Driven torsion pendulum for measuring the complex shear modulus in a steady shear flow |
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Authors: | Dr D van den Ende J Mellema C Blom |
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Institution: | (1) Rheology Group, Faculty of Applied Physics, University of Twente, P.O. Box 217, 7500 AE Enschede, The Netherlands |
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Abstract: | To investigate the viscoelastic behavior of fluid dispersions under steady shear flow conditions, an apparatus for parallel superimposed oscillations has been constructed which consists of a rotating cup containing the liquid under investigation in which a torsional pendulum is immersed. By measuring the resonance frequency and bandwidth of the resonator in both liquid and in air, the frequency and steady-shear-rate-dependent complex shear modulus can be obtained. By exchange of the resonator lumps it is possible to use the instrument at four different frequencies: 85, 284, 740, and 2440 Hz while the steady shear rate can be varied from 1 to 55 s–1. After treatment of the theoretical background, design, and measuring procedure, the calibration with a number of Newtonian liquids is described and the accuracy of the instrument is discussed.Notation
a
radius of the lump
-
A
geometrical constant
-
b
inner radius of the sample holder
-
c
constant
-
C
1, C
2
apparatus constants
-
D
damping of the pendulum
-
e
x
, e
y
, e
z
Cartesian basis
-
e
r
, e
, e
z
orthonormal cylindrical basis
-
E
geometrical constant
-
E
t
, 0
E
t
,
t
relative strain tensor
-
f
function of shear rate
-
F
t
relative deformation tensor
-
G
(t)
memory function
-
G
*
complex shear modulus
-
G
Re(G
*
)
-
G
Im(G
*
)
-
h
distance between plates
-
H
*
transfer function
-
,
functional
-
i
imaginary unit: i
2= – 1
-
I
moment of inertia
-
J
exc
excitation current
-
J
0
amplitude of J
exc
-
k
* = k – ik
complex wave number
-
K
torsional constant
-
K
fourth order tensor
-
l
length of the lump
-
L
mutual inductance
-
M
dr
driving torque
-
M
liq
torque exerted by the liquid
-
0
M
liq,
liq
steady state and dynamic part of Mliq
-
n
power of the shear rate
-
p
isotropic pressure
-
Q
quality factor
-
r
radial position
-
R,R
0, R
c
Re(Z
*, Z
0
*
, Z
c
*
)
-
s
time
-
t, t
time
-
T
temperature
-
T, 0
T,
stress tensor
-
u
velocity
-
U
lock-in output
- 0
velocity
-
V
det
detector output voltage
-
V
sig, V
cr
signal and cross-talk part of V
det
-
x
Cartesian coordinate
-
X , X
0, X
c
Im(Z
*, Z
0
*
, Z
c
*
)
-
y
Cartesian coordinate
-
z
Cartesian coordinate, axial position |
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Keywords: | Superimposed oscillatory shear complex shear modulus torsion pendulum nearly viscometric flow |
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