This presentation was given in Cardiff at the European Society of Rheology Conference in 2009. The presentation is about research in "extreme" areas of rheology and includes work on measuring the viscoelasticity of low viscosity fluids and the limiting extensional viscosity of high viscosity fluids.

1. ESR Cardiff
April 2009
Extreme Rheology; beyond normal boundary conditions
by
Malcolm Mackley,
With acknowledgement to
Damien Vadillo, Tri Tuladhar* Dietmar Auhl **, Lino Scelci and Tim Lord
Department of Chemical Engineering and Biotechnology, Cambridge
*Xaar plc
** University of Leeds
mrm5@cam.ac.uk
1

2. Areas of extremes!
• Linear Viscoelasticity
Low viscosity, weakly viscoelastic fluids
example; ink jet fluids.
• Non Linear viscoelasticity
a) Low viscosity, weakly viscoelastic fluids
example; ink jet fluids
b) High viscosity, limiting large strain, extensional viscosities
example; polymer melts
2

3. WLF; The frequency domain escape route
http://web.mst.edu/~wlf/Mechanical/timetemp.html
3

4. Xaar DOD Printhead
Platform III : Side shooter
Multipulse grey scale printhead (1001 series)
4

5. DOD drop formation
1% PS70
Eff: 1.08
30 micron drops, ms timescale
Photo, courtesy of Dr Steve Routh
5

6. The Cambridge MultiPass Rheometer (MPR)
Pressure variation Rheology flow Cross-slot Filament stretch
mode mode flow mode mode
6

7. The Cambridge Multipass Rheometer (MPR4)
Top
section
Test
section
Bottom
section
7

8. MPR as capillary rheometer
Diethyl phthalate (DEP) Supplier: Sigma Aldrich
BP = 294-296°C; ρ = 1118 kg/m3 ;
σ 20°C = 36 mN/m; η25°C = 10 mPa.s
Polystyrene: Supplier: BASF – Polystyrol VPT granule
M.W ~ 195000
100
ARES data MPR data
90
Apparent viscosity, η (mPa.s)
80
70 DEP
DEP + 1.0 wt% PS
60
DEP + 2.5 wt% Ps
50 DEP + 5.0 wt% PS
40
30
20
10
0
1 10 100 1000 10000 100000 1000000
Shear rate (/s)
8

9. Measurement of Linear Viscoelasticity (LVE)
Piezo Axial Vibrator (PAV)
Developed by Prof Wolfgang Pechold
University of Ulm. Germany
Upper lid
Sample
Gap (steel ring foil)
Lower plate with
overflow ditch
Probe head
Piezoelectric (PZT)
elements stuck on a
square copper tube
Section of PAV
9

10. Mechanical equivalent model of PAV
2R The lower plate oscillates with force F (∝
m1
x1
excitation volt Uref) for a given frequency.
K* Sample d
With blank test: Dynamic compliance of the
m0
x0 lower plate is measured.  ∆x  ~ U eiδ
K1 K1
  0
2 2
 F 0 U ref
K01 F
With sample: Modulated compliance of the
x2 *
K02 m2 sample is measured  ∆x  U iδ
  ~ e
 F  U ref
Mechanical equivalent
model of the PAV. Complex squeeze stiffness K* of the material can
be calculated from the ratio of ∆x0 and ∆x*.
K02 K01 K* K1
m2 m0 m1
For linear viscoelasticity
2 d3  ρω 2 d 2 
F G* = K * 1 +
 + ....

x2 x0 x1 3π R 4  10G * 
G '2 +G"2
Mechanical representation G * (ω ) = G ' (ω ) + iG" (ω ) η* =
10 ω
with springs.

11. High frequency linear viscoelastic data of DEP-10% PS210 at 25°C
Parallel plate rheometer
1000 10000
η*
Complex viscosity, η*, (mPa.s)
Elastic (G') and Viscous (G")
1000
100
modulus, (Pa)
100
10
10 G"
1
G'
1 0.1
0.1 1 10 100 1000 10000
Frequency (Hz)
11

12. High frequency linear viscoelastic data of DEP-10% PS210 at 25°C
Parallel plate rheometer PAV data
1000 10000
η*
Complex viscosity, η*, (mPa.s)
Elastic (G') and Viscous (G")
1000
Open: ARES
100 Close: PAV
modulus, (Pa)
100
10
10 G" Mind the gap!
1
G'
1 0.1
0.1 1 10 100 1000 10000
Frequency (Hz)
12

13. Effect of Polymer on the Linear Viscoelastic response of ‘model’
fluid containing different polymer concentration
Polystyrene MW = 210k in Diethyl Phthalate (DEP) solvent
1000 1000
0.1%
100 100 0.2%
Loss Modulus
G’’ 0% Elastic Modulus 0.4% C%
10 G’ 10 1%
Pa 0.1% 0%
0.2% Pa
1 1
0.4%
1%
0.1 0.1
1 10 100 1000 10000 100 1000 10000
Frequency (Hz) Frequency (Hz)
2.00E-02 1
0%
0.1% 0%
Complex 1.75E-02 0.2% Modulus ratio 0.1%
viscosity 0.4%
1%
G’/ G* 0.2%
0.4%
C%
η∗ 1.50E-02 0.1
Pa.s 1%
1.25E-02
1.00E-02 0.01
1 10 100 1000 10000 100 1000 10000
Frequency (Hz) Frequency (Hz)
13
13

14. The Torsion Resonator
(Prof Pechold (again!) ; University of Ulm)
linear viscoelastic behaviour (LVE)
Connector
cylinder
2 shear piezo 2 shear piezo
for detection for excitation
PT 100
Photograph of the Tube shrunk on inner cylinder
Torsion resonator
Schematic (side and top view) of the
Torsion resonator
14
14

15. The Torsion Resonator
(Damien Vadillo)
The linear viscoelastic behaviour (LVE)
Resonance curve of the mechanical response of the TR
4
The piezoelectric sensor oscillates at fs fe
resonance frequencies, 26kHz and 3
De
U (mV)
77kHz respectively. 2
Ds
1
With blank test: Determination of the
apparatus constant temperature 0
correction coefficient for each 25000 25020 25040 25060 25080 25100
Frequency (Hz)
frequency
K  ∆D  2 
 − ( ∆f ) 
2
With sample: Measure the resonance G = '

ρ sample  2 
 

frequency shift ∆f (=fs-fe) and the
damping shift ∆D (=Ds-De) at each K
G '' = ∆D.∆f
resonance frequency. ρ sample
15
15

16. The Torsion Resonator (TR)
(Damien Vadillo)
Proof of concept (DEP + 2.5%wt PS110)
10000
G’’(77 kHz)
G’ and G’’
(Pa)
G’’(25 kHz)
G’(77 kHz)
1000
G’(77 kHz)
100
10 20 30 40 50 0.1
η1 (25 kHz)
Experiment number η1 (77 kHz)
η1 and η2 0.01
η2 (25 kHz) η2 (77 kHz)
(Pa.s)
0.001
10 20 30 40 50
Experiment number
16
16

17. The Torsion Resonator
(Damien Vadillo)
Proof of concept (DEP + 2.5%wt PS110)
PAV TR
10000
1000 G’’
G’ and G’’
100
(Pa)
10
1
G’
0.1 PAV TR
10 100 1000 10000 100000
0.1
Frequency (Hz)
η* (Pa.s)
• PAV and TR match
• “Shear thinning” at high 0.01
frequency 10 100 1000 10000 100000
Frequency (Hz)
17
17

18. Extensional non linear
measurements
a) Medium viscosity fluids
18

19. Filament thinning
A.V.Bazilevsky, V.M. Entov and A.N.Rozhkov
3rd European Rheology Conference 1990 Ed D.R.Oliver
The “Russian Rheotester”
C A
B
E
15 cm
D
19

20. Liang and Mackley (1994)- Extensional Rheotester
Newtonian modelling
•
τzz τ zz = − p 0 + 2η γ zz = 0 (11)
•
τrr
Top plate
τ rr = − p 0 + 2η γ rr = −2σ / D (12)
•1•
D= εD (13)
Bottom plate
2
•
Extensional rheotester τ E = τ zz − τ rr = 3η ε = 2σ / D (14)
• 2σ
ε= (15)
3ηD
•
η E = τ E / ε = 3η (16)
σ
D(t ) = D0 − t (17)
3η
20

21. Liang and Mackley (1994)- Viscoelastic fluid
S1 fluid First approximation
 1  (18)
D (t ) = D0 exp −
 3λ t

 R 
Viscoelastic modelling
•
τ E = 3η ε d = 2σ / D (19)
• • •
τ E = g ε s = −2σ D/ D 2 (20)
• •
PIB solutions
εd = εs (21)
•
D/ D = − g / 3η (22)
 g 
D (t ) = D0 exp − t 
 3η  (23)
 
21

22. Extensional non linear
measurements
b) Low viscosity fluids
22

23. The Cambridge MultiPass Rheometer (MPR)
Pressure variation Rheology flow Cross-slot Filament stretch
mode mode flow mode mode
23

24. MPR Filament stretch Rheometer
Vp
D
R(z,t)
Top Piston
Lf Rmid(t)
L0
Bottom Piston
Vp
(a) Test fluid positioned (b) Test fluid stretched uniaxially (c) Filament thinning and break up
between two pistons. at a uniform velocity. occurrence after pistons has stopped.
t<0 t≥0
24

25. MPR Filament stretching and thinning of DEP solution
DEP
DEP + 5.0 wt% PS
1.2 mm
Piston diameter = 1.2 mm
Initial stretch velocity = 200 mm/s
Initial sample height = 0.35 mm
Final sample height = 1.35 mm
(piston displaced by 0.5 mm each side)
25

26. The CambridgeTrimaster
A dream turning into a reality
Toothed belt Linear guide rail
timing pulley
Carrier
Timing belt
Replaceable top and
bottom plate
Stepper motor
attached to a pulley
26
Graphics courtesy of James Waldmeyer

27. Drive
belt
Piston
Linear
traverse
Motor
drive
a b
High speed camera
Fibre optic light
Cambridge Trimaster
27

28. Piston response
5000
4000 10 mm/s
100 mm/s
Top piston 500 mm/s
position (µm) 3000
2000
1000
c
0
0 100 200 300 400 500 600
Time (ms)
28

29. The ‘TriMaster’ Filament stretch and break up apparatus
piston
sample
belt
pulley
Initial gap ≈ 0.2 mm, Final gap ≈ 1.2 mm
29
Piston diameter ≈ 1.2 mm, Piston velocity ≈ 1 m/s

30. Filament thinning
a
5.3ms 5.8 6.3
6.8 7
7.2 7.7
(a) DEP,
b
(b) DEP + 0.2% PS110,
(c) DEP + 0.5% PS110,
(d) DEP + 1% PS110, 5.3ms 6 6.7
7 7.15
(e) DEP + 2.5% PS110. 7.3 7.6
c
Initial gap size: 0.6mm,
Stretching distance:0.8mm, 5.3ms 6.15 7
7.5 7.65
7.8 8
Stretching velocity:150mm/s
d
5.3ms 7 7.85
8.7 9.6
10.4 10.6
e
5.3ms 8.2 10.2 13.5
15.2
17
16.8 30

31. Mid filament diameter time evolution
1200
1000 0%
0.50%
800 1%
2.50%
D 5%
600
(µ m )
400
200
0
0 10 20 30 40
Time (ms )
31

32. 250
DEP-0%PS
DEP-0.5%PS
200
DEP-1%PS
Trouton ratio DEP-2.5%PS
150 DEP-5%PS
ηE
η0 100
50
0
0 2 4 6 8 10
D 
Hencky strain, 2 ln 0 
D 
 t
The transient extensional rheology of DEP solutions as a function of relaxation Hencky strain
for different PS concentrations.
Initial distance 0.6mm, final distance: 1.4mm.
The line represent are obtained from the exponential fitting of the evolution of the thinning of the diameter.
The geometrical factor “X” is fitted using the experimental data at low Hencky strain.
32

33. Breakup a
0ms 4 5.5
6.2 6.35 6.5
7
DEP, b
DEP + 0.5% PS110,
0ms 5 6.5
DEP + 1% PS110, 7.7 8.2 8.35
DEP + 2.5% PS110, 8.5
DEP + 5% PS110.
tial gap size: 0.6mm, c
etching distance: 1.6mm,
etching velocity: 150mm/s
0ms 6.7 8
9.35 9.85 10.15
10.35
d
0ms 7.5 10.65 14.15 17
17.15 17.35
e
0ms
38
10.7
39.15
22.35 31
39.35
33

34. Extensional non linear
measurements
c) High viscosity fluids
34

35. The Cambridge MultiPass Rheometer (MPR)
Pressure variation Rheology flow Cross-slot Filament stretch
mode mode flow mode mode
35

36. The Cross-Slot
• Generates a hyperbolic
• pure shear flow
pattern
• as shown.
• Near centre.
• Essentially uniform
extensional
• flow with residence
time, which is
equivalent to strain, 36

37. Typical Result
-Dow PS680E
-Piston velocity of 0.5
mm/s (maximum
extension rate =4.3/s).
-Inlet slit
width=1.5mm
-Section depth=10mm
- T=180°C.
37

38. Stagnation Point flows as rheometers
Dr Dietmar Auhl et al,
Leeds University 2008
6
elongational viscosityµ(t), Pas 10
0.3
. -1 1 0.1 0.03 0.01
ε0 [s ]
shear viscosity η(t), Pas
3 0.003
10 0.001
5
10 . -1
γ0 [s ]
0.001
0.01
0.1
4 0.5
10 1
2
5
LDPE
T = 150°C 10
3
10
-1 0 1 2 3
10 10 10 10 10
time t, s
38

39. η E ,st (ε) = (σ xx − σ yy ) st / εst steady-state elongational viscosity
at the stagnation point
0
ε
 =
principle
ε

0
∆ n = SOC (σX xx − σ yy ) + 4σ xy
• 2 2
ε st = A x V piston -4 -2 0 2 4
39

40. Dr Dietmar Auhl et al , Leeds University
40

41. Conclusions
Piezo Axial Vibrator (PAV) and Torsion Resonator (TR)
Can quantify LVE high frequency response of low
viscosity viscoelastic fluids
Cambridge Trimaster
Can follow transient extensional viscosity and
filament break up process of low viscosity fluids
MPR Cross slot
Can measure limiting extensional viscosities of polymer melts
Acknowledgments
EPSRC and industrial partners in Next Generation Ink Jet Consortium
41