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Institute for Chemical Technology and Polymer Chemistry [email protected] http://www.itcp.kit.edu/wilhelm/
KIT – Universität des Landes Baden-Württemberg und nationales Forschungszentrum in der Helmholtz-Gemeinschaft www.kit.edu
Introduction to Rheology
Prof. Dr. Manfred Wilhelm
private copy 2019
Contents (overview)
Motivation, Literature, Journals
First principles
Simple models: Maxwell, Voigt, Burger, Carreau, Ostwald - de Waele
Glossary
Rheological hardware
Examples: Dispersions (response and phenomena), Polymer melts, ...
Fourier-Transformation
FT-Rheology
I
Contents
Literature: ................................................................................................................................... 1
Books ...................................................................................................................................... 1
Journals ................................................................................................................................... 2
Internet .................................................................................................................................... 2
Definition of the term “Rheology” ............................................................................................. 3
Typical examples of daily live: ( motivation) ........................................................................ 3
1) Brush with paint on a wall ................................................................................................. 3
2) Piston in an engine ............................................................................................................. 4
Why can we assume that Hooke’s law could be correct? ........................................................... 5
Hooke for polymers (rubber elasticity) ...................................................................................... 8
Why can we assume that Newton’s law could be correct? ....................................................... 10
Gedankenexperiment ............................................................................................................ 11
Linear models: Hooke, Newton, Maxwell, Kelvin-Voigt … ................................................... 14
Detailed analysis of Maxwell model ........................................................................................ 16
Without any mathematics: step experiments (step in stress or step in strain) .......................... 27
Memory (Gedächtnis) ............................................................................................................... 28
Multimode models .................................................................................................................... 29
Glossary .................................................................................................................................... 31
a) Lamellar flow ................................................................................................................... 31
b) Reynolds number ............................................................................................................. 31
c) Cox-Merz-rule .................................................................................................................. 32
d) Lissajous figures ............................................................................................................... 33
e) Shear thinning ................................................................................................................... 34
Ostwald-de Waele (example for 2 parameter model) ....................................................... 34
Carreau (example for 3 parameter model) ........................................................................ 35
4 parameter models:.......................................................................................................... 35
Thixotropy shear thinning + long memory ( Hysteresis) ......................................... 36
Shear thickening ( rheopex dilatancy) ....................................................................... 36
Anti-thixotropy shear thickening + memory ( Hysteresis) ....................................... 36
Rheopexy .......................................................................................................................... 37
Dilatancy ........................................................................................................................... 37
II
Bingham plastic ................................................................................................................ 37
Dimensionless groups ........................................................................................................... 38
Deborah number ............................................................................................................... 39
Péclet number ................................................................................................................... 40
Taylor vortex .................................................................................................................... 40
What do we expect for (p,T)? ................................................................................................... 42
Gases ..................................................................................................................................... 42
Viscosity of liquids, temperature dependence ...................................................................... 44
Stress-strain tensor and normal forces ...................................................................................... 46
Definition of the extra stress tensor (right handed system!) ................................................. 48
Properties of the extra stress tensor ...................................................................................... 48
What do normal stress differences mean? ............................................................................ 49
What do we expect for N1,2 0γ,γ ? ..................................................................................... 50
Phenomena where we can directly “see” normal forces ....................................................... 51
a) Rod-climbing ................................................................................................................ 51
b) Secondary flow for rotating disc .................................................................................. 52
c) Extrudate swell ............................................................................................................. 52
Possible measurements (for oscillatory rheometers) and hardware ......................................... 53
1) Detection of onset of non-linearity at fixed frequency .................................................... 53
2) Measurement of G’, G” at T = const., : variable, 0: parameter .................................... 53
3) Temperature dependent measurement .............................................................................. 54
4) Shear rate dependent viscosity ......................................................................................... 55
Hardware: ............................................................................................................................. 55
Couette geometry .............................................................................................................. 55
Hardware .................................................................................................................................. 58
Stress and strain rheometer, typical types of construction: .................................................. 58
Typical hardware specifications (ARES) ............................................................................. 59
Typical pathway of a signal from the torque transducer to G’, G” ...................................... 60
Vane rheometer .................................................................................................................... 61
Melt-flow index .................................................................................................................... 62
Capillary rheometer ( high shear rates) ............................................................................ 62
Elongational rheology, viscosity .......................................................................................... 63
III
Rheology on two specific examples: polymers and dispersions .............................................. 66
Polymers ............................................................................................................................... 66
Reptation theory ................................................................................................................... 66
Typical shape for G’(), G”() for monodisperse linear polymer melts ............................ 69
Time-Temperature-Superposition (TTS) and the Williams-Landel-Ferry (WLF) equation 73
Dispersions ........................................................................................................................... 77
Fourier-Transform-spectroscopy .............................................................................................. 88
Problem of discretisation (ADC, analogue digital converter) .............................................. 89
Some important mathematical relations ............................................................................... 90
Appendix A .............................................................................................................................. 93
Appendix B ............................................................................................................................. 115
Appendix C ............................................................................................................................. 124
1
Literature:
Books
Einführung in Rheologie und Rheometrie (also available in English)
Gebhard Schramm, Gebr. Haake GmbH, Karlsruhe
(easy; book to start with)
Das Rheologie Handbuch (also available in English)
Thomas Mezger, Vincentz Verlag, 2000
(easy, covers lots of practical problems, nice hardware section)
Rheology for Chemists, an Introduction
J. W. Goodwin and R. H. Hughes, Royal Society of Chemistry 2000
(easy)
A Handbook of elementary Rheology
Howard A. Barnes, University of Wales, Institute of Non-Newtonian Fluid Mechanics,
Aberystwyth 2000, (good overview, very elaborated literature at the end)
The structure and rheology of complex fluids
Ronald G. Larson (Head of society of rheology), Oxford University Press 1999
(more advanced)
Rheology Principles, Measurements and Applications
Ch. W. Macosko, Wiley-VCH 1994
(more advanced)
Engineering Rheology
R. J. Tanner, Oxford University Press 2000
(for mechanical engineers)
2
Rheological measurements
A. A. Collyer and D.W. Clegg, Chapman & Hall 1995
(Hardware)
Rheology: A Historical Perspective
R. I. Tanner and K. Walters, Elsevier 1998
(lots about people and phenomena)
Journals
Journal of Applied Rheology
http://www.ar.ethz.ch/
(incl. Jobs!, Hardware guide, reports about upcoming and previous conferences)
Rheologica Acta (Springer)
http://www.springerlink.com/
Journal of Rheology (The Society of Rheology)
http://scitation.aip.org/joro/
Journal of Non-Newtonian fluid mechanics
http://www.elsevier.com/
Internet
www.rheologie.de
www.rheology-esr.org
www.rheology.org
3
Definition of the term “Rheology”
Rheology is the science of deformation and flow of matter.
Side conditions:
conservation of energy, conservation of mass, symmetry constraints, incompressibility;
Analysis of:
Deformations: strain (shear), stretch (elongation);
stress (torque); normal forces;
Typical examples of daily live: ( motivation)
1) Brush with paint on a wall
v = 1 m/s = 1000 mm/s
d = 0.2 mm
What is the relevant quantity?
Assumption: layered structure
i
i
d
v= constant for all i!
v
d
n layers
vn
vi
di
4
s
1 5000
mm 0.2
mm/s 1000
d
v
d
v
d
v
n
n
i
i
shear rate d
vγ [1/s]
Why is this a rate and not a frequency?
Frequency is only used with respect to periodic phenomena, otherwise: rate!
both: [1/s] !
2) Piston in an engine
frequency:
min
16000
2π
ω1
(=rpm, rotations per minute)
s
1100
2π
ω1
stroke (german: Hub):
s = 10 cm = 0.1 m
s
m 20
s
m 20.11002s
2π
ωv 1
s
m 30
s
m 20)2/(
maxv why 2/ ?
m1020μm 20d 6
s
1101.5
m 1020s
m 30
d
vγ 6
6max
max
think about: - shower lotion
- lipstick
- coating of paper
- extrusion of fibres (clothing)
d
1
s v
5
unit of γ : [1/s] = inverse time
comparison: pλγ
1
pλγ is a unitless quantity
Why can we assume that Hooke’s law could be correct?
Do we “buy” this law?
Hooke: xkF
Possible reasons:
a) (x)
!
(x,...) FFF (assumption)
Taylor-expansion (Taylor around x = 0 MacLaurin series)
2
0x2
2
0x0)(x(x) x
x
F
2!
1x
x
FFF
b)
Interaction Potential for vibrational (IR) spectroscopy (beside Hooke):
Morse-Potential U(x) (Potential, not force!)
FdxUFx
U
Polymer molecule with relaxation time p
F
0 x
0,
because
at equil.
k vanishing for
small x
linear nonlinear part
CC
C CC
CCC C
d
v
6
Potential has units of energy!
Morse: 20(x) )))xβ(xexp(A(1U
set: x0 = 0
Taylor: ...3!
xβ
2!
xββx1e
3322βx
2222(x) xAβ...)xA(β...))xβ(1A(1U
xkx2Aβx
UF 2
a) + b) no proof, but we “buy” Hooke’s law
exercise: prove Hooke’s law for the finite extendable nonlinear elastic interaction (FENE),
frequently used in computer simulation, for x R0
X0=0 x
U/A
Note: A: Dissoziation energy
x
U/A
1
U/A
x
1
2
0(x) R
x1lnAU
k
exp(-(x-x0))
1
X0=0
X0=0
1-exp(-(x-x0))
(1-exp(-(x-x0)))2
7
Remark:
If we remember typical force-constant from IR ( spectroscopy books)
k = 500 N/m ( m
kω )
and we remember typical area needed for a chain, e.g. polyethylene:
orthorhombic
a = 7.5 Å
b = 5 Å
c = 2.5 Å
2 chains per unit cell
2202o
2o2o
m1020A202
A37.5
2
A57.5
chain
A
Renormalization to area + relative change in length
L
xEσ
A
F + E: unit: pressure [ 1Pa = 1 N/m2 ]
5 Å
7.5 Å
stress
E-module
F
F
F
different A
different x and L
L
8
upper limit:
L
xEσ
A
xk
A
F
GPa 250Pa10250mm
mN
20
1010500EL
A
k 92
1020
Tungsten (W): 150 GPa
But: bending modes are weaker only several GPa
Hooke for polymers (rubber elasticity)
Start of chain in coordinate origin, where is end?
GaussW(r)
2
2
(r) 2σ
μ)(xexp
σ2π
1W , here = 0
Boltzmann: 22 xkC))ln(exp(-x k C ln(W) k S
with 0 H S,T - H G
) xk - (C T - G 2
G: units of energy:
)Fdx(W F,x
ΔG
x2k T F (temp. + elongation!)
only needed: 0 H ; GaussW(r)
See analogy for Gauss in crystallography! Debye-Waller factor!
C in unit cell 10-10 m (one bond)
CCC
9
10
Why can we assume that Newton’s law could be correct?
Newton: γηd
vησ
A
F
Why not: xA
F ?
x a a,F ?
a2
πv F 3
1 ??
why?
We need proportionality between viscous force and velocity: vFviscous
Remember:
Law from Stokes: vrη6πF (F: e.g. gravity)
Note: unit of :
sPa ηm
smPa
v
dp
dv
AF
η
, old: Poise: 1Pa s = 10 P; 1 cP = 1 mPas (Poiseuille);
typical values: blood 100 – 4 mPas
(thicker than water!; shear thinning)
Glycerin 0°C 10,000 mPas
20°C 1,400 mPas
60°C 60 mPas
Oil, SAE 10 30°C 200 mPas
H2O 1 mPas ( memorize!)
Air 0.02 mPas
Do we “buy” this?
rough surface
2r F
v Sphere in viscous media
T
x
Note: In Rheology Newton’s law is associated with γησ ,
not with his other law: amxmF .
11
Gedankenexperiment ( Prof. Sillescu, article Lord Rayleigh 1891! see Appendix A, p. 93-114)
Tube; big mass M; lots of particles with small mass m strike on mass M; M is moved with
speed vM; What force is needed?
For M: ΔtvΔl MM
For m: ΔtvΔl , same velocity for all small particles!
After time t:
Average density of particles m: Δl
N ρ , number density, not mass density!
The mass M is hit by the following number of particles during t:
NΔlρΔlΔlρ2
1ΔlΔlρ
2
1NN MM
M m
vM vm = v - vm = - v
l
50:50 probability that particles fly in correct direction
direction of particles
particles lM
l
lM
l
l - lM
but N- > N+ !!
m
12
if we define a clash-rate:
vρZZZ
v
vvZ
2
1vvρ
2
1Z
v
vvZ
2
1vvρ
2
1Z
vρZv
Z
Δtv
N
ΔtΔl
ΔtNρ
Δt
N Z
MM
MM
each particle transfers elastic impact onto mass M with relative momentum v2mp
M
M
vv2mp
vv2mp
In one time unit t, this balances the outer force F needed to push mass M with velocity vM.
MM
2MM
22MM
2
MMMM
MM
MM
v4mZ4vvv
mZ
v2vvvv2vvvv
mZ
vvvvvvvvv
mZ
vv2mv
vvZ
2
1vv2m
v
vvZ
2
1F
pZpZF
M
vF
Friction is proportional to the velocity of the mass M.
mv
mv
13
Sir Isaac N
ewton
1642 – 1727 “N
ewton w
as the first to formulate a hypothesis regarding the
magnitude of the force required to overcom
e viscous resis-tance and to treat a case of m
otion in a viscous fluid.” C
fP
rincipia
Lib
iiS
ectIX
Em
ilHatschek
14
Linear models: Hooke, Newton, Maxwell, Kelvin-Voigt … incl. oscillatory excitation and response
Hooke – spring
γ G σ γG σ
dtd
Newton – dash-pot
Math. def. of linear models:
A linear model is a mathematical description of the relation between stress and strain
(respective: strain rate) where only linear terms of 1γ or 1γ are used. Further more G and
are constant.
Experimental def.:
Linear response can be assumed if the response (stress, strain, strain rate) is large enough to
be detected but still in a regime where G and are not affected by the measurement.
,
Pa no unit
γγσ,
dt
dγ γ ,γ η σ
15
The non-linear regime should be avoided for linear response measurements:
Dash-pot (DP) and spring (S) can be arranged in series or in parallel:
G’’
rate sweep: γη
γ
asymptotic deviation!
linear regime
G
0
strain sweep: 00 γG,γG
fixed frequency linear regime
G’
DPS σσ
viscosity with a
bit of elasticity
(long term)
DPS
DPS
γγ γ
γγ γ
σ
η ,G ,γ γ, 0
σ
η G, ,γ γ,
DPS γγ
elasticity with
a bit of viscosity
(long term)
γηGγσ
σσ σ DPS
Maxwell model
(for liquids with some
elastic response)
Kelvin-Voigt model
(for solids with some
viscous response)
G’, G” later
16
Detailed analysis of Maxwell model
)1(η
σ
G
σγ
η
σγ γ η σ
G
σγ γ G σ
DP
S
1. step-experiment
at time t > 0, 0 γ (not in the dash-pot, but overall system!)
using (1):
t 0
0
0(t)0t
00(t)
(t)00(t)
0(0)
(0)0
(0)(t)
0
(0)
(t)
0(0)(t)
t
0
0σ(t)
σ(0)
0
0
0
GGlim
t)η
Gexp(GG
γ
σ t)
η
Gexp(γGσ
γGσ
sPa
sPa
G
ητ);
τ
texp(σt)
η
Gexp(σσ
tη
G
σ
σln
tη
G)ln(σ)ln(σ
tdη
Gdσ
σ
1
tdη
G
σ
dσ
η
σGσ
η
σ
G
σ 0
(see: first order kinetic, or Lambert-Beer)
(for short time force is fully in spring)
relaxation time
t)η
Gexp( 0
0G
0
Memory!
t
17
Oscillatory response:
Hooke
distinguish: amplitude elongation!
System has memory ( stored energy storage modulus G’)
Newton
distinguish:
[rad/s] , f [1/s = Hz] 2πT
2π ω
System has no memory ( energy is lost loss modulus G’’)
t
0
t)sin(ω γ γ 0
t)cos(ωω γ γ
t)sin(ω γ γ
0
0
t
γ 0
phase shift
,
, , γ
18
Maxwell:
(t)0
0(t)
γωi t)ωexp(iω)(i γ γ
t)ωexp(i γ γ
after initial time we reach dynamic steady state:
(t)0
0(t)
σωi δ)t(ωexp(iω)(iσ σ
δ)t(ωexp(iσ σ
eq. (1): η
σ
G
σγ
22
22
22
2222
22
22
22
)()(*
*
*
)(*
)(*
(t)
(t)
(t)
(t)
(t)
(t)(t)
τω1
τωGG
τω1
τωGG
τω1
τωi
τω1
τωG
τω1
τωiτωG
τ)ωi-(1τ)ωi(1
τ)ωi-(1τωiGGiGG
τ)ωi-(1:part imag. and real into sep.τωi1
τωiGG
τωi
τωi1
G
G
G
ητ,
τωi
11
G
G
G
1
σ
γ:def.,
ηωi
G1
σ
γG
σωi
G
η
σ
G
σωi γωi
Plot:
Why G’ = storage ?
Why G” = loss ?
G/2
G’, G” (linear scale)
G’, storage
= 1
G”, loss module
log() [rad/s]
G
Note: (a+b)(a-b)=a2-b2
19
222
0ω22
22
0ω0ωωτωlim
τω1
τωlimGlim
2ωG for small
1
0ω220ω0ωωτωlim
τω1
τωlimGlim
1ωG for small
trick to memorize: ba ωG , a + b = 3
Note for “NMR-People”:
FID: shapeLorentz)T
texp( M
2
in
1
1
2
= 1 log
log G
G”
G’
1
Lorentz:
1τω
τω-Im
1τω
τRe
22
2
)(
22)(
in Rheology: 0 !!
Factor missing
G”
G’
)Gˆ(Re
0
) Gˆ( Im
20
GGG 0ω0ω* (module of spring within Maxwell-element)
width of relaxation spectrum for G”:
Set to = 1:
1ω
ωGG
2 ; G = 1
Maximum at:
full width at half maximum, 1, 2 ?
fwhm
1τω
τωGG
22
G”
log = 1
G2
1
11
1GG
1ω :meaningful physically 1ω
ω1
2ω1ω
1ω
2ω1
1ω1ω
2ω
1ω
1
02ω1ω
ω
1ω
10
dω
Gd
1)(ω
2
22
2
2
222
2
2
222
)(
log
G2
1
G4
1 fwhm
04
1ωω
4
1
ω4
1ω
4
11ω
ω
4
1
2
2
2
21
remember:
The full width at half maximum (fwhm) of a single exponential relaxation is 1.14 decades in
frequency space.
)()(*
*
)(* GiG
γ
σG
t)ωexp(iγiωγ t),ωexp(iγ γ;γGσ 00***
γiω
GiGσ )()(*
** Gηiω units: [1/s Pa s = Pa]
[to memorize: “iong”, i omega n equals G]
14.1ω
ωlog :ratio
0.268ω
3.732ω
32
4312
21
41
41411
ω
2a
4acbbx
0cbxax
2
110
2
1
1/2
2
1/2
2
*η
**
22*
*
Gηω
ηηη
ηiηη
22
Phase lag : G
Gtanδ
experimental advantage: G”, G’ extensive quantities
tan intensive quantity
if, e.g. filling factor is “bad”, G’ is wrong, G” is also wrong, but G”/G’ is still accurate
tan is generally very reproducible
%10GG
GG
typical error margin for rheological measurements
23
24
25
26
27
Without any mathematics: step experiments (step in stress or step in strain)
t0
t0
F, ,
Kelvin-Voigt model
t0
ta
η
σγ 0)(t
G
σγ )(ta
F, ,
Maxwell - model
t0
spring elastic part
viscous part spring, elastic part
t
2nd possibility: strain step:
t
τ
texp~
memory?
28
More complex models (but still linear models!):
Memory (Gedächtnis)
The memory of the system might be defined for a step strain experiment as follows:
dt
dG
γdt
dσ
dt
dGM (t)
(t) , minus sign, so that M(t) is positive;
dt γMdσ
td γM-dσσ t
tt
-t) t-(t
σ
0t
Memory depends only on elapsed time: t – t’ = s, dt’ = -ds
dsγM-σ s)-(t
0
(s)(t)
exchange of limits and: dt’ = -ds
t0
t0
Burger - model
1
2
G1
G2
2
1
G1
2 1, G2
29
for infinite small motion:
dsγGσ
tts ,td γGσ
dtγGdtdt
dγGdσ
Gdγdσ
s)-(t
0
(s)
t
-
t)t-(t
if we have a modulus function with an exponential memory:
τ
t-expGG 0(t)
t
-t0t td γ
τ
)t-(t-expGσ
Improvement of this model: several relaxation times
Multimode models
t
) t(
N
1k kk
N
1k kk(t)
td γ τ
texp Gσ
model mode-N , τ
texp GG
picture for multimode Maxwell-model:
Also possible: multimode Kelvin-Voigt (several Kelvin-Voigt models in series);
Under oscillatory shear, a multimode Maxwell-model will respond as follows
(see next page)
remember: fwhm for single Maxwell: 1.14 decades spacing ?! in
not uncommon for polymers: 5 – 7 decades relaxation time distribution
= 1 = 2 = 3 = 4
= 1 + 2 + 3 + 4
F, ,
4 3 2 1
monomodal:
30
H. M
. Laun
31
To reduce the need of maths for a while, a glossary on important rheological terms is inserted:
Glossary
a) Lamellar flow
For this model: (h)γγ , in contrast: tube 22
0r rrv 40rI !!
Shear deformation is equally distributed throughout the sample. For sliding plate geometry:
the points of similar elongation amplitude form lamellae.
For high shear rates, generally instabilities can occur (Reynolds number) and the lamellar
flow profile is disrupted.
Other possibility: plug-flow (tooth paste!)
b) Reynolds number
The Reynolds number describes the ratio between the kinetic energy of a system and the
energy lost by viscous flow.
viscous
kin
E
E Re , for Re > 2000 we find transition between lamellar ( γF ) and turbulent
( 2γF ) flow!
For a capillary (diameter: circle) we find: η
v2rρ Re
r: radius, : density, v: avg. velocity, : viscosity
vh
1
dt
dx
h
1γ
h
xγ
x
h
Hagen-Poiseuille
flux
v = const.
0 r0
r0
32
example:
Aorta (main blood vessel close to the heart):
r = 1 cm = 0.01 m
= 4 mPas = 0.004 N/m2 s ; 1 N = 1 kgm / s2
= 1000 kg/m3
v = 0.3 m/s
1,500smsm kg
mkgmsm
104
1031010 2 Re
3
22
3-
13-2
,close to transition: lamellar turbulent!
c) Cox-Merz-rule
The Cox-Merz-rule is an empirical rule that connects the shear rate dependent viscosity with
the absolute value of the frequency dependent complex viscosity, as calculated by:
)(*
)(* Gηiω
via: ν2πω ; νfrequency ω:note ; ηη
tally)(experimen 1a , ηη
ω*
)γ(
ω)(a*
)γ(
This rule holds only for rheologically simple materials!
ω
GGηγη
2ω
2ω
ω*
e.g. Maxwell-model:
22)(
22
22
)(
τω1
τωGG
τω1
τωGG
G’
= 1
G”
log()
G’ leading term G” leading term
G0
G0/2
33
1
2
ω
*
ω
0
222
0ω
*
0ω
ωω
const.
ω
Glimηlim
ηconst.ω
τω
ω
Glimηlim
for single Maxwell-model
d) Lissajous figures
oscillatory shear different representations
~ cos (t), ~ sin (t)
vector description of circle
general for ellipse:
a
b
2
δtan
δ)tcos(ω
t)cos(ω
y
x
log() = 1
-1
log( )(*η )
t 1
1
max
(t)
σ
σ
max
(t)
γ
γ
x
y
ba
phase lag
0
34
Linear response: ( ellipse) Contains symmetry elements for Lissajous figure:
2 mirror plains + point symmetry
In case of non-linearity: only point symmetry ( I(31), I(51), …)
Note: deviations < 2-3% of sinusoidal response can generally not be seen in Lissajous
figures!! much less sensitive compared to FT-Rheology (see later)
e) Shear thinning
(deutsch auch: Strukturviskos), pseudoplastic
monotonically decaying viscosity flow curves; viscosity as a function of shear rate in steady
state, so no implicit memory involved.
To describe shear-rate dependent viscosity, empirical equations with 1, 2, 3, 4 parameters are
used, e.g.:
1 parameter: Newtons law! = 0
Ostwald-de Waele (example for 2 parameter model) aγbη
a: scaling parameter, shear thinning exponent; a[0, 1] : 0 Newton
1 max. shear thinning exponent
γ log
η log
2
η 0
0η = Newton
c
0
γβ1
ηη
β
1
-a
aγbη
2121 γγfor γηγη
35
if a = 1: bγγbγηF -1
force independent of γ , force is constant!
bη1γat
Carreau (example for 3 parameter model)
c0
γβ1
ηη
also other def. for Carreau:
c0
γβ1
ηη
(not equal!)
c: scaling parameter c[0, 1]
: pivot point (knee),
2
η
11
η
β
1β1
ηη
β
1γ if 0
c
0c
0
4 parameter models:
- further parameter needed to: e.g. model the width of the knee as the next parameter
- introduction of “second Newtonian plateau” for high shear rates
e.g.: d c
0
γβ1
η)γη(
where cd < 1
WHY??
Polymers: increase in Mw/Mn!
γlog
η log
γlog
η log 0η , 1st Newtonian plateau
η , 2nd Newtonian plateau
e.g.: c
0
γβ1
ηηη)γη(
36
Thixotropy shear thinning + long memory ( Hysteresis)
A decrease of apparent viscosity under constant shear rate, followed by a gradual recovery
when the stress or shear rate is removed. The effect is time dependent.
greek: thixis: shake
trepo: changing
in principle we can have two types of hysteresis:
or:
Shear thickening ( rheopex dilatancy)
2121 γγfor γηγη
Anti-thixotropy shear thickening + memory ( Hysteresis)
start of shear time after shear
γ = const.
steady state
γ increase
Plateau cτ
ningshear thin p thixotrolim0τc
η
t
η
t
Newton
γ
σ
η
t
η
t
γ = const. time after shear
37
or:
Rheopexy
Structure is generated without shear so that viscosity or module increases as a function of time
only (not as a result of applied shear).
Dilatancy
Why is wet sand “dry” for a few seconds when we walk barefoot on the beach? dilatancy!
Application of shear changes (reduces) level of liquid in packed spheres (granula). This can
cause shear thickening.
Experiment:
Dilatation: Ausweitung
Bingham plastic
(the “evil” in the ketchup bottle!)
(deutsch: strukturviskose Flüssigkeit mit Fließgrenze = plastisches Fluid)
γ
σ
ordered spheres ”sand”
shear
or shake
disordered spheres need
more volume in beaker to pack
Vsand = const. !
Vwater = const. ! sand level
water level
γ
σ γ = const.
G
yieldσ
yieldγ
yieldyield
yield
γGσ
σγησ
yieldσ Ketchup 20 Pa
Newton
critical stress
below: elastic, solid-like
above: liquid-like
“dry” particles on top
38
Extension of Bingham-model: Herschel-Bulkley
include: powerlaw for viscosity n
HB γkσσ
for measurements vane rheometer (see later)
Dimensionless groups
Reynolds (already covered), Deborah, Péclet, Taylor
For several phenomena in nature only unitless quantities seem to play the important role:
e.g.: 1) Arrhenius group
RT
E-expAk a
r
if Ea << RT kr A
if Ea >> RT kr << A, slow down
2) kinetics
tk-expAA r(0)(t)
kr t >> 1 basically complete reaction
kr t << 1 just started
γ
σ
Newton
Bingham shear thinning (pseudoplastic)
yieldσ
η
unitless
yieldyield
yield
σσfor σγησ
σσfor 0γ
39
Deborah number
[book of judges 5.5, song of Deborah: “Even the mountains flowed before the Lord ...”]
remember:
response liquid n timeobservatio long 1, De
reponse astic viscoel1, De
like solidn timeobservatioshort 1,De
n timeobservatio
timerelaxation internal
t
τ
tG
ηDe
τG
η t,
η
G
De
1
tη
GexpγGσ (t)
if we take: γt
1
under oscillatory shear: t)sin(ωγωγ
t)cos(ωγγ
0
0
τγωτγDe 0 e.g. longest relaxation time in polymer
Note: Generally Deborah-nr. is not precisely defined ( ωγor ,γωγ 0 ), and there is
confusion with Weissenberg-nr.: τγWi
Weissenberg normally used in the context of: γ =const., steady shear
with respect to normal forces
Pipkin diagram:
dimensionless
step in strain
sol i d
yield ( Bingham plastic)non linear response
viscoelastic
N e w t o n
l i qui d
0γ
τωor τ,γω De, 0
ω γ0 = const.
1
(osc. shear) (Cox-Merz)
40
Péclet number
Stokes: vξF
vrη6πF
Stokes-Einstein for diffusion coefficient D: rη6π
Tk
ξ
Tk D
Time needed to displace object by distance r: 322
rTk
η6π
Tk
rrη6π
D
rt
Tk
σr6πγr
Tk
η6πγtPe
33
frequently used in context with colloids;
Taylor vortex
locityangular ve :Ω
density :ρ
3400γη
RRRΩρTa
2i
3
io22
2r F
γησ
less sensitive to Taylor vortices
more sensitive to Taylor vortices
secondary flow caused by inertia generates vortices
in addition to shear Ri
Ro
moving bob
moving cup
41
Units for Ta (only check, no proof):
1
N
1
s
mkg
Nsm
mkg
sm
Nsm
mkgTa
2
2
2242
42
2
2
26
42
unitless quantity
-=- END OF GLOSSARY -=-
42
What do we expect for (p,T)?
Gases Mean free path length: l
Cross-section 22 r4π2rπA
2
2
r4πV
N1
Aρ
1L
r4πL
1
AL
1
ΔV
1ρ
normal conditions (Gas, 1 bar, 300 K):
nm 300μm 3
1m
103
1
m10103
m 1L
m10m104πA
m103m1022.4
106
V
nρ
l22.4ˆMol 1 nRTpV
621925
3
2192210-
32533
23
mean free path length
clash rate:
s
110
Δt
1 :rateclash typical
s 10
1
s
m 330
m 103
v
LΔt
RT2
3mv
2
1:precise more
s
m 330 v,
Δt
Lv
Δt
1
9
9
6-
2
L >> r
mean distance
2r
v
in physics: (confusing for rheology)
o
A1r
volume
one particle
43
velocity distribution (Maxwell-Boltzmann)
(see Physical Chemistry books for details)
onsdistributifor truegenerally ,vv :note kT2
3vm
2
1
m
3kTv:picture simple
m
kT2.54
mπ
8kTv
dvv2kT
mvexp
kT2π
m4πdvP
___22
___2
2
1
2
1
2
1
222
3
(v)
model: L: mean free path length
e.g. xi: bottom layer, xi+1: top layer, gap of width L
momentum transport if particle leaves layer xi+1 to go to layer xi :
dx
dvLm s
Number of particles n leaving layer xi+1 in unit time in direction xi :
NtvAρtvA
N
LA
N
V
Nρ particle ofdensity n
0n
only half fly in correct direction:
vV
N
2
1vρ
2
1ntvAρ
2
1
0
nn
1st moment 2nd moment
x
z
y
xi
xi+1
L
unit area A
vs: shear velocity
v: particle velocity
vs
dx
dvLv s
s
volume average velocity!
unit time, unit area
44
in unit time, unit area the following momentum p
is transferred:
dx
dvLmv
V
N
2
1
Δt
Δp s
0
this must be equal to the force
dx
dvηF s
mT3kTmrπ8
1η
m
3kT v:using
rπ8
vm1
r4πNV
VvmN
2
1η
r4πN
VL :using
Lvρ2
1Lvm
V
N
2
1η
2
2
1
220
0
20
0
Viscosity of gases is: - independent of density!
- therefore independent of pressure! TTp, ηη !
- a function of mass and temp. of particles! Tη !
Viscosity of liquids, temperature dependence
- no shear: Boltzmann distribution for particles
making transition from left to right ;
typically 1vacancy per shell (= 12 neighbours)
5-10% free volume
no shear (density diff. amorphous crystall!)
energy activation :E* , RT
*Eexp
h
kTN
E*
45
- shear: force on single molecule, typical distance r
2rσAσF
apply this force for half distance r
2
Vσ
2
rσ
2
rrσE m
32
Vm: average occupied volume per molecule,
VM: volume per NL molecules
So effective jumps N are jumps to the right N minus jumps to the left N
(1)
RT
2
Vσ*E
expRT
2
Vσ*E
exph
TkNNN
MM
B
The shear rate is the effective number of jumps in one second divided by layer thickness in
unit time:
(2) Nr
Nrγ
d
v
(2) in (1):
RT
*Eexp
V
hη
k
V
R
NV
R
V σ
η
1γ
σ2 RT
V2
RT
*Eexp
h
Tkγ
x/2x1x1lim/2eelimsinh(x)lim
/2ee sinh(x) :remember
2 RT
Vσexp
2 RT
Vσexp
RT
*Eexp
h
Tkγ
m
0)γ(
B
mLmM
MB
0x
xx
0x0x
xx
MMB
r
r/2
2
rσ*E
3
2
rσ*E
3
46
- Arrhenius for T-dependence
- increase of free volume reduces viscosity (hopping probability )
- Ea ; Ea
- Pressure dependence via average volume per molecule weak p-dependence
Stress-strain tensor and normal forces (Why might we need a tensorial property?!)
So far we have used F and x,v
as collinear (parallel) vectors
scalar description
If we would like to extend this, what happens if F and x
are not parallel?
We need a transformation between F x
.
This transformation should:
1. transform a vector into a vector
2. transform a plane into a plane
3. have a fixed origin in both systems
1-3 define an affine coordination transformation. This transformation is linear if the new
system )y,y,(yy 321
is generated out of the old system )x,x,(xx 321
by a linear set of
equations:
T
(T)
liquid
RT
Eexp a
T
gases
x
F
47
3332321313
3232221212
3132121111
xaxaxay
xaxaxay
xaxaxay
if we introduce matrix (3 by 3 matrix, second rank tensor)
333231
232221
131211
a a a
a a a
a a a
A , we can write: xAy
Example for a simple rotation of a vector x
in 2 dimensions:
)(αsin
)(α cosy ,
αsin
α cosx
use of addition theorems:
sin αsin cosα cos
...isin αsin cosα cosRe
sin i cosαsin iα cosReeeReeRe)(α cos iiααi
analogue for the sine (using the imaginary part):
cosαsin sinα cos...)(αsin
αsin
α cos
cossin
sin cos
)(αsin
)(α cosy
x1
x2
x
α
y2
y1
y
Rotation around
origin by angle
in math. positive sense
(counterclockwise)
x
A
48
Definition of the extra stress tensor (right handed system!)
This results in the following extra stress tensor:
333231
232221
131211
τ ττ
τ ττ
τ ττ
τ
The stress-tensor σ is the sum of the extra stress tensor plus the hydrostatic pressure. The
hydrostatic pressure acts equally along the 332211 τand τ,τ components.
100
010
001
runit tenso:E pressure,:p , τE-pσ
Properties of the extra stress tensor
- The tensor is symmetric (like many in quantum mechanics, see e.g. Fermi’s golden rule):
jiij ττ
reduction from 9 variables to 6 variables
- forces that pull have positive prefactor
- forces that push have negative prefactor
The tensor has properties that are invariant under transformation of coordinates:
1st invariant: Trace of the tensor A
332211
n
1iii1 aaaaAtr I
indices ij :
i: the force acts on a plane that is
normal to the basis vector i
j: the force acts in the direction of
the basis vector j 1
3
2
21
11 23
22
13
12
49
(see also quantum mechanic books
1iii α α )
2nd invariant: 2
2
2 Atr Atr 2
1I
3rd invariant:
333231
232221
131211
3
a a a
a a a
a a a
Adet I
Due to the first invariant 0τττ 332211 the trace of the extra stress tensor has only two
variables. N1 = 11 - 22
N2 = 22 - 33
What do normal stress differences mean? - assume shear stress along 21
22 : force that pushes plates apart
33 : force that pushes material into plate-plate geometry
N1 = 11 - 22 : first normal stress difference, generally positive
N2 = 22 - 33 : second normal stress difference, generally negative 21 NN
to memorize: Na = aa - a+1, a+1
normal stress differences
1
3
2
21
11
22
33
50
What do we expect for N1,2 0γ,γ ?
- N1,2 should only be a function of γ due to kinetic nature of the phenomenon, e.g.:
constant:cb,a, ...γcγbaN 21,2
- if we do not apply a shear rate N1,2 should be 0 a = 0
- if we apply a shear rate, the force N1,2 should be independent of the direction
γ-NγN 1,21,2
even function with respect to nγ
We expect equation like: 21,2 γcN as first approximation
23322
22
2
22211
21
1
γ
ττ
γ
Nψ
γ
ττ
γ
Nψ
1 : first normal stress coefficient
2 : second normal stress coefficient
1 : generally positive, 21 ψψ (typical factor: 10);
N1 can be as high or even higher than 12 !
2 : generally small and negative
1 + 2 can be measured separately using both:
cone-plate plate-plate
Information: N1 Information: superposition N1 and N2
N1 + N2 γN1 γ
&
51
Typical examples for extra stress tensor:
a) ideal viscous fluid b) viscoelastic liquid
000
00τ
0τ0
τ 21
12
33
2221
1211
τ00
0ττ
0ττ
τ 5 unknown
12 = 21 12 = 21 , 11 + 22 + 33 = 0
3 degrees of freedom , 1 , 2
The first normal stress coefficient can be estimated using:
ωγfor ω
G2γψlim
2
)(1
0ω
Phenomena where we can directly “see” normal forces
a) Rod-climbing
Parabola, f(r) 2r
centrifugal forces
(e.g. water)
Newtonian fluid
leading term, f(r) 4r !
ω,...)ρ,,ψ,ψr,f(R, 21
Non-Newtonian fluid
climbing effect is called
Weissenberg effect
52
b) Secondary flow for rotating disc
c) Extrudate swell
Newtonian fluid Non-Newtonian fluid
centrifugal
forces
see: cover page
book: Tanner
Newton or not?
up!
Newtonian fluid Non-Newtonian fluid
parabola
die swell
film blowing(e.g. plastic bags)
53
New chapter:
Possible measurements (for oscillatory rheometers) and hardware
1) Detection of onset of non-linearity at fixed frequency
a: Problem: torque too low, hard to get sensitivity
b: onset of non-linearity, but: depends on accuracy of detection!
c: in filled materials sometimes an overshoot in G” is detected: Payne-effect
typical values: polymer melts: 0 < 0.05 – 0.3
solutions: 0 < 0.1 – 1
cross-linked rubber: 0 0.01
we know linear regime for a fixed frequency ( 10max ω2πγγ )
we can assume: linear if 2 < 1
perhaps non-linear if 2 > 1
2) Measurement of G’, G” at T = const., : variable, 0: parameter
2
= 1 log
log G
G”
G’
1
G”
b
c
linear regime
log 0
log G
G’
1 = const., fixed
a
Frequency dependent module
distribution of relaxation times
e.g. via Multimode-Maxwell models
see section about polymers (later)
(reptation, rubber plateau, TTS)
difference!
(name!: not pain)
54
Typical range: 10-2 < < 60,
dynamic range: 4 decades of hardware due to mechanical device!
ωγγ
t)cos(ωωγγ
t)sin(ωγγ
0max
0
0(t)
adjust 0 every 1 - 2 decades for best performance
3) Temperature dependent measurement
0: fixed but parameter, T: variable, : fixed
Instrument for this: DMTA, (Dynamic mechanical thermo analyser)
cheap due to limited -range
(sometimes also E-module measurements)
adjust 0 !,
torque will change for an increase
in by 104 by up to 104!
log
log G
1
T
G’ or G” [Pa]
log
G’ or G”
if 1/T
Arrhenius
109
105 - 106
55
4) Shear rate dependent viscosity
Fit with: 1 - 4 parameter model (see before)
Hardware:
Couette geometry
preferentially: static bob, moving cup (because of Taylor vortices!)
inside moving: Searle-type, outside moving: Couette-type
experimental linear regime
e.g. 10% reduction
relative to 0
log γ
log asymptotic behaviour in principle no linear regime!
0
(t) (t) (t)
cup
bob
Mooney-Ewart
N(t): torque
or or
double couette
for low viscosity
materials,
e.g. water
Haake-type
Air bubble low friction at lower end
56
To prevent evaporation of water:
If plate - geometries are used:
Dodecane, C12H26
H2O sample
H2O
or
saturated H2O vapor
water trap
H2O
r0
20rArea
Torque at infinitesimal area:
1rΔNrFΔNΔ
total torque 30rN
e.g. change from 50 mm plate-plate geometry to
8 mm plate-plate geometry : torque reduction by
2448
503
, 2.5 decades reduction
plate-plate
non homogeneous γ,γ0 !
r0
cone-plate
homogeneous γ,γ0 !
h
30rN
typical values for : 0.02 - 0.1 rad
1.14 –5.73° very small!
or
truncated cone
easier to manufacture
Advantage of plate-plate (cone-plate) vs. Couette:
- less sample volume (e.g. 0.1ml vs. 10ml)
Disadvantage of plate-plate:
- leakage (low viscosity material)
- less area less sensitivity for low viscosity materials
- heterogeneity of shear rate
57
5) Creep experiment
G(0,t) measured
Wagner-Ansatz (Manfred Wagner, Prof. in Berlin):
γhtGtγ,G
1. 1γhlim0γ
, linear response
2. decreasing strictly monotonic as a function of time, 0γhlimγ
typical examples:
2
212211
γa1
1γh
1ff , γn-expfγn-expfγh
meltPEfor 0.18n e.g. ,γn-expγh
0
t
t
G(t)
log G(0,t)
log t
1 < 2 < 3 < 4
(modulus measured, not stress!)
Overlay via h()
4 2
3
1
damping function
Doi-theory
58
Hardware
Stress and strain rheometer, typical types of construction:
motor, Volt + Amp. torque, (t)
optical encoder, (t) or position sensor (capacity)
air bearing
frame
geometry, e.g. plate-platesample
disc stress-rheometer,
stress is given, strain measured
strain is measured via optical encoder
controlled stress is imposed
motor, (t)
plate-plate
disc
disc
air bearing strain-rheometer (A),
strain is given, stress measured
optical encoder or position sensor
position sensor, (t)
rigid spring, deflection < 1°, otherwise problem with Bingham fluid
strain-rheometer (B),
ARES-type
nominal actual value comparison, feedback loop
sample
ball (cheap) or air bearing (expensive); for normal forces air bearing needed
air bearing
motor acts as rigid spring
motor
magnetic suspension, seal + normal forces
position sensor
position sensorfeedback (t)
feedback: “stand still”
stress detection force rebalance transducer (FRT) e.g. our ARES: 2K FRT N1 (2K = 2000 gcm, N1 = normal forces can be measured)
strain application
59
Typical hardware specifications (ARES)
Magnets, transducer: Al Ni Co - alloy 0.01% / °C
Magnets, motor: Nd (Neodym) 0.1% / °C
(compare: Cu: 0.39% / °C)
Optical encoder: 30,000 lines + interpolation
0.0810-6 radian resolution = 810-8 rad
Alternative: capacitive encoding: LC
1ω
d
1C LC
dynamic range of transducers:
newest: 1K FRT N1 (Rheometrics, also Haake)
Nmax / Nmin = 106, e.g. 10-1Nm to 10-7Nm!
Typical prices (2002):
Stress rheometer: 15k – 40k € (Haake, Bohlin, TA, Rheometrics, …)
ARES (strain): 60k – 100k € (3 types of motors, diff. types of temp.
control, 7 diff. transducers, …)
+ cooling (N2): 10k €
+ dielectric option: 30k €
+ birefringence, dichroism option: 30k €
up to 180k €
Geometry: 1.5 – 3k €
d
1 km
0.08 mm !810-8 rad
60
Typical pathway of a signal from the torque transducer to G’, G”
Torque transducer:
ADC: discrete in time ( dwell-time) and in intensity
(k-bit ADC, 2k slots)
t
V
0 t
0
“smoothing” analogue filter
“low-pass filter”
e.g. integration or
“oversampling”
t 0
Autobias a
a
“zero-frequency-artefact”
t 0
autorange
“NMR: RGA”
(receiver gain autorange)
Dynamic range of
ADC adjusted
t 0
4 bit ADC,
24 = 16 slots
dwell-time
Imax
Imin
dynamic range
V
V V
average
61
Typical acoustic ADC’s: 16 bit = 216 = 65,536 dynamic range: 1: 65,536
(remark: limit for S/N in FT-Rheology!)
dwell-time: 10s, sampling rate: 100 kHz
105 6.55 104 = 6.55 109 decisions per second
corrections: e.g. inertia of geometry, stress transducer, motor,
phase lag due to lumped circuit
Vane rheometer (German: Schaufel, Flügelrad)
useful for determination of yield stress
( Bingham fluid) in concentrated
suspensions, greases or food (yoghurt!);
especially if the history of loading should
be avoided.
Approximation:
yv
v3vm σ
3
2
R
LR2πT
Tm: torque maximum
y: yield stress
position signal
torque signal
corrections + geometric factors
cross correlation or Fourier-Transformation G’, G”
Rv
Lv
62
Melt-flow index
cheap + robust version of a capillary rheometer (see later)
- uncontrolled, non-homogeneous flow
- relative measurement (“index”)
typical parameters:
T = 190°C
M = 2.16 kg
pressure 3 105 Pa
MFI: flow of polymer in [g] per 10 min
rough measure of average MW
Capillary rheometer ( high shear rates)
Model system for e.g. polymer extrusion process
(see also: melt-flow index)
important shear rates:
0.8 mm
9.57 mm
M
0.209 mm
condition “E”
-4 -2 0 2 4 γ [s-1]
oscillatory / vibrational
rotational
capillary
elongational
processing
63
Set-up:
Dominantly viscous properties of the material are determined,
pressure loss at entrance can be corrected “Bagley-Correction”
Information: m(t), ds, pn, …, p1 for different T, M, v, d, L
Elongational rheology, viscosity
important for: fibre spinning, blow moulding, flat film extrusion, film blowing
ratestretch :ε , xεv(t)dt
dx1
L d:L 1:30 for steady state, developed streamlines
M, v
d
ds die swell, extrudate swell normal forces
m(t)
pressure sensors
constant force or constant velocity
1 3
2
L0
L(t)
mass
64
if specimen is stretched with constant rate ε
tεL
Lln
const.ε ; dt εdxx
1
εdtx
dx
0
L
L
t
0
1
0
[Hencky worked for many years in Mainz-Gustavsburg!
See Appendix B, p. 115 - 123]
tensile viscosity: ε
ση E
E
without proof: 3η
ηlim
0
E
0ε
for simple liquids
experimental apparatus:
Hencky-strain, sample length teL !
Trouton’s ratio
sample
thickness + elongation
monitored via camera
Prof. Meissner
Zürich
ex BASF
M
Prof. Münstedt
Erlangen
ex BASF
sample oil: + compensates gravity
+ temp. control
- can act as plasticizer
in the sample
A B
65
Mu
ensted
t, Lau
n, R
heol. A
cta, 18, 492, 1979
66
Rheology on two specific examples: polymers and dispersions
Polymers
End-to-end distance R
, bond length b, N monomers
n
1iirR
Gauß:
NbRNbRR 222
contour length: L = Nb (“odometer”)
e.g. high Mw-PE, N = 100,000, b = 1.5 Å
contour length 15 m (in principle visible!), R = 47 nm
simplified model: 6
RR g
Reptation theory
basic idea:
one-dimensional stochastic process
of chain along contour (reptate: reptile)
simplified
tube with diameter d and other chains are static,
typical distance of other chains: s d
typical d 30 - 80 Å
one-dimensional Fick-equation, for chain distribution probability
2
2
1d x
PD
t
P
R
d
s
67
Solution for P(x, t): Gauß-statistics
tD4
xexp
tD4π
1tx,P
1d
2
1d
Mean square displacement ( second moment)
t2Ddxtx,Pxx 1d22
If we assume stochastic friction coefficient ’, where this friction coefficient ’ is
proportional to N, therefore also M
’ = N : friction per monomer unit
Using the Einstein-relation for the 1-d. diffusion:
11d M
Nξ
kT
ξ
kTD
The time needed to diffuse along L will allow a fully different conformation, so that all
memory of the other chains (static) is erased
3
31
2
1d
2
1d2
Mλ
MM
M
2D
Lλ
2DλL
The self-diffusion coefficient Ds is given by the time t to move the center of mass by a typical
coil diameter R (3-dimensional problem!).
2s
23
2
s
2
MD
MM
M
λ6
RD
3 :here lity,dimensiona :n ,2nDt r
Gauß: 2
2
2σ
x
2nDtrσ 22
n: dimensionality
R
t
68
assuming a Maxwell-model:
0
3
MG
Mλ
G withληG
ηλ
3Polymer Mη , DeGennes 1971, exp.: 3.4
Polymer Mη
for non-entangled: 1Polymer Mη friction of polymer-contour
Rule of thumb for flexible monomers with 2 carbons per polymer backbone (so not true for
PPP, poly-paraphenylene persistence length)
ne 100 – 200 monomers
contourlength between entanglements: 150 3Å = 45 nm, nR e 3Å 3 - 4 nm
examples, Me: PE: 828 g/mol PS: 13 kg/mol
PDMS: 12.3 kg/mol PIB: 7.3 kg/mol
PMMA: 10 kg/mol 1,4 PBd: 1.8 kg/mol
1,4 PI: 5.4 kg/mol
might differ depending on lit. sources
molecular weight independent given by temporary entanglements,
“mesh-length”
Ml
3.4
1
Mc 1/10 Mc log M
log Mc 3 Me
“3 fingers needed to hold a stick”
entangled
69
Typical shape for G’(), G”() for monodisperse linear polymer melts
related length scales:
related time scales:
d R e s
(d: disengagement, R: Rouse time, e: entanglement, s: segmental motion)
Zone I:
2ωG
1ωG
flow-zone, viscosity and the dissipation dominates response
length scale probed Rg
longest relaxation time: = 1 for tan = 1
(Maxwell-model) or
= 1
Maxwell-model
log
log G
G”
G’
1
IV III II I
tan minimum
G’p
2
Rg, 10-50 nm 2-3 nm
glass transition
5 - 10 nm Re, distance between entanglements
see Maxwell-model
log
log G’,
log G”
extrapolated crossing point
for 2ωG and 1ωG
d = 1
70
Zone II: (rubber-plateau)
After G’ exceeds G”, G’ levels off. Response is dominated by elastic spring (G’!) of
physically cross linked entanglements (see motivation for Hooke-solid!). Maximum of
relative elastic response is reached for tan = Minimum, corresponding G’p (p: plateau).
length scale probed:
It is possible to calculate from G’p the entanglement molecular weight:
pe G
TRρM
: density
R: Gas constant
T: temperature
Assuming typical values for a polymer melt:
= 1,000 kg/m3, R = 8.3 J/(mol K), T = 450 K, Me 150 70 g/mol = 10.5 kg/mol
Pam
Nm103.5G
1Nm1J , kgKmolm
molKJkg
10.5
4508.31,000G
35
p
3p
typical plateau value: 105-106 Pa ( memorize!)
(high Tg + low Me increase in rubber plateau modulus)
for higher cross link density in chemical cross linked systems we expect higher modules
Mn does not affect G’p, but 3n
3n Mλ ,Mη ; so increase in molecular weight by factor
10 103 shift in for plateau length;
5-10 nm
71
Strobl, The Physics of polymers
at 3 Me we start to see plateau
increase in Mn by 100
shift in by 1003 = 106
72
Zone III:
G” exceeds again G’.
Strong increase as a function of frequency.
Transition zone towards glass plateau;
Zone IV: (glass plateau)
High torque and high frequency regime, experimentally difficult to obtain.
e.g. small sample diameter ( 5 mm), use of TTS (see later)
Length scale probed in dimension of typical length scale of polymer glasses, e.g. 2-3 nm.
shear rate dependent viscosity (or measured by ω*ηγη , Cox-Merz), typical shape:
Mn2
11 = 1 log
log G’
log G”
3 decades
Mn1
22 = 1
105
plateau length (width)
log γ or
log
log
1λγ : longest relax-
ation time
typical slope for linear polymers: - 0.8 0.1
3.4
n0 Mη
Mn2 10 Mn1
73
Time-Temperature-Superposition (TTS) and the Williams-Landel-Ferry (WLF) equation
Assumption:
The internal mobility of a polymer is monotonically (+ continuously) changed via the
temperature. The changes keep the ratio (not the difference!) between the different relaxation
time distributions and relative strength. This is related to the concept of the “internal clock”
that is only affected by the temperature (McKenna). Obviously this assumption must fail if
phase transitions (e.g. first order: crystallisation, second order: glass transition or TODT) are
involved. If we set a reference temperature T2, where we know or have measured G’((T2)),
G”((T2)), we can predict G’((T1)), G”((T1)).
Maths:
Modification of Arrhenius law Vogel-Fulcher equation:
(1)
VF
a0 TTR
EexpηTη
note: Ea for flow of linear polymer melts 25-30 KJ/mol (typical value)
no information about TVF yet, except:
- if TVF = 0 Arrhenius
- if T = TVF singularity in
therefore we expect:
a) 1T
TVF
for typical temperatures T 300-500 K, because it is only a correction!
b) fixed difference of TVF relative to Tg due to the assumption of similar mobility of
different polymers at Tg; using (1):
temp.ref.:T ,TT ;
η
η :
Tη
Tη221
2
1
2
1
Maxwell G
η τ;
ω
1τη ;
ω
ω
η
η
1
2
2
1
: characteristic “frequency” of motion
74
11
2
1 Tηfη
η
2: fixed value at reference temperature T2 (not defined yet)
elogTTR
Ef log
ω
ωlog
VF1
a
1
2
=: - C1 (no units!) 0.434
! temp.ofunit :C ; CC :elogR
E221
a
22VF CT :T ; choice of T2 will change C1 and C2!
212
211
221
211
1
2
TTC
TTC-
CTT
CC-C
ω
ωlog
only diff. to ref. T2 important!
212
211T
1
2
TTC
TTC-a log :
ω
ωlog
WLF-equation
shift factor
If we choose the reference temperature as T2 = Tg (other choices also possible!):
g12
g11
1
g
TTC
TTC-
ω
Tωlog
For these conditions (T2 = Tg) and for typical polymers it is found:
C1 17.4 C2 51.6 K
(C1 7.6 C2 100 K for T2 = Tg + 50 K ; rem.: C1 C2 const. ( 900 K))
apparent activation energy:
0
T
1for kJ/mol 17.5
elog
RCCE 21
a
75
If we assume (Tg) 0.1 rad/s ( 0.01 Hz) as the typical jump rate (motion) at the glass
transition temperature for a spatial entity of several monomerunits (e.g. 100) -
relaxation. We do not look at side chain motion -relaxation (typically pure Arrhenius) or
–CH3 1012 Hz (at room temperature)
17.4
T51.6
T17.4
TT51.6
TT17.4
ω
0.1loglim
g
g
TTT
g2
C1 is related to prefactor
15.6
16.4
17.4
10ν
ν2π
ω ; 10ω
10ω
0.1
further: TVF = T2 – C2 T2 = Tg
TVF = Tg – 51.6 K
log
1/T1/Tg
Arrhenius
slope ~ apparent activation energy,
differs as a function of temperature! -1
0
singularity at: Tg - C2
Prefactor A – B stretch, IR frequency typically 1012 - 1014 1/s
76
so: Tm > Tg > TVF
rule of thumb for polymers:
3
2
T
T
m
g (in Kelvin!, absolute energy scale)
Tg – C2 = TVF (C2 50 K)
ΔT51.6
ΔT17.4
ω
0.1loga log
(T)T
T2 = Tg:
T aT (T) [rad/s]
0 1 0.1
5 10-1.5 3.5
10 10-2.8 60
20 10-4.8 7103
30 10-6.4 2.5105
50 10-8.6 3.6107
100 10-11.5 31010
don’t take this table to literally!
16.4
mT
1
log
Ea = 17.5 kJ/mol
WLF
if crystalline
-1
curve defined via:
1) axis intercept Arrhenius
2) slope aT
ET
1lim
3) singularity at TVF
gT
1
VFT
1
T
1
C2!
3 deg 1 decade change in
mobility, close to Tg
15 deg 1 decade change
singularity
77
Glossary: (to relax from maths for a second)
Boger fluid: To study the relaxation of high Mn polymers (e.g. normal forces, G’, G”) the
very long relaxation times are shifted to more “practical” values via low Mn
solvents.
Dispersions
Definition: lat.: dispersio, fragmentation
A system built of several phases where one is a continuous and at least one
more phase is fine fragmentated within the continuous phase. If the size of
the dispersed phase is < 0.2 m (visibility!) they might be classified as
colloids or colloidal dispersions.
continuous phase dispersed phase name example
solid solid vitreosol ruby glass
solid liquid solid emulsion butter
solid gas solid foam pumic-stone (Bims)
liquid solid colloidal sol dispersion of Au, S in H2O
liquid liquid emulsion milk, pharmaceutic
or cosmetic emulsion
liquid gas foam soap-foam
gas solid smoke NH4Cl, carbon black smoke
gas liquid fog, mist natural mist
gas gas --- ---
why?!
78
Zero-shear viscosity as a function of solid content:
Einstein 1906: (see Appendix C, p. 124-144 for original work)
0.1for 2.51ηη s
s : viscosity solvent
: volume fraction
lit.: A. Einstein, Ann. Physik, 1906, 10, 289
1911, 34, 591
General behaviour for higher concentrations:
yet definednot :O , ...O2.51ηη 22s
Intrinsic viscosity (relative change of viscosity normalised to solvent viscosity):
tcoefficienEinstein , 2.5ηlim
η
ηηη
0
s
s
Extension of Einstein, O(2) Batchelor (1977)
shear O6.22.51η
η 32
s
extension O6.72.51η
η 32
s
t
shear + extension already anisotropic!
Shear can deform liquid particles to prolate or oblate shape if surface tension, mobility and
shear rate are sufficient (e.g. blood):
flow field,
hard, rotating particle
Idea:
a
b
c a
b
c
a = b < c prolate a < b = c oblate
In both cases: aspect ratio: 1c
a
79
High aspect ratio more excluded volume (see liquid crystals: Onsager theory)
Zero-shear viscosity as a function of volume fraction (no information about γη !) for higher
fractions:
d
d η 2.5ηη ss
(1) dη 2.5dη s
At certain volume fraction the addition of d leads to an increase in d that is expected to be:
02.5η
ηln
d2.5η
dη
dη 2.5dη
s
0
η
ηs
2.5expηη s Ball, Richmond 1980
Taylor:
...
42!
25
2
51ηη 2
s too low increase in 2!
Other idea:
Addition of small amount of particles d to the volume fraction (1-) of remaining fluid,
raises volume fraction by:
1
d
in analogy to (1):
physicalnot 1at y singularit ,11
1
η
η
ln11
1ln
1
1ln
η
ηln
1
d
2
5
η
dη
η1
d
2
5dη
2.52.5
s
2.52.5
0
2.5
s
80
If we assume a maximum filling fraction m, we find:
m-2.5
ms
1η
η
, Krieger-Dougherty (1959)
Maximum filling factor ( crystallography, inorganic chemistry), examples:
Simple cubic, sc m = 0.52
Hexagonal packed sheet m = 0.605
(colloids high shear rates)
random close packing m = 0.637
body-centered cubic, bcc m = 0.68
face-centered cubic (fcc) / m = 0.74
hexagonal close packing
Max. possible value
for monodisperse!
81
Best theoretical equation for ():
3
1
m
3
1
m
s
18
9
η
η
Frankel and Acrivos, with m 0.62 - 0.64
experimental:
16.6exp102.710.052.51η
η 32
s
Zero-shear-(rate)-viscosity can drastically be influenced by multimodal distribution:
e.g. r1 : r2 = 5 : 1
60% ! η50
1η total50:50pure2
1 15%2 60%60% pure2, ηη 15% increase in solid content, same viscosity
picture:
Viscosity as a function of shear-rate for colloids:
log
Shear thickening,
layered structure
2nd Newtonian plateau
scaling law
cγ : critical shear stress
log γ
filling of voids
0, 1st Newtonian plateau
82
Scaling-law behaviour (+2-Newton) can be described by:
- Ostwald de Waele -BγAγη , B[0,1]
- e.g. Ellis-model, using the Péclet-number as universal parameter
kT
rγη
kT
rσ6Pe
Peb1
1
ηη
ηη
3s
3
p0
The critical shear-rate can be estimated for a 0.50 = mixture:
s
nm10γd
27
c2 ; d [nm] , cγ [1/s]
Understanding of related forces F(x) and potentials ( energies) xVFdx for colloidal
particles:
Vtotal = Vvan-der-Waals + Velectrostatic + Vdepletion + Vsteric
DLVO-theory
DLVO: Derjaguin – Landau – Verwey – Overbeek (1941 + 1948)
101
103
101
105
10-3
10-1
102 103 104 105 particle diameter [nm]
cγ [1/s]
e.g.: d = 100 nm, cγ 103 1/s
83
Van-der-Waals:
Attractive force between atoms, molecules and particles caused by induced electrical dipols of
electron cloud.
Permanent dipol interaction 3r
1rV ,
induced 6
2
3 r
1
r
1rV
, minus prefactor, because attractive!
For two quadratic surfaces with side length L one finds:
22
Lr12π
ArV
A: Hamaker constant [energy]; typical value: 0.4 - 4 10-19 J
For Lennard-Jones potential, also short range repulsion rfrV ,
for hard spheres, one typically finds:
126 r
1b
r
1arV
Velectrostatic:
2
r0
21
rεε4π
qqrF
q1 q2
Coulomb
1st - Maxwell-equation: r0 εε
ρE
, : charge density
eVE , Ve: electric potential ( qVWq;EF e
)
in spherical coordinates:
(1) ρεε
1Vr
dr
d
r
1ΔV
r0e2
2
r
84
Approximation of via a Boltzmann-distribution of screened Coulomb potential (single ions):
kT
rVq1ρ
kT
rVqexpρrρ e
0e
0
homo
ee
0eff VkT
Vqρρ
using (1)
e2
e2
2
e VχVrdr
d
r
1ΔV Eigenvalue problem
( Quantum mechanics ψEψH )
solution:
rχexprεε4π
qrV
r0e
no screening screened potential, screening length1/: Debye length
rχexpr4π
)exp(χrρ
2
1/ : equivalent to Bohr-radius in H-atom, first Laguerre polynom
c
0.304
χ
1rD [nm] for 1:1 electrolyte, c in mol/litre
c = 1 mol/l rD = 3 Å
c = 0.01 mol/l rD = 30 Å
Bjerrum length:
What is the distance l b, where the electrostatic energy of ion is equivalent kT?? (“electrostatic
yardstick”)
+
l b
85
FdxW
bl
2r0
dxxεε4π
eekT , RT = 2.4 KJ/mol , r = 80
l b = 7 Å
for distances smaller 7 Å Manning condensation, opposite charges bound to each other
In case of identical spheres at constant surface potential 0, radius a of spheres:
rχexpr
ψaε4πV
20
2
e
for a < 5
rχexp1lnψaε2πV 20e for a > 5
Vdepletion:
In bimodal systems with large size difference, e.g. polymeric solution plus particle, polymer
does not bind to particle. Potential caused by osmotic pressure.
Vd - , : osmotic pressure
mixed flocculated
areas not accessible for polymer
forces acting!
86
Vsteric:
Influenced by: - number of chains per area
- layer thickness (Mn!)
- solvent quality
- anchor strength
systems: e.g. block copolymers (5 - 50% as anchor)
triblocks: bridging flocculation (e.g. sewage water treatment)
particle particle
non charged surfactants or polymers
87
Book: Israelachvili
88
Fourier-Transform-spectroscopy
[Joseph Baron de Fourier (1768-1830), mathematician and physicist]
In the past (in ESR till today): CW (continuous wave)
excite with single frequency
measure the resonance
change frequency
FT: all signals are acquired simultaneously (“multiplex advantage”)
In words:
A Fourier transform analyses the corresponding frequencies of a given timesignal with respect
to amplitude, frequency and phase (i.e. full information).
pulse
free induction decay (FID)
time
FT, Fourier transform
Ta
0
a
a T
1ν
89
Note:
This method is of special importance in NMR, IR, X-ray, neutrons, QM, ... and Rheology!
Math:
dt tωiexptfωF
complex! complex (can be separated in cos + sin)
real + imaginary part or magnitude + phase
This operation is reversible! (one-to-one)
dω tωiexpωF2π
1tf
In NMR we use a single-sided, complex, discrete Fast-Fourier transform (FFT)
(special algorithm (“butterfly-algorithm”), which needs 2N datapoints)
Problem of discretisation (ADC, analogue digital converter)
scan rate of the signal
(dwell-time, DW)
Signals are not distinguishable!
Nyquist-frequency (cp. solid state-modes, Einstein, Debye model)
Frequency regime in which the signal can be assigned unambiguously (= spectral width, SW):
DW2
1SW
t
FID
FID
90
Some important mathematical relations
1) FT is linear
ωGbωFatgbtfa
i.e. the different signals can be detected independently!
(“proof”: BCACCBA , linearity of the integral)
2) The point in time t = 0 is proportional to the whole integral of the absorptive spectrum.
Proof:
dω tωiexpωFtf
dω ωFdω 0ωiexpωF0tf
3) Timesignal and spectrum are inverse to each other with respect to the full width at half
maximum. (units: s, 1/s !)
Heisenberg’s uncertainty principle: t E h E = h
t 1
t
B A
A
B
b 0 a
FT
t pulse
FT
0
same integral, because
FID(t=0) is the same!
91
4) important Fourier-pairs:
a) exponential Lorentzian
exp(-t/) (cp. Relaxation etc.)
22
22
0
00
τ
1ω
ωi
τ
1ω
τ
1
ωiτ
1
ωiτ
1
τ
1ωi
1t
τ
1ωiexp
τ
1ωi
1
dt tτ
1ωi expdt tωi exp
τ
texp
Magnitude: 22 ImRe “broader feet” ( test it! MC)
b) box sinc
real part
absorptive
imaginary part
dispersive
real (absorptive)
imaginary (dispersive)
0
πT
1
2
FT
FT
1
t0 0 -t0
FT
cp. single slit diffraction pattern
x -x q-vector
92
proof:
(Euler) 01πiexp ; tωsinitωcostωiexp :remark
ω
tωsin2tωsini2
ωi
1
tωsinitωcostωsinitωcosωi
1
tωiexptωiexpωi
1
tωiexpωi
1dttωiexp
00
0000
00
t
t
t
t
0
0
0
0
c) Gaussian Gaussian (without proof)
2
σωexpσ2πtωiexp
2σ
texp
2t
2
t2t
2
5) convolution
Multiplication of a timesignal t(t) with a function g(t) corresponds to a convolution in
Fourierspace.
In NMR the measured timesignals are often multiplied with exp(-kt) or exp(-k’t2).
(This is equal to a convolution with a lorentzian or a gaussian.)
Example:
sinc()
FT
S
N
S/N 5/1 S/N 5/(1/3) =15 but: broader peak!
N 1/3
Convolution with a box-
function
averaging, new function
93
Appendix A
Phil. Mag. 32 (5. Series), 424 (1891)
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
Appendix B
116
117
118
119
120
121
122
123
124
Appendix C
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
