### 旋转流变仪及其测量技术简介----张宝庆

```旋转流变仪及其

2
Linear Viscoelasticity
Viscoelastic relaxation modulus of flexible linear polymers.
3
Polym J. 2009, 41(11), 929.
Stress Relaxation (Transient Test)
strain g
unit area
 = G(t) g
unit
height
G t 

g
4
Superposability of Stress
Just for g1
1(t) = G(t-t1)g1
g
g2
g1
t1
t
t2
Just for g2
2(t) = G(t-t2)g2

1
t1
2
t2
1+2
t
For g1 + g2
1+2(t) = 1(t) + 2(t)
= G(t-t1)g1 + G(t-t2)g2
5
Boltzmann Principle
For infinitesimal strain dgi at time
g
 G t - t g
 (t ) 
i
i
i
dgi
t
dt'
for strain g(t) of arbitrary history
 (t ) 

di

t
-
G ( t - t ' )g ( t ' ) dt '
g ( t ') 
t
ti
t
d g ( t ')
dt '
6
Boltzmann Superposition Principle
The principle of linear superposition of stresses and/or deformations：
• The response to any event is linear；
• All consequent events lead to independent responses.
The material reacts to the next action as if no former action
took place!
Rheology: Concepts, methods and applications. Page 61.
7
Linear Viscoelasticity (Oscillatory Shear)
Input
g ( t )  g 0 sin( t )
 ( t )   0 sin ( t   )
Output
G
*
 0 /g0
G ( )  G  ( )  iG ( )
*
G   G sin 
G   G cos 
*
*
tan   G  G 
 ( )   ( ) - i ( )
*
G   G 
2

*

2
η*: complex viscosity

  ( )  G  ( ) / 
  ( )  G  ( ) / 
8
Frequency Defined
g ( t )  g 0 sin(  t )
Test Input: strain (g), frequency (), and gap (H).
Measure: torque (M) and phase angle ().
9
Frequency Sweep
PDMS
1.000E6 1.000E5
1.000E6
1.000E5
1.000E5
10000
10000
10000
|n*| (Pa.s)
1000
100.0
100.0
10.00
G '' (Pa)
G ' (Pa)
1000
10.00
1000
1.000
1.000
0.1000
0.1000
PDMS Extended frequency sweep-0001o, Frequency sweep step
0.01000 100.0
1.000E-51.000E-41.000E-3 0.01000 0.1000
frequency (Hz)
1.000
10.00
0.01000
100.0
The amplitude of the perturbation can be freely chosen for each frequency,
and dynamic modulus measurement is so far the most common method of
10
linear viscoelastic characterization currently.
Stress Relaxation vs. Frequency Spectrum
G(t) vs. t
G'(ω) vs. ω
A is monodisperse with M<Mc; B is monodisperse with M>>Mc and C is polydisperse
LVE response is very sensitive to the molecular structure of the polymers
11
Dynamic
Compliance
J*(ω)
Algebraic
Equations
Fourier
Transforms
Creep
Compliance
J(t)
Fourier
Transforms
Integral
Equations
Laplace
Transforms
Retardation
Time
Distribution
L(τ)
Dynamic
Modulus
G*(ω)
Relaxation
Modulus
G(t)
Laplace
Transforms
Integral
Transforms
Relaxation
Time
Distribution
H(τ)
Polymeric liquids and networks – Dynamic and rheology. Page 122.
12
Time-Temperature Superposition (TTS)
WLF (Williams-Landel-Ferry) equation
log a T 
C 1 (T - T g )
C 2  (T - T g )
13
Time-Temperature Superposition (TTS)
Thermorheologically simple
Master curve of the linear viscoelastic moduli
 Ea 1
1 
a T (T )  exp 
( - )
 R T T0 
J Rheol. 2011, 55(5), 987.
bT  T 0  0 T 
14
Creep – Creep Recovery
J (t )  g (t )  0
Recoverable
Non-Recoverable
Principle of a creep-recovery experiment
J Rheol. 2014, 58(3), 565.
15
Dynamic
Compliance
J*(ω)
Algebraic
Equations
Fourier
Transforms
Creep
Compliance
J(t)
Laplace
Transforms
Dynamic
Modulus
G*(ω)
Fourier
Transforms
Integral
Equations
Relaxation
Modulus
G(t)
Laplace
Transforms
Retardation
Relaxation
Integral
Time
Time
Transforms
Distribution
Distribution
L(τ)
H(τ)
Polymeric liquids and networks – Dynamic and rheology. Page 122.
16
Prog Polym Sci. 2001, 26(6), 895.
17

18
ARES
ARES-G2

（SMT）
Separate Motor and Transducer
AR-Series
Hybrid-Series
Aton Paar
Malvern

（CMT）
19
FRT
Torque
Measurement
is Unaffected
by Motor
Inertia &
Friction
Motor/
Transducer
Primary
Moving
Elements
Motor Inertia
& friction
Involved in
Torque
Measurement
Motor

（SMT）
20

（CMT）
20
Strain vs. Stress controlled
Strain Controlled
•
•
Good for oscillatory
measurements
Good for fixed shear rate/strain
measurements (Stress relaxation)
•
Motors are really good - good for
weak materials
•
Very sensitive torque transducers
Stress Controlled
•
•
OK for oscillatory measurements
•
•
Good for creep measurements
•
EC motors often have more
inertial effects
•
Often assumes certain type of
material response
Good for fixed stress
measurements
Drag cup motors often cannot do
low stresses well

21
 Torque range
（扭矩范围）
 Angular Resolution
（角位移分辨率）
 Angular Velocity Range
（角位移速率范围）
 Frequency Range
（可测频率范围）
 Normal Force
（法向力范围）
 Motor type
（驱动马达类型）
22

g ( t )   g 0 , g 1 ,   , g n ,   , g N -1 
 ( t )   0 ,  1 ,   ,  n ,   ,  N -1 
From the time into the frequency domain
Discrete Fourier transformation (DFT)
g
1
N

1
N
g
*

2
2
g   g 
N
 g n cos(
- 2 n
n0
N
N
- 2 n

n
cos(
 g  arctan(



*
g
*
g
N
  
)
N
n0
G
g  
)
N
1
g
)

*

sin(
1
N

n
sin(
    
- 2 n
)
N
n0
2
)
N
n0
N
g 
n
- 2 n
2
   - g
   arctan( 
23

)
S /N
n
1
2
24
Testing Geometries
Parallel Plates
○ 用量少(~0.5 - 3 mL)
○ 非均匀应变
○ 制样简单
○ 可用于变温测试
○ Gap可变，用于界面滑移

○ Gap可变，shear rate随之

Cone Plate
Concentric Cylinder
Single/double-gap
○用量少(～ 1 mL)
○ 均匀应变 （真实粘度)
○ 第一法向应力差测试
○ 不适用于较大粒子的分散体

○ 对间距设置更敏感
○ 不适用于变温测试
○ 高粘度流体制样有困难
○ 适用于低粘度样品
○均匀应变场
○样品用量大(~9 mL)
○清洁困难
○末端效应校正
25
Testing Geometries
M (扭矩) — τ (应力)，ω(角速度) — g (剪切速率）
26
Extensional Viscosity Fixture (EVF)
0 
L 0 w 0 h 0  L 0 A0  const .
 E (t ) 
F (t )
A (t )
1 dL ( t )
L
dt
A ( t )  A0 e

F (t )
A0
e
 0t
-  0t
 E (t ) 
h ( t )  h0 e
 E (t )
0
27
1
-  0t
2

28

29
Rheological Measurements
 Oscillation tests





Frequency sweep
Time sweep
Strain/stress sweep (LVE)
Temperature ramp
Temperature/Frequency sweep
(TTS)
 Fast Sampling
 Multiwave
 LAOS
 Strain-Rate Frequency
Superposition (SRFS)
 Flow tests
 Constant shear rate
 Continuous stress/rate ramp and
down
 Steady state shear rate sweep
 Flow temperature ramp
 Flow reversal
 Transient tests
 Stress relaxation
 Creep & creep recovery
 others
 Elongational test
30
Slow Relaxation Behavior of Linear Chains
7
10
relaxation time t ~ M3.4±0.2
o
T0 = 40 C
10
6
6
G " (P a )
G' (Pa)
10
410K
207K
100K
44K
5
10
10
10
5
Ze262
Ze132
Ze63
Ze28
4
4
10
-2
10
-1
10
0
10
1
10
2
10
3
10
4
10
5
10
6
10
7
10
8
10
9
10
10
-2
10
-1
10
0
10
1
10
2
10
3
10
4
10
5
10
6
10
7
 (ra d /s)
Delay of orientation/stress relaxation due to
entanglement of uncrossable chains
31
10
8
Slow Relaxation of Star-branched Chains
PBD:
Linear Mw=160K
6-arm star Ma=77K
Relaxation time
t ~ exp(0.6Marm/Me)
Much stronger delay
for star chain
cf. t ~ M3.4±0.2
for linear chain
32

Example for the extension of the frequency range using the retardation
spectrum obtained from creep-recovery tests (recover time up to 104 s).
J Rheol. 2014, 58(3), 565.
33

UHMWPE
DFreq
SR
ARES-G2
Fourier
Relaxation Transforms Dynamic
Modulus
Modulus
G(t)
G*(ω)
34
Oscillation Time Sweep
Re-entanglement kinetics of freeze-dried polymers
(a) Buildup of modulus in polystyrene samples with time.
(b) Equilibrium entanglement time of samples freeze-dried from solutions with different
original concentrations.
35
Macromolecules. 2012, 45 (16), 6648 .
Oscillation Time Sweep
Effect of thermally reduced graphite oxide (TrGO) on the polymerization
kinetics of poly(butylene terephthalate)
Polymer. 2013, 54 (6), 1603.
36
Multiwave Oscillation
 The total strain amplitude should not exceed the linear viscoelastic
regime
 The test time is the same as the dynamic single point experiment under
the fundamental frequency
37
Multiwave Oscillation
Evolution of the loss tangent during a curing reaction. The gel point
is the point, when tan δ becomes independent of frequency.
38
Oscillation Temperature Ramp
39
Oscillation Temperature Ramp
Phase separation temperature of polymer blends
PS/PVME
with big difference in Tg
PB/PI
with big discrepancy
in viscoelasticity
Miscible Metastable Phase-separated
Dynamic temperature s ramp for a 50:50 PS
38K/PVME-23K blend
J Phys Chem B. 2004, 108 (35), 13220.
40
Physics Today. 2009, 62(10), 27.
41
Shear Reversal
Results of flow reversal studies of a 4.80 wt %
PP/clay hybrid nanocomposite.
Macromolecules. 2001, 34 (6), 1864.
42
Elongational Test-1
Polylactide with long-chain branched structure

Strain-hardening coefficient:
 E (t ) 

0
 (t ) 
 E (t ,  )

3 0 ( t )
N

 i G i (1 - e
- t i
)
i 1
Ind Eng Chem Res. 2014, 53(3), 1150.
43
Elongational Test-2
(a) Chewing and (b) bubble gum behavior during start-up of uniaxial extension
J Rheol. 2014, 58(4), 821.
44
Prog Polym Sci. 2001, 26(6), 895.
45
The Rheology Handbook-For Users of Oscillatory Rheometers ( 3rd
ed.)
Thomas G. Mezger
2013
Structure and Rheology of Molten Polymers:
From Structure To Flow Behavior and Back Again
John M. Dealy , Ronald G. Larson.
2006
46
Melt Rheology and Its Applications in the Plastics Industry
John M. Dealy , Jian Wang
2013
Colloidal Suspension Rheology
Norman J. Wagner, Jan Mewis.
2012
47
Viscoelastic Properties of Polymers (3rd Revised)
John D. Ferry
1980
Rheology: Principles, Measurements, and Applications
Ch. W. Macosko
1994
48
 Journal of Rheology
 Rheologica Acta
 Journal of Non-Newtonian Fluid Mechanics
 Applied Rheology
 Korea-Australia Rheology Journal
 Nihon Reorogi Gakkaishi (Journal of Society of Rheology Japan)
 Macromolecules
 Langmuir
 Soft Matter
 Physical Review Letters
 Physical Review E
 Journal of Chemical Physics
49
50
51
52
53
Rheology needs a lot of experience. Modern rheometers will give
you numbers, no problem, but the question is always whether they
are correct. That and the optimization of the parameters to minimize the noise and do what you want to the material (destroy or
not destroy a structure) is what sets a good rheologist apart from
an inexperienced one.
54
```