Rheology and Rheometry

Paula Moldenaers Department of Chemical Engineering Katholieke Universiteit Leuven W. De Croylaan 46, B-3001 Leuven

Tel. 32(0)16 322675 Fax 32(0)16 322991 http://www.cit.kuleuven.ac.be/cit. ρειν = flow

Rheology = The Science of Deformation and Flow

Why do we need it? - measure fluid properties - understand structure-flow property relations - modelling flow behaviour - simulate flow behaviour

of melts under processing conditions Processing conditions shear rate [s-1]

Sedimentation 10-6 –10-4 Leveling 10-3 Extrusion 100-102 Chewing 101 -102 Mixing 101-103 Spraying, brushing 103 -104 Rubbing 104 -105 Injection molding 102 -105 coating flows 105 -106 How about Newtonian behaviour?

1. Constant viscosity

2. No time effects

3. No normal stresses in shear flow

4. η(ext)/η(shear) = 3

Symbols: ε° ⇒ D σ⇒T s ⇒σ Newton’s law: T = -pI +η 2D Contents

1. Rheological phenomena

2. Constitutive equations 2.1. Generalized Newtonian fluids 2.2. Linear visco-elasticity 2.3. Non-linear viscoelasticity

3. Rheometry

4. Parameters affecting rheology 1. Rheological Phenomena:

Do real melts behave according to Newton’s law?

Deviation 1:

Mayonnaise: resistance (viscosity) decreases with increasing shear rate: shear thinning

Starch solution: resistance increases with increasing shear rate: shear thickening

This is a non-linearity: η(γ&) Do real melts behave according to Newton’s law? Deviation 2: example silly putty

t < τ t > τ

The response of the material depends on the time scale: * Short times: elastic like behaviour * Long times: liquid-like behaviour

VISCO-ELASTIC BEHAVIOUR G(t) or G(t,γ)

Linear non-linear Do real melts behave according to Newton’s law? Deviation 3: Weissenberg effect Die Swell

Normal stresses Do real melts behave according to Newton’s law? deviation 4: Ductless Syphon

η(ext) > > η(shear) In summary: Newtonian behaviour real behaviour

1. Constant viscosity 1. Variable viscosity

2. No time effects 2. Time effects

3. No normal stresses in shear flow 3. Normal stresses

4. η(ext)/η(shear) = 3 4. Large η(ext)

3 dim: T = -pI + η (2D) ? Simple shear: σ = η dγ/dt Contents

1. Rheological phenomena

2. Constitutive equations 2.1. Generalized Newtonian fluids 2.2. Linear visco-elasticity 2.3. Non-linear viscoelasticity

3. Rheometry

4. Parameters affecting rheology 2. Constitutive equations 2.1 Generalized Newtonian fluids

Examples of non-Newtonian behaviour:

Macosko, 1992 2.1 Generalized Newtonian fluids Non-Newtonian behaviour is typical for polymeric solutions and molten polymers

e.g. ABS polymer melt (Cox and Macosko) 2.1 Generalized Newtonian fluids

T = −pI + f1 (II2D ) ⋅2D

The viscosity is now replaced by a function of the second invariant of 2D

for a shear flow this becomes:

2 σηxy =⋅1(γ&&)γ

and different forms for this function have been proposed

2 2 With: II2D = 1/ 2 (tr 2D – tr (2D) ) tr = trace = sum of the diagonal elements Model 1: Power Law

= (n−1)/ 2 2 parameters T = −pI + k ⋅ II2D ⋅2D

100 n n−1 σxy = kγ& η = kγ& n=0.4, k=1 n=0.8, k=1 10 n=1.4, k=0.01

300 η 0.1 250

0.01 200

0.001 150 σ

0.001 0.01 0.1 1 10 100 1000 100 shear rate (s-1)

50 n<1 : shear-thinning 0 0 200 400 600 800 1000 1200 n>1: shear-thickening

shear rate (s−1) Model 2: Ellis model

η 1 = 3 parameters η ()1−n 0 1+⋅k γ&

n−1 1000 ⎡ II ⎤ η ( σ ) =+1 ⎢ ⎥ log η η ⎢ k ⎥ 0 0 ⎣ ⎦ 100

n-1

η 10

0.1 0.001 0.01 0.1 1 10 100 1000

shear rate (s-1) Model 3: CROSS model

ηη− ∞ 1 = 4 parameters ηη− ()1−n 0 ∞ 1+⋅k γ&

1000

log η0 3D 100

n-1

ηη− 1 η 10 ∞ = 2 ()12−n / ηη− ∞ 0 1+⋅(kII2D ) 1

log η∞

0.1 1e-3 1e-2 1e-1 1e+0 1e+1 1e+2 1e+3 1e+4 1e+5

shear rate (s-1) Special case: plastic Behaviour

Yield stress σ < σy → γ& = 0 , σ = G ⋅ γ No flow, only deformation

σσ>→y γ& ≠0 What have we gained in generalized Newtonian?

Newtonian behaviour real behaviour

1. Constant viscosity 1. Variable viscosity

2. No time effects 2. Time effects

3. No normal stresses in shear flow 3. Normal stresses

4. η(ext)/η(shear) = 3 4. Large η(ext)

3 dim: T = -pI + η (2D) T = -pI + η(II2D) (2D)

Simple shear: σ = η dγ/dt Contents

1. Rheological phenomena

2. Constitutive equations 2.1. Generalized Newtonian fluids 2.2. Linear visco-elasticity 2.3. Non-linear viscoelasticity

3. Rheometry

4. Parameters affecting rheology 2. Constitutive equations 2.2 Linear visco-elasticity

Reference: 2 extremes

Hooke’s Law Newton’s Law (solid mechanics) (fluid mechanics)

σ = G.γ σ= η .dγ/dt

G = modulus (Pa) η = viscosity (Pa.s) Material property material property Time effects (linear visco-elastic phenomena): Example 1: creep

Apply constant stress σ

Compliance

γ()t Jt()= σ0 Time effects (linear visco-elastic phenomena): Example 2: stress relaxation upon step strain

Apply constant strain

Modulus G(t) = σ(t)/γ Example for molten LDPE [Laun, Rheol. Acta, 17,1 (1978)] How to descibe this behaviour? Example: differential models

Phenomenological models

Hookean spring

σ1 = G0γ1

Newtonian dashpot σ = η ⋅ γ Maxwell Kelvin-Voigt 2 0 & 2 Maxwell model: γ = γ12+ γ

σσ==12σ

γγ&&=+12γ&

σσ& 1 2 γ& =+ G0 η0

⎛ η0 ⎞ σ + ⎜ ⎟ση& = 0γ& ⎝G0 ⎠

στ+=σ& η0γ& τ = relaxation time Example: Stress relaxation upon step strain for a Maxwell element

10 γ =≥γ 0 t 0

τ ) dσ t 6 στ+=0 G(

dt 4

tG==0;σγ00 2

0 σ()t 0.01 0.1 1 10 ==Gt( ) G0 ⋅exp(−t/ τ) γ ω [rad/s] Generalized Maxwell model to describe molten polymers:

σTOT = ∑ σi i

στii+=σ&&iηiγ

Gt( ) =−∑G0i exp( t/τi ) i Relaxation functions: For a simple Maxwell model: single exponential (single relaxation time τ):

Gt()= G0 exp(−t/ τ)

For a generalized Maxwell fluid (discrete number of relaxation times τI:

Gt( ) =−∑Giiexp( t/ τ )

We can replace the discrete relaxation times by a continuous spectrum:

∞ ⎛ −t ⎞ Gt()= ∫ F(τ)exp⎜ ⎟dτ 0 ⎝ τ ⎠ Or based on a logaritmic time scale : the relaxation spectrum is defined by:

∞ ⎛ −td⎞ τ Gt()= ∫ H(τ)exp⎜ ⎟ 0 ⎝ τ ⎠ τ Time effects (linear visco-elastic phenomena): Example 3: Oscillatory experiments

STRAIN:

γγ= 0 sin(ωt)

γγ= 0 exp(itω)

STRAIN RATE

γγ& ==00ωcos(ωtt) γsin(ω+90°)

γω& = ieγ0 xp(itω)

HOOKEAN SOLID

σγ==GGγ0 sin(ωt) σγ= Gi0 exp( ωt)

NEWTONIAN FLUID

ση==γ& ηγ0ωsin(ωt +90°) ση==iiγ0ωexp( ωt) ηiγω VISCOELASTIC MATERIAL

σσ=+el σvisc σγ=+Giωηγ

σω=+()Giηγ complex modulus:

σγ= G∗

∗ σγ=+Gt0 sin(ωδ)

∗ σγ=+Gt0[]sin(ω).cos(δ) cos(ωt).sin(δ) ∗∗ σδ=⋅(Gtcos( )) γ00sin(ω) +(Gtsin(δ))⋅γcos(ω)

σω=⋅[]Gt'sin()+G"⋅cos()ωtγ0

σγ=+('GiG") G'' Storage and Loss modulustanδ= G' Dynamic moduli of a Maxwell fluid

10 G0

22 G ωτ 1 τ G'( ) 0 ω = 22 1+ ωτ a] P [ 0.1 G ωτ G"( ) 0 Moduli ω = 22 1+ ωτ 0.01 G' G"

0.001

0.01 0.1 1 10 100 w [rad/s] G’ and G” for a generalized Maxwell fluid:

∞ ∞ N ⎛ −τi s ⎞ GG"(=⋅ωω∫ s)cos(s)ds=ω∫ ⎜ ∑Gie⎟ ⋅ cos(ωs)ds 0 0 ⎝ i =1 ⎠ N ωτ G i = ∑ i 2 i =1 1+ ()ωτi

∞ GG'(=⋅ωω∫ s)sin(s)ds 0

N ()ωτ 2 G i = ∑ i 2 i =1 1+ ()ωτi What have we gained in linear visco-elasticity? Newtonian behaviour real behaviour

1. Constant viscosity 1. Variable viscosity

2. No time effects 2. Time effects

3. No normal stresses in shear flow 3. Normal stresses

4. η(ext)/η(shear) = 3 4. Large η(ext)

3 dim: T = -pI + η (2D) G(t), H(τ),… Simple shear: σ = η dγ/dt fully descibes linear VE Contents

1. Rheological phenomena

2. Constitutive equations 2.1. Generalized Newtonian fluids 2.2. Linear visco-elasticity 2.3. Non-linear viscoelasticity

3. Rheometry

4. Parameters affecting rheology 2. Constitutive equations 2.3. Non-linear visco-elasticity Example 1: Steady state shear flow

σ η = xy σ xy γ&

N1 = σ xx −σ yy σ xx −σ yy Ψ1 = 2 N2 = σ yy −σ zz γ&

σ yy −σ zz Ψ2 = 2 LDPE, Laun et al. PIB in decalin, Keentok et al. γ& Example 2: Non-linear stress relaxation upon step strain

LDPE, Laun et al. Example 3: Stress evolution upon inception of shear or elongational flow What have we gained in non-linear visco-elasticity? Newtonian behaviour real behaviour

1. Constant viscosity 1. Variable viscosity

2. No time effects 2. Time effects

3. No normal stresses in shear flow 3. Normal stresses

4. η(ext)/η(shear) = 3 4. Large η(ext)

3 dim: T = -pI + η (2D) no universal model

Simple shear: σ = η dγ/dt Contents

1. Rheological phenomena

2. Constitutive equations 2.1. Generalized Newtonian fluids 2.2. Linear visco-elasticity 2.3. Non-linear viscoelasticity

3. Rheometry

4. Parameters affecting rheology 3. Rheometry

Why ? 1. Input for Constitutive Equations 2. Quality control 3. Simulate industrial flows

A rheometer is an instrument that measures both stress and deformation. Ù indexer Ù viscometer

What do we want to measure? - steady state data - small strain (LVE) functions - large strain deformations Introduction: classifications

ÎKinematics : shear vs elongation

Îhomogeneous vs non-homogeneous vs complex flow fields

Î type of straining:

+ - - small : G’(ω), G”(ω), η (t), η (t), G(t), σy - large : G’(ω,γ), G”(ω,γ), η+(t,γ), η-(t ,γ), G(t ,γ), η(t,ε) γ& γ& - steady : η( ), Ψ1( )

Î shear rheometry : Drag or pressure driven flows. Shear flow geometries

Drag flows Pressure driven flows

Sliding plates

Capillary Couette

Slit Cone and Plate

Annulus Parallel plates

Macosko, 1992 Drag flows : Cone and plate

Probably most popular geometry Mooney (1934)

1. Steady, laminar, isothermal flow 2. Negligible gravity and end effects 3. Spherical boundary liquid

4. vr=vz=0 and vφ(r,θ) 5. Angle α < 0.1 radians

Equations of motion:

∂σr 2 2 1 ( rr ) σσθθ + φφ v r: − =−ρ θ r 2 ∂rr r 1 ∂σ()sinθ cotθ θ: rθ −⋅σ =0 rrsinθ ∂ rθθ

12∂σθφ φ:c+⋅otθσ =0 rr∂θ θφ Drag flows : Cone and plate geometry

Boundary conditions

1. vφ(π/2) = 0 2. vφ(π/2-α)=ωrsin(π/2-α)≈ωr Shear stress

⎫ 12d()σθφ C φ:c+⋅otφσθφ =0 →σθφ =≅Cte⎪ r drθ sin2 θ ⎪ ⎪ 3M ⎬ σθϕ = 2π ⋅ r 3 2π R ⎪ 2 ⎪ Mr= ∫ ∫ σφθφdrd 0 0 ⎭⎪

Independent of fluid characteristics because of small angle! Drag flows : Cone and plate geometry

Shear rate: is also constant throughout the sample: homogeneous!

vr ω⋅r ω ω γ& == =≅ hr() rt⋅ g(α) tg(α) α

Normal stress differences: total trust on the plate is measured Fz

πR2 F =−()σσ z 2 θθ φφ

2Fz N1 = πR2 Drag flows : Cone and plate geometry

+ - constant shear rate, constant shear stress -homogeneous! - most useful properties can be measured - both for high and low viscosity fluids - small sample - easy to fill and clean

- - high visc: shear fracture - limits max. shear rate -(low visc : centrifugal effects/inertia - limits max. shear rate) - (settling can be a problem) - (solvent evaporation) - stiff transducer for normal stress measurements - viscous heating Drag flows : Parallel plates

Again proposed by Mooney (1934)

1. Steady, laminar, isothermal flow 2. Negligible gravity and end effects 3. cylindrical edge

4. vr=vz=0 and vφ(r,z)

Equations of motion:

1 ∂σ(r ) σ v 2 r: rr −=θθ −ρ θ r ∂rr r ∂σ() θ: θz = 0 ∂z ∂σ() z: zz = 0 ∂z Drag flows : Parallel plates

Shear rate: is not constant throughout the sample

vr ω⋅r γ& == h h

Shear stress M ⎡ dMln ⎤ σ =+1 3 ⎢ ⎥ 2πR ⎣ d ln γ& R ⎦

Normal stresses F ⎡ dFln z⎤ NN−=z 2 + 12 2 ⎢ ⎥ πR ⎣ d ln γ& R ⎦ Drag flows : Parallel plate geometry

+ - preferred geometry for viscous melts - small strain functions - sample preparation is much simpler - shear rate and strain can be changed also by changing h - determination of wall slip easy

-N2 when N1 is known - edge fracture can be delayed

- - non-homogeneous flow field (correctable) - inertia/secondary flow - limits max. shear rate - edge fracture still limits use - (settling can be a problem) - (solvent evaporation) - viscous heating Pressure driven flows : capillary rheometry

Q R 1. Steady, laminar, isothermal flow

2. No slip at the wall, vx =0 at R=0 3. vr=vθ=0 4. Fluid is incompressible, η≠f(p)

θ r Equation of motion:

Z 1 ∂()rσrz ∂p L z: −=0 r ∂r ∂z

Only a function of r Only a function of z

1 dr()σ dp rz = r dr dz 1 dr()σ PP− rz = o L r dr L PP− r C σ = oL⋅+2 rz L 2 r

C2 =≠00because σrz ∞@ r =

PP− R σσ()R ==oL⋅ rz w L 2

r σσrz ()r = w R Note : independent of fluid properties R Qv= 2π ∫ z ()rrdr Shear rate calculation from Q 0

R R 2 dv Qv=−22ππr rz dr Integration by parts + assume :no slip z 0 ∫ dr 0 R R substitute r and dr r ==σ and dr dσ σwwσ

2 R σ 2 dv w ⎛ RR⎞ dv Qr=−22ππzzdr =− ⎜ σ⎟ dσ Typical “trick” to deal with inhomogeneous flows: ∫ dr ∫ dr 0 0 ⎝ σw ⎠ σw changing of variables

3 σ 2 3 Qσ w 2 dv 3Qσwwσ dQ 2 dv w =− ()σσz d +⋅=−σ z 3 ∫ dr 3 3 w dr πR 0 πRRπ dσw σw

Differentiate with respect to σ w dv 3Q 3Q dQln using Leibnitz’s rule γ =+−=z ⋅ & w dr 33 σw 4ππR 4 R d lnσw

γ& a ⎛ dQln ⎞ γ& w =+⎜3 ⎟ 4 ⎝ d lnσw ⎠ γ& a ⎛ dQln ⎞ Weissenberg-Rabinowitsch “Correction” γ w =+⎜ 3 ⎟ 4 ⎝ d ln σ w ⎠

“Correction” factor accounts for material behaviour

Physical meaning: 2 n=1 n=0.7 n=0.3 e.g. power law fluid n=0.01 > z

Shear rate is increased with respect to 0 0 1 the Newtonian case: r/R 4Q γ& a = πR3 What pressure drop do we really measure?

Cannot be measured! P − P R σ = o L ⋅ w L 2

∆Pen

∆P ∆Ptot

Exit

∆Pex z=0 z=L Reservoir Capillairy BAGLEY plots

LDPE @190°C

10s-1 Pressure -1 30s drop -1 40 50s γ -1 1 100s 200s-1 γ 300s-1 2 ] 30 a 500s-1 γ

∆Pe 3 p [MP ∆ 20

10 L/D e 0 0 102030405060 L/D ∆P ⋅R σ = ∆pe can be used to estimate w 2(L + eR) the elongational viscosity (contraction flow) Are these conditions met ? 1. Steady, laminar, isothermal flow 2. No slip at the wall, vx =0 at R=0 3. vr=vθ=0 4. Fluid is incompressible, η≠f(p) Problem 1: Melt distortion

5 Melt fracture typically occurs at σw 10 Pa Are these conditions met ? 1. Steady, laminar, isothermal flow 2. No slip at the wall, vx =0 at R=0 3. vr=vθ=0 4. Fluid is incompressible, η≠f(p) Problem 2: Viscous heating

. ατR2 γ N = a 4k Are these conditions met ? 1. Steady, laminar, isothermal flow 2. No slip at the wall, vx =0 at R=0 3. vr=vθ=0 4. Fluid is incompressible, η≠f(p) Problem 3: Wall slip 4v γ = γ + s @ σ = cst & a & R Are these conditions met ? 1. Steady, laminar, isothermal flow 2. No slip at the wall, vx =0 at R=0 3. vr=vθ=0 4. Fluid is incompressible, η≠f(p) Problem 4: Melt compressibility, η=f(p)

10 s-1 60 30 s-1 -1 50 50 s 100 s-1 40 200 s-1 (MPa)

essure drop -1

r 300 s P 30 500 s-1 1000 s-1 20 2000 s-1 10

0 0 102030405060 L/D Capillary rheometry : conclusions

+ - Simple yet accurate! - High shear rates possible - Sealed system, can be pressurized - Process simulator - Entrance flows and exit flows can be used -MFI

- - Non-homogeneous flow field (correctable)

- Only viscosity data, some indications for N1, ηe - Lot of data required (Bagley plots) - Melt fracture limits shear rate - Wall slip can be a problem - Viscous heating - Shear history / degradation Melt Flow index Shear rheometers: the verdict

Couette + low η, high rates - end corrections + can be homogeneous - high visc. : too difficult

+ settling - no N1

Cone and Plate + best for N1 - edges, low shear rate + homogeneous - free surface + transient meas. - alignment

Parallel Plate + easy to load - edges + G’,G” for melts - non-homogenous + vary h! - free surface

Capillary + high rates - corrections: + accuracte - non-homogenous

+ sealed - no N1 Contents

1. Rheological phenomena

2. Constitutive equations 2.1. Generalized Newtonian fluids 2.2. Linear visco-elasticity 2.3. Non-linear viscoelasticity

3. Rheometry

4. Parameters affecting rheology 4. Parameters affecting rheology

• Chemistry • Molecular weight • Molecular weight distribution • Molecular architecture (branching) • Fillers/additives • Temperature • Pressure •… EEffect of molecular weight on the viscosity curve

Example: narrow MW polystyrenes

From Stratton Effect of the MW on the zero shear viscosity for linear molten polymers

3.4 η ∝ Mw

PS, 217°C

η ∝ M w

2.Mw,e=39000

Log η + ct

Berry and Fox Log Mw + ct Effect of molecular weight on moduli ⎢

Onogi et al., 1970

Mw = 581000

Mw = 8900 Effects of molecular weight distribution

Viscosity:

Shear thinning sets in earlier with increasing molecular weight distribution

From Uy and Greassley EEffect of chain architecture (branching) Low shear rates higher shear rates

O = linear and ∆ = branched

Kraus and Gruver EEffect of fillers

Chapman and Lee EEffect of temperature: time-temperature superposition

-C1 (T-Tr)

WLF-relation: log aT = ------

C2 + T - Tr Effect of pressure on viscosity

⎛ d lnη ⎞ β = ⎜ ⎟ ⎝ dP ⎠ ÖRelevant for injection moulding T ÖAdd-on ‘pressure chamber’ on Gottfert capillary rheometer “Corrected” viscosity for PMMA (210° C)

Mean pressures: 80 MPa 70 MPa 55 MPa

) 40 MPa

s 1e+4 a 30 MPa P 0.1 MPa ( scosity i V

1e+3

10 100 Shear rate (s-1) Viscoity curves for PαMSAN and LDPE

Mean pressures: 80 MPa 70 MPa 55 MPa 40 MPa ) s 1e+4 30 MPa a

P 0.1 MPa ( PαMSAN scosity i V

1e+3 LDPE

10 100 Shear rate (s-1) Determination of β at several shear rates

11 ⎛ d lnη ⎞ Shear rates: β = ⎜ ⎟ 6 s-1 -1 ⎝ dP ⎠T 13 s 10 33 s-1 70 s-1 108 s-1 146 s-1

η -1 9 223 s ln 260 s-1 290 s-1

7 0 20406080100120

Pmean (MPa) β at several shear rates

18 PMMA 16 PαΜSAN LDPE 14 PMMA PαMSAN LDPE ) 12 -1

Pa 10 (G γ

β 8

2 10 100 1000

Shear rate (s-1) Determination of β at several shear stresses (PMMA)

⎛ d lnη ⎞ β = ⎜ ⎟ 11 ⎝ dP ⎠T

η Stresses:

ln 80 kPa 8 100 kPa 140 kPa 206 kPa 7 270 kPa 363 kPa 430 kPa 500 kPa 6 0 20406080100120

Pmean (MPa) β at several shear stresses (PMMA)

) 30 -1

25 (GPa σ β

15 PMMA 10 PαMSAN LDPE 5 0 100 200 300 400 500 600 Shear stress (kPa) Useful references concerning rheology

Rheology: Principles, Measurements and Applications C. Macosko Ed, VCH (1994)

Understanding Rheology F. Morrison, Oxford University Press (2001)

Structure and Rheology of Molten Polymers J.M. Dealy and R.G. Larson, Hanser (2006)