 Pa

6

10 s

2

10− s 10

fast retraction along tube contour reptation (disengagement)

2

N ∝

3

N ∝

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Linear response of an entangled polymer melt

Step strain :

t=0 rest, t>0 constant strain log time log shear modulus

macro

γ

Pa

9

10 Pa

6

10 s

2

10− s 10

retraction reptation

2

N ∝

3

N ∝

molecular mass distribution

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Linear response of an entangled polymer melt

Step strain :

t=0 rest, t>0 constant strain log time log shear modulus

macro

γ

Pa

9

10 Pa

6

10 s

2

10− s 10

retraction reptation

2

N ∝

3

N ∝

3

G

3 3θ

G

2

G

2 2θ

G

1

G

1 1θ

G ...

i i

Gθ

Spectrum fitted using conventional rheometers

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Steady shear

Shear rate: Drag (affine deformation)/relaxation balance → stress: Viscosity: Shear thinning: rotation of segments

d

θ γ / 1 <

• d

N

G θ γ τ

• =

d N

G θ η =

d

θ γ / 1 >

• θ

N

G

log viscosity log shear rate

• 1

d

θ / 1

Doi-Edwards 1986

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Rheometric tools

Cone-plate rheometer: mechanical spectroscopy (linear regime) Capillary rheometer: steady shear viscosity

Piston Pressure transducer Die

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Sodium source λ = 589nm Picture polarizer 45° λ/4 135° λ/4 analyzer

Robert et al. 2003

Flow-induced birefringence: stress measurements

Cw kλ σ = ∆

Stress-optical law:

Schweizer et al., Rheo Acta 2005

w

Used to test pertinence

• f constitutive equations

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Constitutive equation (3D)

Stress: (autocorrelation) B conformation tensor: At rest: Affine deformation: Because when no relaxation

3 3

2

B G R R R G

N T N

= =

• τ

( ) ( )

3 2 2

R d R R R R R R R

T R T

• ∫

= ψ

ds R n R R G ds n f d

T N 2

3

• =

=τ

( )

v B B v R R v R R R R v dt dB

T T T

• ∇

+ ∇ = ∇ + ∇ =

2 2

d dt ψ

• R

( )d

• R3

( ) = 0

I B 3 1 =

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Constitutive equation (3D)

Stress: B conformation tensor: At rest: Affine deformation: Relaxation:

3 3

2

B G R R R G

N T N

= =

• τ

( ) ( )

3 2 2

R d R R R R R R R

T R T

• ∫

= ψ

ds R n R R G ds n f d

T N 2

3

• =

=τ

( )

v B B v R R v R R R R v dt dB

T T T

• ∇

+ ∇ = ∇ + ∇ =

2 2

      − − ∇ + ∇ = I B v B B v dt dB

T

3 1 1 θ

• I

B 3 1 =

Upper-convected Maxwell model

Why choosing differential models for B ?

Maxwell model: closed form from Smoluchowski equation But general kinetic theory models (Likthman et al., Öttinger & Kröger et al, etc ..) involve too many refinements for polymer processing and require closure approximations Integral models (KBKZ 1962-63, Doi-Edwards 1986, Wagner et al. 1976-2012) require lots of computational ressources to be solved in practice

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Constitutive equation (3D)

Evolution equation for , tube model:

• take Maxwell model
• trace(B) is mean square chain stretch:
• rescale B so that trB=1

      − − ∇ + ∇ = I B v B B v dt dB

T

3 1 1 θ

• 2

R R R B

T

• =

( ) ( )

1 1 2 − − ∇ = trB B v tr dt dtrB θ

• (

)

      − − ∇ − ∇ + ∇ = I B B B v tr v B B v dt dB

T

3 1 1 2 θ

• Larson 1988

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2

. R R R trB

• =

Test of reptation differential model

Evolution equation for , tube model:

• Marrucci & Ianniruberto (1996): convective constrain release

relaxation is accelerated by retraction

• linear viscoelasticity spectrum + β fitted from steady shear

2

R R R B

T

• =

( )

      − − ∇ − ∇ + ∇ = I B B B v tr v B B v dt dB

T

3 1 1 2 θ

• 41

1 θ = 1 θ +2β tr ∇uB

( )

Valette, Mackley, Hernandez 2006

θ

N

G

log viscosity log shear rate slope

1 θ = 1 θd + ∇u

Piston 1 Piston 2

1 2 3 4 5 6 7 8 9 10 11 12 13 0,25 0,5 0,75 1 1,25 1,5 1,75 2 2,25 2,5 2,75 3 3,25 3,5 3,75

Temps (s) Perte de charge (bars)

Flow:

• start
• flow
• relax

Measure:

• birefringence
• pressure drop
• strain rate ~1/θd

Valette, Mackley, Hernandez, JNNFM 2006

Equations : momentum equation incompressibility transport equations for conformation tensors Splitting : Perturbed Stokes problem ← extra-stress Stabilization : « solvent » part (discretization) SUPG (transport) Time : implicit Euler Approximation : P, Conformation (linear continuous) Velocity (quadratic continuous)

Numerics: finite elements

1 2 3 4 5 6 7 8 9 10 11 12 13 0,25 0,5 0,75 1 1,25 1,5 1,75 2 2,25 2,5 2,75 3 3,25 3,5 3,75

Temps (s) Perte de charge (bars)

a : 0.06s b : 0.28s c : 0.84s d : 1.20s e : 2.85s f : 2.89s g : 2.96s h : 3.40s

|

a

|

b

|

c

|

d

|

e

|

f

|

g

|

h

polymer symmetry

½ gap

Valette, Mackley, Hernandez, JNNFM 2006

Valette, Mackley, Hernandez, JNNFM 2006

Valette, Mackley, Hernandez, JNNFM 2006

Solving it with suitable numerical methods, then compute birefringence

0.06s 0.28s 0.84s 1.20s 2.85s 2.89s 2.96s 3.40s

Valette, Mackley, Hernandez, JNNFM 2006

Solving it with suitable numerical methods, then compute pressure drop

1 2 3 4 5 6 7 8 9 10 11 12 13 0,25 0,5 0,75 1 1,25 1,5 1,75 2 2,25 2,5 2,75 3 3,25 3,5 3,75

Temps (s) Perte de charge (bars)

Carreau, compressible Rolie Poly, incompressible Rolie Poly, compressible 1 2 3 4 5 6 7 8 9 10 11 12 13 2,8 2,85 2,9 2,95 3 3,05

Temps (s)

1 2 3 4 5 6 7 8 9 10 11 12 13 0,05 0,1 0,15 0,2 0,25

Temps (s) Perte de charge (bars)

Carreau, compressible Rolie Poly, incompressible Rolie Poly, compressible

• - Viscous compressible

__ VE incompressible __ VE compressible

• - Viscous compressible

__ VE incompressible __ VE compressible

Predictions of tube-based differential models are ≈ ≈ ≈ ≈ OK

Flows were not as strong as polymer processing flows: Birefringence:

• measure of stress
• not linked to a measure of strain (rate)

d

θ γ / 1 >>

Flows stronger than 1/θ θ θ θd

Take chain retraction dynamics into account:

• stress is still
• partition of B:
• not really satisfactory predictions

3GB = τ S B

2

λ =

( )

      − − ∇ − ∇ + ∇ = I B B B v v B B v dt dB

d T

3 1 1 : 2 θ

• McLeish & Larson 1998

( ) ( )

1 1 − − ∇ = λ θ λ

r

B v tr dt d

• Koventry, Valette, Mackley 2004

Planar elongational flow Coventry, Mackley, Valette 2004

Piston 1

« Purely » elongational :

• 3 s-1
• birefringence
• polystyrene
• compare different models

Piston 2

Planar elongational flow Coventry, Mackley, Valette 2004

Same physics, different expressions (closure approximation) :

• Rolie-Poly
• Pompom
• DXPompom

Too many adjustable parameters !!!!

Flows stronger than 1/θ θ θ θd

Take chain retraction dynamics and finite extensibility into account:

• stress is still
• finite extensibility: chains are no more Gaussian:
• rest dimension
• unfolded chain length
• max extension square
• modulus
• single equation partition for B:

3GB = τ

Gf trB N N G G = − − → 1

( )I

ftrB I trB B f v B B v dt dB

r T

1 3 1 3 − −       − − ∇ + ∇ = θ θ

• Marrucci & Ianniruberto 2003

b N R =

Nb R =

max

( )

N R R =

2 max /

( ) ( )

1 1 1 1 1 1 1 − + −         − + = ftrB ftrB

d r d

β β θ θ θ θ

• Measure Vx (axial)

Spatial resolution 35 μ μ μ μm, velocity resolution 50μ μ μ μm/s

Test constitutive equations using LDV

• D. Hertel, PhD, LSP Erlangen

Computation Velocity +gradients Stress ?

Rheometry : Linear regime: spectrum Capillary: steady shear

Optical rheometer

Boukellal, Durin, Valette, Agassant 2011

Optical rheometer: on symmetry axis get velocity

Boukellal, Durin, Valette, Agassant 2011

Optical rheometer: on symmetry axis deduce strain rate

Boukellal, Durin, Valette, Agassant 2011

Optical rheometer: compute stress and compare

Boukellal, Durin, Valette, Agassant 2011

Conclusion

Models are OK for moderately fast flows and simple chain topologies No adjustable parameters (modulo uncertainties) Predictive constitutive models for complex topologies and faster flows remain a challenge

Polymer processing flows: instabilities

Very strong flows

r

θ γ / 1 >>

• Combeaud, Demay, Vergnes 2004

Polymer processing flows: instabilities

flow instabilites interfacial instabilities

Polymer processing flows: coextrusion instabilities

interfacial instabilities

Valette, Laure, Demay, Agassant 2003

Polymer processing flows: coextrusion instabilities

interfacial instabilities inside flow cell / after cooling

Valette, Laure, Demay, Agassant 2004a

Polymer processing flows: coextrusion instabilities

interfacial instabilities: solve direct model -> convective instability forced system: Level-set + SUPG method

Valette, Laure, Demay, Fortin 2002

Polymer processing flows: coextrusion instabilities

interfacial instabilities: solve direct model -> convective instability wavepacket: Discontinuous Galerkin method

Valette, Laure, Demay, Fortin 2001

Polymer processing flows: coextrusion instabilities

interfacial instabilities: linear stability of 2 layer viscoelastic Poiseuille flow

Valette, Laure, Demay, Agassant 2004b

Polymer processing flows: instabilities

flow instabilites interfacial instabilities