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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° ⇒ Dσ ⇒Ts ⇒ σ
Newton’s law: T = -pI +η 2D
Contents
1. Rheological phenomena
2. Constitutive equations2.1. Generalized Newtonian fluids2.2. Linear visco-elasticity2.3. 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:
)(γη &
Do real melts behave according to Newton’s law?
G(t) or G(t,γ)
Deviation 2: example silly putty
The response of the material depends on the time scale:* Short times: elastic like behaviour * Long times: liquid-like behaviour
VISCO-ELASTIC BEHAVIOUR
t > τt < τ
Linear non-linear
Weissenberg effect Die Swell
Normal stresses
Do real melts behave according to Newton’s law?Deviation 3:
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 equations2.1. Generalized Newtonian fluids2.2. Linear visco-elasticity2.3. Non-linear viscoelasticity
3. Rheometry
4. Parameters affecting rheology
2. Constitutive equations2.1 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
DI D 2)II(fpT 21 ⋅+−=
The viscosity is now replaced by a function of the second invariant of 2D
for a shear flow this becomes:
and different forms for this function have been proposed
( )σ η γ γxy = ⋅12& &
With: II2D = 1/ 2 (tr22D – tr (2D)2)
tr = trace = sum of the diagonal elements
Model 1: Power Law
1nnxy
2/)1n(2
kk
2IIkpT
−
−
γ=ηγ=σ
⋅⋅+−==
&&
DI D
2 parameters
shear rate (s-1)
0.001 0.01 0.1 1 10 100 1000η
0.001
0.01
0.1
1
10
100
n=0.4, k=1n=0.8, k=1n=1.4, k=0.01
shear rate (s−1)
0 200 400 600 800 1000 1200
σ
0
50
100
150
200
250
300
n<1 : shear-thinningn>1: shear-thickening
Model 2: Ellis model
( )
ηη γ
ηη
σ
0
0
11
1
1
1
=+ ⋅
= +⎡
⎣
⎢⎢
⎤
⎦
⎥⎥
−
−
k
II
k
n
n
& ( ) 3 parameters
shear rate (s-1)
0.001 0.01 0.1 1 10 100 1000
η
0.1
1
10
100
1000
log η0
n-1
Model 3: CROSS model
η ηη η γ
−−
=+ ⋅
∞
∞ −0
11
1 k n& ( ) 4 parameters
shear rate (s-1)
1e-3 1e-2 1e-1 1e+0 1e+1 1e+2 1e+3 1e+4 1e+5
η
0.1
1
10
100
1000
log η0
n-1
log η∞
( )η η
η η−−
=+ ⋅
∞
∞−
0
1
1 22
1 2k II
nD
( )/
3D
Special case: plastic Behaviour
σ σ γ σ γ
σ σ γ
< → = = ⋅
> → ≠
y
y
G& ,
&
0
0
No flow, only deformation Yield stress
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 equations2.1. Generalized Newtonian fluids2.2. Linear visco-elasticity2.3. Non-linear viscoelasticity
3. Rheometry
4. Parameters affecting rheology
2. Constitutive equations2.2 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
J t t( ) ( )=
γσ0
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
Newtonian dashpot
101 G γ=σ
202 γ⋅η=σ &Maxwell Kelvin-Voigt
γ γ γ
σ σ σ
γ γ γ
γσ σ
η
ση
σ η γ
σ τσ η γ
= +
= =
= +
= +
+⎛⎝⎜
⎞⎠⎟ =
+ =
1 2
1 2
1 2
1
0
2
0
0
00
0
& & &
&&
& &
& &
G
G
Maxwell model:
τ = relaxation time
ω [rad/s]
0.01 0.1 1 10
G(t)
0
2
4
6
8
10
12
τ
G0
Example: Stress relaxation upon step strain for a Maxwell element
γ γ
σ τσ
σ γ
σγ
τ
= ≥
+ =
= =
= = ⋅ −
0
0 0
0
0
0
0
t
ddt
t G
t G t G t
;
( ) ( ) exp( / )
Generalized Maxwell model to describe molten polymers:
σ σ
σ τ σ η γ
τ
TOT ii
i i i i
ii
iG t G t
= ∑
+ =
= −∑
& &
( ) exp( / )0
Relaxation functions:For a simple Maxwell model: single exponential (single relaxation time τ):
G t G t
G t G t
G t F t d
G t H t d
i i
( ) exp( / )
( ) exp( / )
( ) ( )exp
( ) ( )exp
= −
= −∑
=−⎛
⎝⎜⎞⎠⎟∫
=−⎛
⎝⎜⎞⎠⎟∫
∞
∞
0
0
0
τ
τ
ττ
τ
ττ
ττ
For a generalized Maxwell fluid (discrete number of relaxation times τI:
We can replace the discrete relaxation times by a continuous spectrum:
Or based on a logaritmic time scale : the relaxation spectrum is defined by:
Time effects (linear visco-elastic phenomena): Example 3: Oscillatory experiments
STRAIN
t
i t
STRAIN RATE
t t
i t
HOOKEAN SOLID
G G tG i t
NEWTONIAN FLUID
ti i t i
:
sin( )
exp( )
& cos( ) sin( )
& i exp( )
sin( )exp( )
& sin( )exp( )
γ γ ω
γ γ ω
γ γ ω ω γ ω
γ ωγ ω
σ γ γ ωσ γ ω
σ ηγ ηγ ω ωσ η γ ω ω η γω
=
=
= = + °
=
= ==
= = + °= =
0
0
0 0
0
00
00
90
90
[ ]( ) ( )[ ]
VISCOELASTIC MATERIAL
G i
G i
complex ulus
G
G t
G t tG t G t
G t G t
G iG
el viscσ σ σ
σ γ ωηγ
σ ωη γ
σ γ
σ γ ω δ
σ γ ω δ ω δσ δ γ ω δ γ ω
σ ω ω γ
σ γ
= +
= +
= +
=
= +
= += ⋅ + ⋅
= ⋅ + ⋅
= +
∗
∗
∗
∗ ∗
( )
mod :
sin( )
sin( ).cos( ) cos( ).sin( )cos( ) sin( ) sin( ) cos( )
' sin( ) " cos( )
( ' " )
0
0
0 0
0
Storage and Loss modulustan ' ''
δ =GG
Dynamic moduli of a Maxwell fluid
w [rad/s]
0.01 0.1 1 10 100
Mod
uli [
Pa]
0.001
0.01
0.1
1
10
G'G"
G0
τG G
G G
'( )
"( )
ωω τω τ
ωωτ
ω τ
=+
=+
02 2
2 2
02 2
1
1
G’ and G” for a generalized Maxwell fluid:
G G s s ds G e s ds
G
G G s s ds
G
iis
i
N
ii
ii
N
ii
ii
N
" ( ) cos( ) cos( )
( )
' ( ) sin( )
( )( )
= ⋅ =⎛⎝⎜
⎞⎠⎟
⋅
=+
= ⋅
=+
∞−
=
∞
=
∞
=
∫ ∑∫
∑
∫
∑
ω ω ω τ ω
ωτωτ
ω ω
ωτωτ
0 10
21
0
2
21
1
1
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 equations2.1. Generalized Newtonian fluids2.2. Linear visco-elasticity2.3. Non-linear viscoelasticity
3. Rheometry
4. Parameters affecting rheology
2. Constitutive equations2.3. Non-linear visco-elasticityExample 1: Steady state shear flow
22
21
γσσ
γσσ
γσ
η
&
&
&
zzyy
yyxx
xy
−=Ψ
−=Ψ
=
zzyy
yyxx
xy
N
N
σσ
σσ
σ
−=
−=
2
1
LDPE, Laun et al.PIB in decalin, Keentok et al.
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 equations2.1. Generalized Newtonian fluids2.2. Linear visco-elasticity2.3. Non-linear viscoelasticity
3. Rheometry
4. Parameters affecting rheology
3. Rheometry
Why ? 1. Input for Constitutive Equations2. Quality control3. Simulate industrial flows
A rheometer is an instrument that measures both stress and deformation.
indexerviscometer
What do we want to measure?- steady state data- small strain (LVE) functions- large strain deformations
Introduction: classifications
Kinematics : shear vs elongation
homogeneous vs non-homogeneous vs complex flow fields
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)- steady : η( ), Ψ1( )
shear rheometry : Drag or pressure driven flows.
&γ &γ
Shear flow geometries
Pressure driven flowsDrag flows
Sliding plates
Couette
Cone and Plate
Parallel plates
Capillary
Slit
Annulus
Macosko, 1992
Drag flows : Cone and plate
( )
( )
rr
r
r rvr
r r r
r r
rr
r
:
:sin
sin cot
: cot
1
1 0
1 2 0
2
2 2∂ σ
∂
σ σρ
θθ
∂ σ θ
∂θ
σ
φ∂σ
∂θθ σ
θθ φφ θ
θθθ
θφθφ
−+
= −
− ⋅ =
+ ⋅ =
Probably most popular geometry Mooney (1934)
1. Steady, laminar, isothermal flow2. Negligible gravity and end effects3. Spherical boundary liquid4. vr=vz=0 and vφ(r,θ)5. Angle α < 0.1 radians
Equations of motion:
Drag flows : Cone and plate geometry
Boundary conditions1. vφ(π/2) = 02. vφ(π/2-α)=ωrsin(π/2-α)≈ωr
Shear stress
( )φ
σ
θφ σ σ
θ
σ φ
σπ
θφθφ θφ
θφ
π
θϕ
: cotsin
1 2 03
2
2
2
00
23
r
d
d rC Cte
M r drd
Mr
R
+ ⋅ = → = ≅
= ∫∫
⎫
⎬
⎪⎪⎪
⎭
⎪⎪⎪
=⋅
Independent of fluid characteristics because of small angle!
Drag flows : Cone and plate geometry
Shear rate: is also constant throughout the sample: homogeneous!
&( ) ( ) ( )
γω
αωα
ωα
= =⋅
⋅= ≅
vh r
rr tg tg
r
Normal stress differences: total trust on the plate is measured Fz
( )F R
N
z
FR
z
= −
=
πσ σθθ φφ
π
2
12
2
2
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 flow2. Negligible gravity and end effects3. cylindrical edge4. vr=vz=0 and vφ(r,z)
Equations of motion:
( )
( )
( )
rr
rr r
vr
z
zz
rr
z
zz
:
:
:
1
0
0
2∂ σ
∂σ
ρ
θ∂ σ
∂
∂ σ
∂
θθ θ
θ
− = −
=
=
Drag flows : Parallel plates
Shear rate: is not constant throughout the sample
&γω
= =⋅v
hr
hr
Shear stressσ
π γ= +
⎡
⎣⎢
⎤
⎦⎥
MR
d Md R2
13lnln &
Normal stressesN N
FR
d Fzd
z
R1 2 2 2− = +
⎡
⎣⎢
⎤
⎦⎥
π γlnln &
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
1. Steady, laminar, isothermal flow2. No slip at the wall, vx =0 at R=03. vr=vθ=04. Fluid is incompressible, η≠f(p)
Equation of motion:r
Z
θ
L
RQ
zr
rr
pz
rz: ( )1 0∂ σ∂
∂∂
− =
Only a function of rOnly a function of z
1r
d rdr
dpdz
rz( )σ=
1
2
0 0
2
2
2
rd r
drP P
L
P PL
r Cr
C because r
RP P
LR
r rR
rz o L
rzo L
rz
rz wo L
rz w
( )
@
( )
( )
σ
σ
σ
σ σ
σ σ
=−
=−
⋅ +
= ≠ ∞ =
= =−
⋅
=Note : independent of fluid properties
( )
Q v r rdr
Q v r r
r R and dr R d
Q r R R d
QR
d
zR
zR dv
drdr
R
w w
dvdr
drR
w
dvdr
w
w dvdr
z
z zw
zw
= ∫
= − ∫
= =
= − ∫ = −⎛⎝⎜
⎞⎠⎟∫
= − ∫
2
2 2
2 2
0
02
0
2
0
2
0
3
32
0
π
π π
σσ
σσ
π πσ
σσ
σ
σ
πσ σ
σ
σ
( )
3
34
34
43
2
3
3
32
3 3
QR R
dQd
QR
QR
d Qd
d Qd
w w
ww
dvdr
wdvdr w
wa
w
z
w
z
w
σ
π
σ
π σσ
γπ π σ
γγ
σ
σ
σ
+ ⋅ = −
= + ⋅
= +⎛⎝⎜
⎞⎠⎟
− =&ln
ln
&& ln
ln
Integration by parts + assume :no slip
substitute r and dr
Typical “trick” to deal with inhomogeneous flows:changing of variables
Shear rate calculation from Q
Differentiate with respect to σwusing Leibnitz’s rule
r/R
0 1
v z / <
v z>
0
1
2
n=1n=0.7n=0.3n=0.01
γγ
σwa
w
d Qd
= +⎛⎝⎜
⎞⎠⎟
& lnln4
3Weissenberg-Rabinowitsch “Correction”
“Correction” factor accounts for material behaviour
Physical meaning:
e.g. power law fluid
Steepness of the velocity profile changesShear rate is increased with respect to the Newtonian case:
γγ
wa
n= +⎛
⎝⎜⎞⎠⎟
&
43 1
&γπ
aQR
=4
3
What pressure drop do we really measure?
2R
LPP Lo
w ⋅−
=σCannot be measured!
z=0 z=LCapillairyReservoir
Exit
∆Pen
∆P ∆Ptot
∆Pex
L/D
0 10 20 30 40 50 60
∆p
[MPa
]
0
10
20
30
40
50
10s-1 30s-1 50s-1 100s-1 200s-1 300s-1 500s-1
BAGLEY plots
L/De
Pressuredrop
γ3
γ1
γ2
∆Pe
LDPE @190°C
)eRL(2RP
w +⋅∆
=σ ∆pe can be used to estimatethe elongational viscosity (contraction flow)
Are these conditions met ? 1. Steady, laminar, isothermal flow2. No slip at the wall, vx =0 at R=03. vr=vθ=04. Fluid is incompressible, η≠f(p)
Problem 1: Melt distortion
Melt fracture typically occurs at σw 105 Pa
Are these conditions met ? 1. Steady, laminar, isothermal flow2. No slip at the wall, vx =0 at R=03. vr=vθ=04. Fluid is incompressible, η≠f(p)
Problem 2: Viscous heating
NaRk= ατ γ2
4
.
Are these conditions met ? 1. Steady, laminar, isothermal flow2. No slip at the wall, vx =0 at R=03. vr=vθ=04. Fluid is incompressible, η≠f(p)
Problem 3: Wall slip
cst@Rv4 s
a =σ+γ=γ &&
Are these conditions met ? 1. Steady, laminar, isothermal flow2. No slip at the wall, vx =0 at R=03. vr=vθ=04. Fluid is incompressible, η≠f(p)
Problem 4: Melt compressibility, η=f(p)
L/D
0 10 20 30 40 50 60
Pres
sure
dro
p(M
Pa)
0
10
20
30
40
50
60
70
80
10 s-1
30 s-1
50 s-1
100 s-1
200 s-1
300 s-1
500 s-1
1000 s-1
2000 s-1
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 equations2.1. Generalized Newtonian fluids2.2. Linear visco-elasticity2.3. 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
4.3wM∝η
PS, 217°C
2.Mw,e=39000
wM∝η
Effect of the MW on the zero shear viscosity for linear molten polymers
Log η+ ct
Log Mw + ctBerry and Fox
Effect of molecular weight on moduli⎢
8900Mw =
581000Mw =
Onogi et al., 1970
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
Relevant for injection moulding
Add-on ‘pressure chamber’ on Gottfert capillary rheometer
TdPd
⎟⎠⎞
⎜⎝⎛=
ηβ ln
“Corrected” viscosity for PMMA (210° C)
Shear rate (s-1)
10 100
Visc
osity
(Pas
)
1e+3
1e+4
80 MPa70 MPa55 MPa40 MPa30 MPa0.1 MPa
Mean pressures:
Viscoity curves for PαMSAN and LDPE
Shear rate (s-1)
10 100
Visc
osity
(Pas
)
1e+3
1e+4
80 MPa70 MPa55 MPa40 MPa30 MPa0.1 MPa
Mean pressures:
PαMSAN
LDPE
Determination of β at several shear rates
Pmean (MPa)0 20 40 60 80 100 120
ln η
7
8
9
10
11
6 s-1
13 s-1
33 s-1
70 s-1
108 s-1
146 s-1
223 s-1
260 s-1
290 s-1
Shear rates:
TdPd
⎟⎠⎞
⎜⎝⎛=
ηβ ln
β at several shear rates
Shear rate (s-1)
10 100 1000
β γ(GPa
-1)
2
4
6
8
10
12
14
16
18PMMAPαΜSANLDPEPMMAPαMSANLDPE
Determination of β at several shear stresses (PMMA)
Pmean (MPa)0 20 40 60 80 100 120
ln η
6
7
8
9
10
11
80 kPa100 kPa140 kPa206 kPa270 kPa363 kPa430 kPa500 kPa
Stresses:
TdPd
⎟⎠⎞
⎜⎝⎛=
ηβ ln
β at several shear stresses (PMMA)
Shear stress (kPa)0 100 200 300 400 500 600
βσ(
GP
a-1)
5
10
15
20
25
30
35
40
45
PMMAPαMSANLDPE
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)