Institute for Chemical Technology and Polymer Chemistry
[email protected] http://www.itcp.kit.edu/wilhelm/
Introduction to Rheology
Prof. Dr. Manfred Wilhelm
private copy 2019
KIT – Universität des Landes Baden-Württemberg und nationales Forschungszentrum in der Helmholtz-Gemeinschaft www.kit.edu
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
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
I 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 γ, γ0 ? ...... 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
II 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
III 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)
1 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
2 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
v = 1 m/s = 1000 mm/s d = 0.2 mm d
What is the relevant quantity?
Assumption: layered structure vn
n layers vi
di
v i = constant for all i! di
3 v v v 1000 mm/s 1 i n 5000 di d n d 0.2 mm s v shear rate γ [1/s] d 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 d frequency: ω 1 s v 1 6000 2π min (=rpm, rotations per minute) ω 1 1 100 2π s stroke (german: Hub): s = 10 cm = 0.1 m ω m m 1 v 1 s2 1000.12 20 2π s s m m v ( / 2)20 30 why / 2 ? max s s d 20 μm 20106 m
30 m vmax s 6 1 γ max 1.510 d 20106 m s
think about: - shower lotion - lipstick - coating of paper - extrusion of fibres (clothing)
4 unit of γ : [1/s] = inverse time v
d 1 comparison: λ γ p
Polymer molecule with relaxation time p γ λp is a unitless quantity
Why can we assume that Hooke’s law could be correct?
Do we “buy” this law? Hooke: F k x F
0x Possible reasons:
! a) F F(x,...) F(x) (assumption)
Taylor-expansion (Taylor around x = 0 MacLaurin series)
F 1 2F F F x x 2 (x) (x0) x 2! 2 x 0 x x 0 0, k vanishing for because small x at equil.
linear nonlinear part b) C C C C C C C C C
Interaction Potential for vibrational (IR) spectroscopy (beside Hooke): Morse-Potential U(x) (Potential, not force!) U F U Fdx x
5 Potential has units of energy!
2 Morse: U(x) A(1 exp(β(x x0 )))
U/A U/A
1
1-exp(-(x-x0)) exp(-(x-x0))
x
1
x X0=0 X0=0
U/A
1
2 (1-exp(-(x-x0)))
Note: A: Dissoziation energy x X0=0
set: x0 = 0 β2x 2 β3x3 Taylor: eβx 1 βx ... 2! 3!
2 2 2 2 U (x) A(1 (1β x ...)) A(β x ...) Aβ x U F 2Aβ2x k x x k 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
2 x U A ln1 (x) R 0 6 Remark: If we remember typical force-constant from IR ( spectroscopy books)
k k = 500 N/m ( ω ) m and we remember typical area needed for a chain, e.g. polyethylene: orthorhombic a = 7.5 Å 5Å b = 5 Å c = 2.5 Å
7.5 Å
2 chains per unit cell
o 2 o 2 A 7.55A 37.5A o 2 20A 201020 m2 chain 2 2
F
different A F
different x and L
F
L
Renormalization to area + relative change in length F x σ E + E: unit: pressure [ 1Pa = 1 N/m2 ] A L
stress
E-module 7 upper limit: F k x x σ E A A L k 5001020 1010 N m L E 250109 Pa 250 GPa A 20 mm 2
C in unit cell 10-10 m (one bond) Tungsten (W): 150 GPa
C But: bending modes are weaker only several GPa C C
Hooke for polymers (rubber elasticity)
Start of chain in coordinate origin, where is end?
W(r) Gauss
1 (x μ)2 W exp , here = 0 (r) 2 2π σ 2σ
Boltzmann: S k ln(W) C k ln(exp(-x2 )) C k x 2 with G H - T S, H 0
G - T (C - k x 2 )
G: units of energy: ΔG F, (W Fdx) F T k 2 x (temp. + elongation!) x
only needed: H 0 ; W(r) Gauss
See analogy for Gauss in crystallography! Debye-Waller factor!
8
9 Why can we assume that Newton’s law could be correct?
x F v Newton: σ η η γ rough surface A d
Note: In Rheology Newton’s law is associated with σ η γ ,
not with his other law: F mx ma .
F Why not: x ? A F a, a x ? Do we “buy” this? 1 π F v 3 a ?? 2 why?
We need proportionality between viscous force and velocity: Fviscous v
Remember: Law from Stokes: F 6π ηr v (F: e.g. gravity)
F Sphere in viscous media 2r v
Note: unit of : F pd Pa ms η A η Pa s , old: Poise: 1Pa s = 10 P; 1 cP = 1 mPas (Poiseuille); v v m d typical values: blood 100 – 4 mPas (thicker than water!; shear thinning) Glycerin 0°C 10,000 mPas 20°C 1,400 mPas T 60°C 60 mPas Oil, SAE 10 30°C 200 mPas
H2O 1 mPas ( memorize!) Air 0.02 mPas
10 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?
m m M vm = v vM - vm = - v
For M: ΔlM vM Δt For m: Δl vΔt , same velocity for all small particles!
After time t: lM
particles lM l
l
l - lM
N Average density of particles m: ρ , number density, not mass density! Δl
l
The mass M is hit by the following number of particles during t: 1 1 N N ρΔl Δl ρΔl Δl ρΔl N 2 M 2 M
50:50 probability that particles fly in correct direction
direction of particles but N- > N+ !!
11 if we define a clash-rate: N N Δt N Z Z ρ Z ρ v Δt ΔlΔt vΔt v
1 1 v v Z ρv v Z M 2 M 2 v 1 1 v v Z ρv v Z M 2 M 2 v
Z Z Z ρ v
each particle transfers elastic impact onto mass M with relative momentum p 2mv
p 2mv v mv M p 2mv vM mv
In one time unit t, this balances the outer force F needed to push mass M with velocity vM.
F Z p Z p
1 v v 1 v v F Z M 2mv v Z M 2mv v 2 v M 2 v M Zm v v v v v v v v v M M M M
Zm 2 2 2 2 v 2vvM vM v 2vvM vM v Zm 4vv 4mZ v v M M
F vM Friction is proportional to the velocity of the mass M.
12
Cf tance and to treat a case of motion in a viscous fluid.” magnitude of the force required “Newton was the first to formulate a hypothesis regarding the Princi pia i iSc I Sect ii Lib Sir Isaac Newton 1642 – 1727 – 1642 mlHatschek Emil X to overcome viscous resis-
13 Linear models: Hooke, Newton, Maxwell, Kelvin-Voigt … incl. oscillatory excitation and response
Hooke – spring
, d dt
σ G γ σ G γ
Pa no unit
Newton – dash-pot
dγ σ η γ, γ dt
σ,γ γ
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 γ1 or γ 1 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.
14 The non-linear regime should be avoided for linear response measurements:
asymptotic deviation!
rate sweep: η γ linear regime
γ
G G’ strain sweep: Gγ ,Gγ G’, G” later 0 0 linear regime G’’ fixed frequency
0
Dash-pot (DP) and spring (S) can be arranged in series or in parallel:
σ σ γ γ S DP S DP viscosity with a elasticity with bit of elasticity a bit of viscosity (long term) (long term) γ γ γ σ σ σ S DP S DP γ γ γ σ Gγ ηγ S DP
γ, γ, G, η
σ Maxwell model Kelvin-Voigt model (for liquids with some (for solids with some γ, γ, G 0 , η elastic response) viscous response) σ
15 Detailed analysis of Maxwell model
σ σ G γ γ S G σ σ η γ γ DP η σ σ γ (1) G η
1. step-experiment
0
0 t at time t > 0, γ 0 (not in the dash-pot, but overall system!)
σ σ using (1): 0 G η 0 σ σ G 0 η dσ G 0 dt (see: first order kinetic, or Lambert-Beer) σ η σ(t) 1 G t dσ 0 dt σ(0) σ η 0
G 0 ln(σ(t) ) ln(σ(0) ) t η σ(t) G ln 0 t relaxation time σ(0) η
G 0 t η Pa s σ(t) σ(0) exp( t) σ(0) exp( ); τ s η τ G Pa (for short time force is fully in spring) σ(0) G 0 γ G σ G σ G γ exp( 0 t) (t) G G exp( 0 t) (t) 0 η γ (t) 0 η
limG (t) G 0 t0 G exp( 0 t) η
G 0
Memory!
0 t 16 Oscillatory response:
Hooke
,
γ γ0 sin(ω t)
0
t
distinguish: amplitude elongation! System has memory ( stored energy storage modulus G’)
Newton
, , γ phase shift γ γ sin(ω t) 0 γ γ ωcos(ω t) γ 0 0
t
distinguish: 2π [rad/s] , f [1/s = Hz] ω 2π T System has no memory ( energy is lost loss modulus G’’)
17 Maxwell:
γ(t) γ0 exp(iω t)
γ γ0 (iω)exp(iω t) i ω γ(t) after initial time we reach dynamic steady state:
σ(t) σ0 exp(i(ω t δ)
σ σ0 (iω)exp(i(ω t δ) iωσ(t) σ σ eq. (1): γ G η
iωσ(t) σ G iω γ(t) G η iωσ(t)
G γ(t) G γ(t) 1 1 , def.: * σ(t) i ωη σ(t) G ( ) G 1 η * 1 , τ G ( ) i ω τ G G 1 i ω τ G* iω τ iω τ G* G sep. into real and imag. part :(1-iω τ) 1 iω τ iω τ(1-iω τ) G* G iG G ( ) ( ) (1 i ω τ)(1-i ω τ) ω2 τ2 iω τ ω2 τ2 ω τ G G i 2 2 2 2 2 2 1 ω τ 1 ω τ 1 ω τ ω2 τ2 G G 1 ω2 τ 2 ω τ G G Note: (a+b)(a-b)=a2-b2 1 ω2 τ 2 Plot: G’, G” (linear scale) G’, storage G
G/2 G”, loss module Why G’ = storage ?
Why G” = loss ?
log() [rad/s] = 1
18 ω2 τ2 limG lim limω2 τ2 ω2 ω0 ω0 1 ω2 τ 2 ω0 1
G ω2 for small
ω τ limG lim limω τ ω1 ω0 ω0 1 ω2 τ 2 ω0 1
G ω1 for small trick to memorize: G a ωb , a + b = 3
log G
1 G” 2
G’
= 1 log
Note for “NMR-People”: t FID: M exp( ) Lorentz shape in T2
Lorentz: Re (ˆ G ) τ Re G” ( ) ω2 τ2 1 Factor - ω τ2 missing G’ Im( ) 2 2 ω τ 1 0
Im (ˆ G ) in Rheology: 0 !!
19 * G ω0 Gω0 G (module of spring within Maxwell-element) width of relaxation spectrum for G”: G” ω τ G G 2 2 ω τ 1
= 1 log
Set to = 1: ω G G ; G = 1 ω2 1
dG 1 ω Maximum at: ( ) 0 2ω 0 dω ω2 1 2 2 ω 1 2 1 2ω 2 2 2 ω 1 ω 1 ω2 1 2ω2 1 ω2 1 ω2 1 2ω2 1 ω2 ω 1 physically meaningful: ω 1 1 1 G(ω1) G G 11 2
full width at half maximum, 1, 2 ? fwhm 1 G 2 1 G fwhm 4
log
1 ω 4 ω2 1
1 1 ω2 ω 4 4 1 1 ω2 ω 0 4 4 20 remember: ax 2 bx c 0 b b2 4ac x1/2 2a
1 1 4 1 1 ω 4 4 1/2 1 2 21 3 4
2 3
ω1 3.732
ω2 0.268
ω ratio : log 1 1.14 10 ω 2
The full width at half maximum (fwhm) of a single exponential relaxation is 1.14 decades in frequency space.
* * σ G ( ) G iG γ* ( ) ( )
* * * σ G γ ; γ γ0 exp(iω t), γ iω γ0 exp(iω t)
* G( ) iG( ) σ γ iω
η*
iωη* G* units: [1/s Pa s = Pa] [to memorize: “iong”, i omega n equals G]
η* η iη η* η2 η2
ω η* G*
21 G Phase lag : tanδ G 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
G G 10% typical error margin for rheological measurements G G
22
23
24
25
26 Without any mathematics: step experiments (step in stress or step in strain)
0 t
σ γ (t0) η σ γ (ta ) G
F, ,
0 ta t Kelvin-Voigt model
2nd possibility: strain step:
0 t t
t ~ exp spring τ elastic part memory? spring, viscous part elastic part t
0 t F, ,
Maxwell - model
27 More complex models (but still linear models!):
G1
0 t G2 1
2 1, G2 1 2
2
G1
0 t
Burger - model
Memory (Gedächtnis)
The memory of the system might be defined for a step strain experiment as follows: dσ dG γdt dt
dG (t) M , minus sign, so that M(t) is positive; (t) dt dσ M γ dt
σ tt σt dσ - M(t-t )γ t dt 0 t- Memory depends only on elapsed time: t – t’ = s, dt’ = -ds
σ - M γ ds (t) (s) (t-s) 0 exchange of limits and: dt’ = -ds
28 for infinite small motion: dσ Gdγ dγ dσ G dt Gγdt dt t σ G γ dt, s t t (t-t) t - σ G γ ds (s) (t-s) 0 if we have a modulus function with an exponential memory:
t - t - (t - t) monomodal: G (t) G 0 exp σt G0exp γ t dt τ - τ Improvement of this model: several relaxation times
Multimode models N t G(t) Gk exp , N - mode model k1 τk
t N t σ G k exp γ (t ) dt k 1 τk picture for multimode Maxwell-model: 1 2 3 4
= 1 = 2 = 3 = 4
= 1 + 2 + 3 + 4
F, ,
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
29
H. M.Laun
30 To reduce the need of maths for a while, a glossary on important rheological terms is inserted:
Glossary
a) Lamellar flow
x γ x h
1 dx 1 h γ v h dt h flux r0 2 2 4 For this model: γ γ (h) , in contrast: tube 0 vr r0 r I r0 !! Hagen-Poiseuille r0
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!) v = const.
b) Reynolds number
The Reynolds number describes the ratio between the kinetic energy of a system and the energy lost by viscous flow. E Re kin , for Re > 2000 we find transition between lamellar ( F γ ) and turbulent E viscous ( F γ 2 ) flow! 2rρ v For a capillary (diameter: circle) we find: Re η
r: radius, : density, v: avg. velocity, : viscosity
31 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
2 10-2 103 3101 m s2 m2 kg m Re -3 3 1,500 ,close to transition: lamellar turbulent! 410 kg m sm s
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:
* * iωη ( ) G ( )
* η(γ) η (aω) , a 1 (experimentally) via: * η(γ) η ω ; note : ω frequency ν ; ω 2π ν
This rule holds only for rheologically simple materials!
2 2 * Gω G ω ηγ η ω ω ω2 τ2 G ( ) G 2 2 e.g. Maxwell-model: 1 ω τ ω τ G G ( ) 1 ω2 τ2
G’ G0
G /2 0 G”
= 1 log()
G” leading term G’ leading term 32 2 2 2 * G ω τ lim η lim const. η0 ω0 ω0 ω ω
2 * G const. 1 lim η lim ω ω ω ω ω
* log( ( ) ) 0 η for single Maxwell-model
-1
= 1 log()
d) Lissajous figures
oscillatory shear different representations σ (t) σ max 1
t 1 γ (t) γ max
~ cos (t), ~ sin (t) vector description of circle general for ellipse: y
x cos(ω t) b a y cos(ω t δ) x δ b phase lag tan 2 a
33 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.
log η η 0 = Newton η 0 2
-a η η 0 1 β γ c η b γ a
1 log γ
β
ηγ 1 η γ 2 for γ 1 γ 2
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) η b γ a a: scaling parameter, shear thinning exponent; a[0, 1] : 0 Newton 1 max. shear thinning exponent
34 if a = 1: F η γ b γ -1 γ b force independent of γ , force is constant! at γ 1 η b
Carreau (example for 3 parameter model) η η η 0 also other def. for Carreau: η 0 (not equal!) 1 β γ c 1 β γ c c: scaling parameter c[0, 1] : pivot point (knee), 1 η η η 0 0 0 if γ η c c β 1 11 2 1 β β
4 parameter models: - further parameter needed to: e.g. model the width of the knee as the next parameter
log η η e.g.: η(γ) 0 d 1 β γ c
Polymers: where cd < 1 increase in Mw/Mn! WHY??
log γ
- introduction of “second Newtonian plateau” for high shear rates
log η st η 0 , 1 Newtonian plateau
η η e.g.: η(γ) η 0 1 β γ c
η , 2nd Newtonian plateau log γ
35 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:
start of shear η η time after shear γ = const. γ increase Plateau τc steady state t t or:
σ Newton
lim thixotrop shear thinning τc 0
γ
Shear thickening ( rheopex dilatancy)
ηγ 1 η γ 2 for γ 1 γ 2
Anti-thixotropy shear thickening + memory ( Hysteresis)
η γ = const. η time after shear
t t
36 or: σ
Newton
γ
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: “dry” particles on top
shear Vsand = const. ! sand level Vwater = const. ! water level or shake
ordered disordered spheres spheres need ”sand” more volume in beaker to pack
Dilatation: Ausweitung
Bingham plastic (the “evil” in the ketchup bottle!) (deutsch: strukturviskose Flüssigkeit mit Fließgrenze = plastisches Fluid)
σ critical stress γ = const. σ η γ σ yield
below: elastic, solid-like σ yield G γ yield σ above: liquid-like yield
G σ yield Ketchup 20 Pa
γ γ yield 37
σ shear thinning (pseudoplastic) Bingham σ yield γ 0 for σ σ yield Newton σ η γ σ yield for σ σ yield η
γ
Extension of Bingham-model: Herschel-Bulkley
n include: powerlaw for viscosity σ σ HB k γ 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 - E a k r A exp RT
unitless
if Ea << RT kr A
if Ea >> RT kr << A, slow down
2) kinetics
A(t) A (0) exp- k r t
kr t >> 1 basically complete reaction
kr t << 1 just started
38 Deborah number [book of judges 5.5, song of Deborah: “Even the mountains flowed before the Lord ...”] remember: G step in strain σ (t) G γ exp t η
dimensionless
1 G η t, τ De η G η τ internal relaxation time De G t t observation time
De 1, short observation time solid like De 1, viscoelastic reponse De 1, long observation time liquid response
1 if we take: γ t γ γ cos(ω t) under oscillatory shear: 0
γ ω γ 0 sin(ω t)
De γ τ ω γ 0 τ e.g. longest relaxation time in polymer
Note: Generally Deborah-nr. is not precisely defined ( γ ω γ 0 , or γ ω), and there is (osc. shear) (Cox-Merz) confusion with Weissenberg-nr.: Wi γ τ Weissenberg normally used in the context of: γ =const., steady shear
with respect to normal forces
Pipkin diagram:
γ 0 N l yield ( Bingham plastic) e i non linear response w q s t u o
o i γ 0 ω = const. l n d i d viscoelastic
1 De, ω γ 0 τ, or ω τ
39 Péclet number
F 6π η r v Stokes: 2r F F ξ v
k T k T Stokes-Einstein for diffusion coefficient D: D ξ 6π η r
r 2 6π η r r 2 6π η Time needed to displace object by distance r: t r 3 D k T k T
6π η 6π r 3 σ Pe t γ r 3 γ k T k T
σ η γ frequently used in context with colloids;
Taylor vortex moving bob
Ro secondary flow caused by inertia generates vortices Ri in addition to shear
moving cup
more sensitive less sensitive to Taylor vortices to Taylor vortices
ρ 2 Ω 2 R R 3 R Ta o i i 3400 ηγ 2 ρ : density Ω : angular velocity
40 Units for Ta (only check, no proof): 2 4 2 4 2 kg m kg m kg m 1 Ta 2 2 4 2 2 2 1 6 2 N m s N s N m s 2 s m unitless quantity
-=- END OF GLOSSARY -=-
41 What do we expect for (p,T)?
Gases Mean free path length: l 2r
v L >> r mean distance
Cross-section A π2r 2 4π r 2
in physics: (confusing for rheology)
one particle 1 1 1 ρ ΔV L A L 4π r 2
volume 1 1 L ρ A N 2 4π r V normal conditions (Gas, 1 bar, 300 K): pV nRT 1 Mol ˆ 22.4 l n 61023 ρ 31025 m3 V 22.4103 m3 2 A 4π 10-10 m2 1019 m2
o r 1A
1 m3 1 1 L m μm 300 nm 31025 1019 m2 3106 3 mean free path length clash rate: 1 Δt L m 1 3 v , v 330 more precise : mv2 RT Δt s 2 2
L 310-6 m 1 Δt s v m 9 330 10 s 1 1 typical clash rate : 109 Δt s
42 velocity distribution (Maxwell-Boltzmann) (see Physical Chemistry books for details)
3 2 2 m mv 2 P(v)dv 4π exp v dv 2π kT 2kT 1 1 1 8kT 2 kT 2 3kT 2 v 2.54 simple picture : v π m m m 1 ___ 3 ___ m v 2 kT note : v 2 v 2 , generally true for distributions 2 2 1st moment 2nd moment model: L: mean free path length
unit area A x
dvs vs L xi+1 dx
L vs xi
vs: shear velocity y v: particle velocity z
e.g. xi: bottom layer, xi+1: top layer, gap of width L
dv momentum transport if particle leaves layer xi+1 to go to layer xi : mL s dx
Number of particles n leaving layer xi+1 in unit time in direction xi : N N N density of particle ρn ρn A v t N V0 A L A v t
volume average velocity! only half fly in correct direction: 1 1 1 N
ρ n A v t n ρ n v v 2 2 2 V0
unit time, unit area
43 in unit time, unit area the following momentum p is transferred:
Δp 1 N dv v m L s Δt 2 V0 dx dv this must be equal to the force F η s dx 1 N 1 η m v L ρ v L 2 V0 2
V using : L 0 N 4π r 2
1 N m v V 1 m v η 0 2 2 2 V0 N 4π r 8 π r 1 3kT 2 using : v m 1 η m 3kT T m 8 π r 2
Viscosity of gases is: - independent of density!
- therefore independent of pressure! ηp,T η T !
- a function of mass and temp. of particles! η T !
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!) E*
kT E * N exp , E*: activation energy h RT
44 r - shear: force on single molecule, typical distance r F σ A σ r 2
r/2 apply this force for half distance r r σ r 3 σ V E σ r 2 m σ r3 2 2 2 E * 2
σ r3 Vm: average occupied volume per molecule, E * 2 VM: volume per NL molecules
So effective jumps N are jumps to the right N minus jumps to the left N
σ VM σ VM E * E * k B T 2 2 (1) N N N exp exp h RT RT
The shear rate is the effective number of jumps in one second divided by layer thickness in unit time:
v r N (2) γ N d r
(2) in (1):
k B T E * σ VM σ VM γ exp exp exp h RT RT 2 RT 2
remember : sinh(x) ex ex /2 limsinh(x) limex ex /2 lim1 x 1 x /2 x x0 x0 x0
k T E * V γ B exp 2 M σ h RT RT 2 1 V V N V γ σ M m L m η R R k B
h E * η(γ 0) exp Vm RT
45 - Arrhenius for T-dependence - increase of free volume reduces viscosity (hopping probability )
- Ea ; Ea - Pressure dependence via average volume per molecule weak p-dependence
(T)
E exp a liquid RT gases
T T
Stress-strain tensor and normal forces (Why might we need a tensorial property?!)
So far we have used v,x and F as collinear (parallel) vectors scalar description If we would like to extend this, what happens if x and F are not parallel?
F
x
We need a transformation between x F . 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 (y1,y2 , y3 ) is generated out of the old system x (x1,x 2 ,x 3 ) by a linear set of equations:
46 y1 a11 x1 a12 x 2 a13 x 3
y2 a 21 x1 a 22 x 2 a 23 x 3
y3 a 31 x1 a 32 x 2 a 33 x 3 if we introduce matrix (3 by 3 matrix, second rank tensor) a a a 11 12 13 A a 21 a 22 a 23 , we can write: y A x a 31 a 32 a 33
Example for a simple rotation of a vector x in 2 dimensions:
x2 y2 Rotation around y origin by angle in math. positive sense x (counterclockwise)
α
x1 y1
cos α cos (α ) x , y sin α sin (α ) use of addition theorems: cos (α ) Reeiα Reeiα ei Recos α isin αcos isin Recos α cos sin α sin i ... cos α cos sin α sin analogue for the sine (using the imaginary part): sin (α ) ... cos αsin sin αcos
cos (α ) cos sin cos α y sin (α ) sin cos sin α
x A
47 Definition of the extra stress tensor (right handed system!)
2
22 indices ij :
21 i: the force acts on a plane that is
12 normal to the basis vector i 11 23 j: the force acts in the direction of
13 1 the basis vector j
3
This results in the following extra stress tensor: τ τ τ 11 12 13 τ τ21 τ22 τ23 τ31 τ32 τ33 The stress-tensor σ is the sum of the extra stress tensor plus the hydrostatic pressure. The hydrostatic pressure acts equally along the τ11,τ22 and τ33 components.
1 0 0 σ -p E τ , p : pressure, E : unit tensor 0 1 0 0 0 1
Properties of the extra stress tensor
- The tensor is symmetric (like many in quantum mechanics, see e.g. Fermi’s golden rule):
τij τ ji 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
n I1 tr A a ii a11 a 22 a 33 i1
48 (see also quantum mechanic books αi αi ) i1
2 nd 1 2 2 invariant: I2 tr A tr A 2
a11 a12 a13 rd 3 invariant: I3 det A a 21 a 22 a 23
a 31 a 32 a 33
Due to the first invariant τ11 τ 22 τ33 0 the trace of the extra stress tensor has only two variables. N1 = 11 - 22 normal stress differences N2 = 22 - 33
What do normal stress differences mean?
- assume shear stress along 21
2
22
21
11
1 33
3
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 N1 N2
to memorize: Na = aa - a+1, a+1
49 What do we expect for N1,2 γ, γ0 ?
- N1,2 should only be a function of γ due to kinetic nature of the phenomenon, e.g.:
2 N1,2 a b γ c γ ... a,b,c : constant
- 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
N1,2 γ N1,2 - γ
even function with respect to γ n
2 We expect equation like: N1,2 c γ as first approximation N τ τ ψ 1 11 22 1 γ 2 γ 2
N τ τ ψ 2 22 33 2 γ 2 γ 2
1 : first normal stress coefficient
2 : second normal stress coefficient
1 : generally positive, ψ1 ψ2 (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:
γ N1 γ N1 + N2
&
cone-plate plate-plate
Information: N1 Information: superposition N1 and N2
50 Typical examples for extra stress tensor: a) ideal viscous fluid b) viscoelastic liquid 0 τ 0 τ τ 0 12 11 12 τ τ21 0 0 τ τ21 τ22 0 5 unknown 0 0 0 0 0 τ33
12 = 21 12 = 21 , 11 + 22 + 33 = 0
3 degrees of freedom , 1 , 2
The first normal stress coefficient can be estimated using:
2G( ) limψ1 γ for γ ω ω0 ω2
Phenomena where we can directly “see” normal forces a) Rod-climbing
Parabola, f(r) r 2 leading term, f(r) r 4 !
centrifugal forces f(R,r,ψ ,ψ ,ρ,ω,...) 1 2 (e.g. water) Non-Newtonian fluid Newtonian fluid climbing effect is called Weissenberg effect
51 b) Secondary flow for rotating disc up!
centrifugal see: cover page forces book: Tanner Newton or not?
Newtonian fluid Non-Newtonian fluid
c) Extrudate swell
parabola
die swell film blowing (e.g. plastic bags)
Newtonian fluid Non-Newtonian fluid
52 New chapter:
Possible measurements (for oscillatory rheometers) and hardware
1) Detection of onset of non-linearity at fixed frequency
log G linear regime 1 = const., fixed
a b G’
c G”
log 0 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 (name!: not pain) 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 ( γ max γ0 2π ω1 )
we can assume: linear if 2 < 1
perhaps non-linear if 2 > 1
2) Measurement of G’, G” at T = const., : variable, 0: parameter
difference! log G
Frequency dependent module 1 distribution of relaxation times G” e.g. via Multimode-Maxwell models 2
see section about polymers (later) G’
(reptation, rubber plateau, TTS) = 1 log
53 Typical range: 10-2 < < 60, dynamic range: 4 decades of hardware due to mechanical device!
log G adjust 0 !,
torque will change for an increase
in by 104 by up to 104!
log
γ(t) γ0 sin(ω t) γ γ ωcos(ω t) 0 1
γ max γ0 ω
adjust 0 every 1 - 2 decades for best performance
3) Temperature dependent measurement
0: fixed but parameter, T: variable, : fixed
G’ G’ or or 9 G” 10 G” [Pa] if 1/T 105 - 106 Arrhenius
T log
Instrument for this: DMTA, (Dynamic mechanical thermo analyser) cheap due to limited -range (sometimes also E-module measurements)
54 4) Shear rate dependent viscosity
log asymptotic behaviour in principle no linear regime!
0 experimental linear regime
e.g. 10% reduction relative to 0
log γ
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
N(t): torque
bob
or or
cup
(t) (t) (t)
Mooney-Ewart double couette Haake-type
for low viscosity Air bubble
materials, low friction at lower end
e.g. water
55 To prevent evaporation of water: saturated H2O vapor
H2O Dodecane, C12H26 H2O
or water trap H2O sample
If plate - geometries are used: 2 Area r0 r0 Torque at infinitesimal area: ΔN ΔF r ΔN r1
total torque N r 3 0 e.g. change from 50 mm plate-plate geometry to plate-plate 8 mm plate-plate geometry : torque reduction by 3 non homogeneous γ0 ,γ ! 50 244 , 2.5 decades reduction 8
r0 3 N r0
typical values for : 0.02 - 0.1 rad h 1.14 –5.73° very small!
cone-plate
homogeneous γ0 ,γ !
or
Advantage of plate-plate (cone-plate) vs. Couette: - less sample volume (e.g. 0.1ml vs. 10ml) Disadvantage of plate-plate: truncated cone - leakage (low viscosity material) easier to manufacture - less area less sensitivity for low viscosity materials - heterogeneity of shear rate
56 5) Creep experiment
0
t
G(t)
t
G(0,t) measured
Overlay via h() log G(0,t) 1 < 2 < 3 < 4 3 (modulus measured, not stress!) 1
2 4
log t
Wagner-Ansatz (Manfred Wagner, Prof. in Berlin): Gγ,t G t h γ
damping function
1. limhγ 1, linear response γ0 2. decreasing strictly monotonic as a function of time, limhγ 0 γ typical examples: hγ exp- n γ, e.g. n 0.18 for PE melt
hγ f1 exp - n1 γ f2 exp - n 2 γ , f1 f2 1 1 hγ Doi-theory 1 a γ2
57 Hardware
Stress and strain rheometer, typical types of construction:
motor, Volt + Amp. torque, (t)
optical encoder, (t) disc or position sensor (capacity) stress-rheometer, stress is given, strain measured air bearing strain is measured via optical encoder controlled stress is imposed sample geometry, e.g. plate-plate
frame
rigid spring, deflection < 1°, otherwise problem with Bingham fluid
disc optical encoder or position sensor
air bearing strain-rheometer (A), strain is given, stress measured plate-plate
disc position sensor, (t) nominal actual value comparison, feedback loop motor, (t)
magnetic suspension, seal + normal forces stress detection
position sensor force rebalance transducer (FRT) feedback: “stand still” e.g. our ARES: 2K FRT N1 (2K = 2000 gcm, motor acts as rigid spring N1 = normal forces can be measured)
strain-rheometer (B), air bearing
ARES-type sample ball (cheap) or air bearing (expensive); for normal forces air bearing needed
position sensor strain application
feedback (t) motor
58 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
810-8 rad 0.08 mm ! 1 km 1 1 Alternative: capacitive encoding: ┤ ├ C ω d LC LC d dynamic range of transducers: newest: 1K FRT N1 (Rheometrics, also Haake) 6 -1 -7 Nmax / Nmin = 10 , e.g. 10 Nm to 10 Nm!
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 €
59 Typical pathway of a signal from the torque transducer to G’, G”
Torque transducer: analogue filter “smoothing” V average V “low-pass filter”
0 e.g. integration or 0 t t “oversampling”
Dynamic range of V V “zero-frequency-artefact” ADC adjusted
autorange Autobias a 0 a 0 t t “NMR: RGA” (receiver gain autorange)
Imax dynamic range
4 bit ADC, Imin
0 24 = 16 slots t
dwell-time
ADC: discrete in time ( dwell-time) and in intensity (k-bit ADC, 2k slots)
60 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
position signal
corrections + geometric factors G’, G” cross correlation or Fourier-Transformation torque signal
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 Rv ( Bingham fluid) in concentrated suspensions, greases or food (yoghurt!); especially if the history of loading should be avoided. Lv Approximation:
3 L v 2 Tm 2π R v σ y R v 3
Tm: torque maximum
y: yield stress
61 Melt-flow index cheap + robust version of a capillary rheometer (see later) - uncontrolled, non-homogeneous flow M - relative measurement (“index”) typical parameters: 9.57 mm T = 190°C condition “E” M = 2.16 kg 0.8 mm pressure 3 105 Pa MFI: flow of polymer in [g] per 10 min 0.209 mm
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:
oscillatory / vibrational
rotational capillary
elongational
processing
γ [s-1] -4 -2 0 2 4
62 Set-up: constant force or constant velocity M, v
d pressure sensors L d:L 1:30 for steady state, developed streamlines
ds die swell, extrudate swell normal forces
m(t) mass
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
L0 2
3 1
L(t)
dx v(t) ε x , ε :stretch rate dt 1
63 if specimen is stretched with constant rateε dx ε x1 dt L 1 t dx ε dt ; ε const. x L0 0 L ln ε t L0
Hencky-strain, sample length L et !
[Hencky worked for many years in Mainz-Gustavsburg! See Appendix B, p. 115 - 123]
σ tensile viscosity: η E E ε η without proof: lim E 3 for simple liquids ε0 η0 Trouton’s ratio
experimental apparatus: B A
M sample
oil: + compensates gravity sample thickness + elongation + temp. control monitored via camera - can act as plasticizer in the sample Prof. Meissner Prof. Münstedt Zürich Erlangen ex BASF ex BASF
64 Muenstedt, Laun, Rheol. Acta, 18, 492, 1979
65 Rheology on two specific examples: polymers and dispersions
Polymers
End-to-end distance R , bond length b, N monomers n R ri i1 Gauß: R R R 2 b N R 2 b2 N 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
R simplified model: R g 6
Reptation theory basic idea: one-dimensional stochastic process of chain along contour (reptate: reptile)
simplified tube with diameter d and other chains are static, s typical distance of other chains: s d typical d 30 - 80 Å d one-dimensional Fick-equation, for chain distribution probability P 2P D t 1d x 2
66 Solution for P(x, t): Gauß-statistics x 2 Gauß: 2σ 2 1 x 2 Px,t exp σ 2 r 2 2nDt 4π D1d t 4D1d t n: dimensionality Mean square displacement ( second moment)
x 2 x 2P x,t dx 2D t 1d
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:
kT kT D M 1 1d ξ ξ N
The time needed to diffuse along L will allow a fully different conformation, so that all memory of the other chains (static) is erased
2 L λ 2D1d L2 M 2 λ M3 1 2D1d M
λ M3
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!).
r2 2nDt , n : dimensionality, here : 3 t
2 R M 2 Ds M 6 λ M3 R
2 Ds M
67 assuming a Maxwell-model: η λ G η λ G with λ M3 G M 0 molecular weight independent given by temporary entanglements, “mesh-length”
Ml
3 3.4 ηPolymer M , DeGennes 1971, exp.: ηPolymer M
1 for non-entangled: ηPolymer M friction of polymer-contour
log Mc 3 Me 3.4 entangled
“3 fingers needed to hold a stick” 1
log M 1/10 Mc Mc
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, R e n 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
68 Typical shape for G’(), G”() for monodisperse linear polymer melts
I II III IV log G
G’p
1 G” 2 tan G’ minimum
= 1 log Maxwell-model related length scales:
Rg, 10-50 nm 5 - 10 nm 2-3 nm R , distance between e glass transition entanglements related time scales:
d R e s
(d: disengagement, R: Rouse time, e: entanglement, s: segmental motion)
Zone I: G ω2 see Maxwell-model G ω1 flow-zone, viscosity and the dissipation dominates response length scale probed Rg longest relaxation time: = 1 for tan = 1 (Maxwell-model) or
extrapolated crossing point log G’, for G ω2 and G ω1 log G”
d = 1 log
69 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:
5-10 nm
It is possible to calculate from G’p the entanglement molecular weight: ρR T M e Gp : density R: Gas constant T: temperature
Assuming typical values for a polymer melt:
3 = 1,000 kg/m , R = 8.3 J/(mol K), T = 450 K, Me 150 70 g/mol = 10.5 kg/mol
1,0008.3450 kg J K mol G , 1J 1Nm p 10.5 m3 molK kg 5 Nm Gp 3.510 Pa m3 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
3 3 Mn does not affect G’p, but η M n , λ M n ; so increase in molecular weight by factor 10 103 shift in for plateau length;
70 Strobl, The Physics of polymers
at 3 Me we start to see plateau
increase in Mn by 100 shift in by 1003 = 106
71 plateau length (width)
log G’ Mn2
log G” Mn1
105
Mn2 10 Mn1
3 decades
22 = 1 11 = 1 log
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 ηγ η*ω , Cox-Merz), typical shape:
log 3.4 η0 M n
typical slope for linear polymers: - 0.8 0.1
γ λ 1 log γ or
: longest relax- log ation time
72 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: E (1) η T η exp a 0 RT TVF 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: T a) VF 1 T 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): η T η 1 1 : ; T1 T2 , T2 : ref. temp. ηT2 η2 η ω 1 η 1 2 ; η τ ; τ Maxwell η2 ω1 ω G : characteristic “frequency” of motion
73 η 1 f η1 T1 η2
2: fixed value at reference temperature T2 (not defined yet)
ω E log 2 log f a loge ω1 RT1 TVF
=: - C1 (no units!) 0.434
E a loge : C C ; C : unit of temp. ! R 1 2 2
TVF : T2 C2 ; choice of T2 will change C1 and C2!
ω C C - C T T 2 1 2 1 1 2 log -C1 ω1 T1 T2 C2 C2 T1 T2
only diff. to ref. T2 important!
ω - C T T 2 1 1 2 WLF-equation log : log a T ω1 C2 T1 T2 shift factor
If we choose the reference temperature as T2 = Tg (other choices also possible!): ωT - C T T log g 1 1 g ω1 C2 T1 Tg
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))
C C R 1 apparent activation energy: E 1 2 17.5 kJ/mol for 0 a loge T
74
Prefactor A – B stretch, IR frequency typically 1012 - 1014 1/s log
Arrhenius
slope ~ apparent activation energy,
differs as a function of temperature! -1
0 1/Tg 1/T
singularity at: Tg - C2
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
12 –CH3 10 Hz (at room temperature)
0.1 17.4T T 17.4T limlog g 17.4 T ω 51.6 T Tg 51.6 T T2 Tg
C1 is related to prefactor
0.1 1017.4 ω ω ω 1016.4 ; ν 2π 15.6 ν 10
further: TVF = T2 – C2 T2 = Tg
TVF = Tg – 51.6 K
75 WLF curve defined via: 1) axis intercept Arrhenius log 16.4 1 Ea = 17.5 kJ/mol 2) slope lim E a T T
3) singularity at TVF
-1 C2!
1 1 1 1 T Tm T T g VF if crystalline
singularity
so: Tm > Tg > TVF rule of thumb for polymers: T 2 g (in Kelvin!, absolute energy scale) Tm 3
Tg – C2 = TVF (C2 50 K)
0.1 17.4ΔT log a T log ω(T) 51.6 ΔT
T2 = Tg:
T aT (T) [rad/s] 0 1 0.1 3 deg 1 decade change in 5 10-1.5 3.5 mobility, close to Tg 10 10-2.8 60 20 10-4.8 7103 30 10-6.4 2.5105 50 10-8.6 3.6107 15 deg 1 decade change 100 10-11.5 31010
don’t take this table to literally!
76 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?!
77 Zero-shear viscosity as a function of solid content: Einstein 1906: (see Appendix C, p. 124-144 for original work)
η ηs 1 2.5 for 0.1 Idea:
s : viscosity solvent : volume fraction lit.: A. Einstein, Ann. Physik, 1906, 10, 289 1911, 34, 591 flow field, hard, rotating particle General behaviour for higher concentrations:
2 2 η ηs 1 2.5 O ... , O : not defined yet
Intrinsic viscosity (relative change of viscosity normalised to solvent viscosity): η η η s ηs limη 2.5 , Einstein coefficient 0
Extension of Einstein, O(2) Batchelor (1977) η shear 1 2.5 6.2 2 O 3 ηs ηt extension 1 2.5 7.6 2 O 3 ηs 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):
a c a c b b
a = b < c prolate a < b = c oblate a In both cases: aspect ratio: 1 c
78 High aspect ratio more excluded volume (see liquid crystals: Onsager theory)
Zero-shear viscosity as a function of volume fraction (no information about ηγ !) for higher fractions: d η ηs 2.5 ηs d
(1) dη 2.5 ηs d At certain volume fraction the addition of d leads to an increase in d that is expected to be: dη 2.5 η d η dη 2.5 d η ηs 0 η ln 2.5 0 ηs
η ηs exp2.5 Ball, Richmond 1980
5 25 2 2 Taylor: η ηs 1 ... too low increase in ! 2 2!4
Other idea: Addition of small amount of particles d to the volume fraction (1-) of remaining fluid, raises volume fraction by: d
1 in analogy to (1): 5 d dη η 2 1 dη 5 d η 2 1 2.5 2.5 η 1 1 2.5 ln ln ln ln1 ηs 1 1 0 2.5 η 1 2.5 1 , singularity at 1 not physical ηs 1
79 If we assume a maximum filling fraction m, we find:
-2.5 η m 1 , Krieger-Dougherty (1959) ηs m
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!
80 Best theoretical equation for ():
1 3 9 η m Frankel and Acrivos, with m 0.62 - 0.64 1 ηs 3 81 m experimental: η 1 2.5 10.05 2 2.7 103 exp16.6 ηs
Zero-shear-(rate)-viscosity can drastically be influenced by multimodal distribution:
e.g. r1 : r2 = 5 : 1 1 η η ! 60% pure2 50 50:50 total
ηpure2, 60% η 60% 215% 1 15% increase in solid content, same viscosity picture:
filling of voids
Viscosity as a function of shear-rate for colloids:
st 0, 1 Newtonian plateau log scaling law Shear thickening, layered structure
log γ 2nd Newtonian plateau
γ c : critical shear stress
81 Scaling-law behaviour (+2-Newton) can be described by: - Ostwald de Waele ηγ A γ -B , B[0,1]
- e.g. Ellis-model, using the Péclet-number as universal parameter
η η 1 p η0 η 1 b Pe
6 σ r3 η γ r3 Pe s kT kT
The critical shear-rate can be estimated for a 0.50 = mixture:
nm2 d2 γ 107 ; d [nm] , γ [1/s] c s c
γ c [1/s]
5 10 e.g.: d = 100 nm, γ 103 1/s c 103
101
10-1
10-3
101 102 103 104 105 particle diameter [nm]
Understanding of related forces F(x) and potentials ( ˆ energies) Fdx Vx for colloidal particles:
Vtotal = Vvan-der-Waals + Velectrostatic + Vdepletion + Vsteric
DLVO-theory
DLVO: Derjaguin – Landau – Verwey – Overbeek (1941 + 1948)
82 Van-der-Waals:
Attractive force between atoms, molecules and particles caused by induced electrical dipols of electron cloud. 1 Permanent dipol interaction Vr , r3
2 1 1 induced Vr , minus prefactor, because attractive! r3 r6
For two quadratic surfaces with side length L one finds: A Vr L2 12π r2 A: Hamaker constant [energy]; typical value: 0.4 - 4 10-19 J
For Lennard-Jones potential, also short range repulsion Vr f r , for hard spheres, one typically finds: 1 1 Vr a b r6 r12
Velectrostatic:
q1 q2 q1 r q2 Fr 2 4π ε0 εr r Coulomb
ρ 1st - Maxwell-equation: E , : charge density ε0 εr E Ve , Ve: electric potential ( F E q; W Ve q ) in spherical coordinates: 1 d2 1 (1) ΔV 2 r Ve ρ r dr ε0 εr
83 Approximation of via a Boltzmann-distribution of screened Coulomb potential (single ions):
q Ve r q Ve r ρr ρ0 exp ρ0 1 kT kT homo q V ρ ρ e V using (1) eff 0 kT e
2 1 d 2 ΔVe r Ve χ Ve Eigenvalue problem r dr2 ( Quantum mechanics Hˆ ψ E ψ ) solution: q Ve r exp χ r 4π ε0 εr r no screening screened potential, screening length1/: Debye length
exp(χ 2 ) ρr exp χ r 4π r
1/ : equivalent to Bohr-radius in H-atom, first Laguerre polynom
1 0.304 rD [nm] for 1:1 electrolyte, c in mol/litre χ c
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
84 W Fdx l b e e kT dx , RT = 2.4 KJ/mol , r = 80 2 4π ε0 εr x
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:
4π ε a 2 ψ 2 V 0 exp χ r for a < 5 e r
2 Ve 2π ε a ψ0 ln1 exp χ r 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
forces acting!
areas not accessible for polymer
mixed flocculated
85 Vsteric:
particle particle
non charged surfactants or polymers
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)
86
Book: Israelachvili
87 Fourier-Transform-spectroscopy
[Joseph Baron de Fourier (1768-1830), mathematician and physicist]
free induction decay (FID)
pulse time
Ta
FT, Fourier transform
1 0 ν a Ta
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).
88 Note: This method is of special importance in NMR, IR, X-ray, neutrons, QM, ... and Rheology!
Math:
Fω f t exp i ω t dt complex! complex (can be separated in cos + sin)
real + imaginary part or magnitude + phase
This operation is reversible! (one-to-one) 1 f t Fω exp i ω t dω 2π
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)
FID
FID
t 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): 1 SW 2 DW
89 Some important mathematical relations
1) FT is linear a f t b g t a F ω b Gω i.e. the different signals can be detected independently! (“proof”: A BC AC BC , linearity of the integral)
2) The point in time t = 0 is proportional to the whole integral of the absorptive spectrum.
Proof: f t F ω exp i ω t dω
f t 0 F ω exp i ω 0 dω F ω dω
A FT A B B
t 0 b a
3) Timesignal and spectrum are inverse to each other with respect to the full width at half maximum. (units: s, 1/s !)
pulse t
FT same integral, because FID(t=0) is the same!
0
Heisenberg’s uncertainty principle: t E h E = h t 1
90 4) important Fourier-pairs:
a) exponential FT Lorentzian
exp(-t/) (cp. Relaxation etc.)
t 1 exp exp i ω t dt exp i ω t dt 0 τ 0 τ 1 i ω 1 1 1 exp i ω t τ 1 1 1 i ω τ i ω i ω τ 0 τ τ 1 i ω τ 1 1 ω2 ω2 τ2 τ2
real part imaginary part
absorptive dispersive
real (absorptive)
1
T2 π
0
imaginary (dispersive)
Magnitude: Re2 Im2 “broader feet” ( test it! MC)
b) box FT sinc
1
FT
-t0 0 t0 cp. single slit diffraction pattern -x x q-vector 91 proof:
t t0 1 0 exp i ω t dt exp i ω t i ω t0 t0 1 exp i ω t exp i ω t i ω 0 0 1 cosω t i sin ω t cos ω t i sin ω t i ω 0 0 0 0 1 sinω t 2 i sinω t 2 0 i ω 0 ω sinc()
remark : expi ω t cos ω t i sin ω t ; exp i π 1 0 (Euler)
FT c) Gaussian Gaussian (without proof)
t 2 ω2 σ 2 exp exp i ω t 2π σ exp t 2 t 2 2σt
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: Convolution
with a box- S/N 5/(1/3) =15 S/N 5/1 function but: broader peak! S
N N 1/3
averaging, new function
92 Appendix A Phil. Mag. 32 (5. Series), 424 (1891)
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123 Appendix C
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