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Normal stresses in extremely shear thickening polymer dispersions
ELSEVIER
J. Non-Newtonian Fluid Mech., 54 (1994) 87-108
Normal stresses in extremely shear thickening polymer dispersions * H.M. Laun BASF Aktiengesellschaft, Polymer Research Division, Polymer di Solid State Physics Department, ZKh4 - G201, D-67056 LudwigshafenlRhein, Germany
Received 22 March 1994
Abstract
Measurements of the first and second normal stress differences of an extremely shear thickening polymer dispersion have been performed at imposed shear stress both by means of a Rheometrics RDS and a home-made Normal Force Cell using cone-plate geometry. The sample is a 58.7 vol.% dispersion of styrenelethyl acrylate copolymer particles with 280 nm average diameter in glycol. The spheres are electrostatically stabilized by negative charges due to carboxylic groups. In the regime of strong shear thickening a negative first normal stresses difference N, is found, the absolute value of which is equal to the absolute value of the shear stress T: N, = - 171. Measurements of the normal force acting on a disk of smaller radius than the full plate radius yield for the second normal stress difference N2 the relation N2 = -N, /2. The experimentally observed relation of the apparent shear stress 7a and apparent first normal stress difference N,, in plate-plate geometry, N,, = -17a), can be explained by assuming that only the sample part near the rim is shear thickening whereas the inner part remains in the low viscosity regime. In the latter regime N,, is positive and smaller than 7,. It is further explained why small deviations of the concentricity in Searle geometry create a strong lateral force that pulls the bob towards the cup. Keywords:
Normal stresses; Polymer dispersions; Rotational
rheometry; Shear thickening
* Dedicated to Professor Ken Walters FRS on the occasion of his 60th birthday. 0377-0257/94/$07.00 0 1994 - Elsevier Science B.V. All rights reserved SSDZ 0377-0257( 94)01307-4
88
H.M.
Lam
1 J. Non-Newtoman
Fluid Mech.
54 (1994) 87-108
1. Introduction Concentrated suspensions of nearly monodisperse polymer particles dispersed in a low viscosity Newtonian fluid and stabilized by electrostatic forces exhibit distinct changes of the viscosity with increasing shear rate. Typically, a shearing thinning behaviour is observed at low shear rates. Whereas this low shear rate regime, dominated by Brownian forces, has been investigated widely (e.g. Refs. [l-4]) there exists a much smaller number of publications dealing with the high shear rate behaviour and the occurrence of shear thickening [4-121. A better understanding of the phenomenon of dilatancy is of importance for the design of passively switching mechanical elements like clutches and dampers [ 131. Extremely shear thickening dispersions exhibit a reversible step-like shear stress increase at a critical shear rate which reaches values higher than lo5 Pa! An investigation of the related particle structures in the flowing dispersions was performed by using neutron scattering [14]. The work on normal stresses of the dispersions described in this paper was motivated by observations during measurements using concentric cylinder geometry: it was found difficult to maintain the concentricity of the cylinders in the regime of shear thickening. Strong lateral forces on the bob caused a twist of the driving shaft of our Physica Rheolab MC 20 rheometer when the DIN 24 cup with 14 mm diameter was used.
2. Sample The sample investigated here is the glycol model dispersion C5G5 as described in Ref. [14]. It consists of styrene/ethyl acrylate copolymer particles prepared by emulsion polymerization, styrene being the dominant monomer. Small amounts of mono- and dicarboxylic acids were added to the monomers to incorporate carboxylic groups on the particle surface, a partial dissociation of which causes a negative particle charge in polar fluids. The average particle diameter of latex 18/157 as determined by quasi-elastic light scattering (QELS) is 295 nm [14]. For comparison Fig. 1 shows the integral and differential partical size (mass) distribution measured by means of the analytical ultracentrifuge (AUC) [ 151. The distribution is rather narrow. The peak of the differential distribution yields an average particle diameter of 280 nm. Sample C5G5 was prepared by redispersion of the freeze-dried latex powder in glycol at a solid concentration of 58.7 vol.%.
3. Viscosity function Fig. 2 shows the viscosity vs. shear rate as measured in a Rheometrics Dynamic Stress Rheometer (DSR) at 27°C in a shear stress ramp experiment. Both coneplate (cone angle CI= 0.1 rad, radius R = 12.5 mm) and plate-plate (gap h = 1 mm, R = 12.5 mm) geometries were used. A logarithmic sweep of the shear stress up to 5000 Pa within 3 min was applied. In the low shear rate range a weakly shear
H.M.
Lam
/J. Non-Newtonian
Fluid Mech.
54 (1994) 87-108
89
latex IS/157
0.0 100
150
200
250
diameter Fig. 1. Size distnbution
of the latex particles
300
350
400
[nm]
from the analytical
ultracentrifuge
(AUC)
thinning behaviour is found. At the beginning, with increasing shear stress, the curve remains on a quasi-Newtonian plateau of viscosity ( z 1.2 Pa s) up to a shear rate of about 55 SC’ (shear stress 7.2 Pa). Subsequently the shear rate decreases to the equilibrium critical shear rate j, = 22 s-’ (cone-plate) or rim shear rate compare Eq. (12), below, respectively. ?Lc = 26 SC’ (plate-plate), At r > 7.2 Pa the sample has switched to the high viscosity regime. Here, the average shear rate remains constant while the shear stress rises by about a factor of 40. The upper limit of the viscosity step is about an order of magnitude higher and cannot be measured in the DSR. The observation of a metastable range of low viscosity as well as fluctuations of the equilibrium critical shear rate for an extremely shear thickening dispersion have been described before [ 121. It has also been shown that the critical shear rate depends on the geometry of the gap [ 121. A repetition of the shear stress ramp for plate-plate geometry immediately after the first run reproduced the measurement shown in Fig. 2(b). In the case of cone-plate geometry, however, the low viscosity level was distinctly lower in the second run. An inspection of the sample showed that part of the material has been sheared out of the gap during the application of high torque levels. After refilling the gap the measurement represented in Fig. 2(a) could be reproduced.
4. Limitations
of imposed shear rate rheometers
For normal stress measurements on concentrated dispersions our shear strain or shear rate controlled Rheometrics Dynamic Spectrometer (RDS) equipped with a force rebalance transducer would in principle be the pertinent instrument. On these are only reasonable in the strongly shear thickening samples, however, subcritical regime since the possibility of a large and rapid change of the shear stress (torque) at shear rates 3 2 3, could damage the transducer.
90
H.M.
Lam / J. Non-Newtonian
- Dispersion DSR
Fluid Mech. 54 (1994) 87-108
C5G5
(shear stress ramp)
102 -
- cone-plate _ a= O.lrad
77 GJ
k
R = 12.5 mm T= 27°C
F h .% z 5: ‘5
101 7
100 -
I 100
10’ shear rate
L
- Dispersion DSR
7 5 $ B E? z ‘5
y [s-l]
C5G5
(shear stress ramp)
102:
_ plate-plate _ h= 1llllTi _
R = 12.5 mm 21°C T=
10’ -
5 9 B 100 7
I 10’
100 (b)
rim shear rate
Fig. 2. Viscosity function measured in the Rheometrics cone-plate geometry; (b) using plate-plate geometry.
YR [s-l]
Dynamic
Stress Rheometer
(DSR):
(4 using
Fig. 3 shows a constant shear rate test (time interval 30 s < t < 60 s) at 3 = 4 SC’ using cone-plate geometry. Although the shear rate is still subcritical, positive spikes higher than the average shear stress plateau are observed. These spikes indicate a partial switching to the high viscosity state, albeit for a short period only. The magnitude and frequency of spikes become higher the closer the imposed shear rate approaches j,. It should be mentioned that our instrument uses a ball bearing for the driving shaft. The bearing may cause small fluctuations of sample geometry during shear. It was not possible to measure normal stresses with cone-plate geometry in the subcritical regime.
H.M. Lam 1 J. Non-Newtonian Fluid Mech. 54 (1994) 87-108
60
r-
I
- Dispersion RDS
t 0
.
C5G5
(imposed
I
I
1’ cone-plate
shear rate 9
I
I
u=O.oQrad R= 25mm T= 21°C
.I
,I
*
50 time
Fig. 3. Constant
91
100
t [s]
shear rate test in a Rheometrics
Dynamic
Spectrometer
(RDS).
5. Poor man’s torque controlled rotational rheometer
To avoid difficulties due to the viscosity discontinuity, experiments should be performed at controlled shear stress. Therefore, the RDS drive motor was disconnected from the Rheometrics control board. Instead, a controlled motor current was imposed by means of an external current source (Fig. 4). The relation between the resulting torque M and the motor current I should be linear if the current IF necessary to overcome the friction torque remains independent of the speed and angle of rotation of the shaft (cM torque constant): M = c,(l
- IF).
(1)
A
z 3 s
3
e s
L!!M=cM.( I-IF)
IF
=’
motor current I [A]
Fig. 4. Poor man’s constant
stress rheometer
for the measurement
of normal
force.
92
H.M. Laun / J. Non-Newionian
Fluid Mech. 54 (1994) 87-108
16OCGcone-plate u= O.lrad
R= 125mm
/
Rheometrics
RDS t
/
D $
8000-
*m B 4OOC
a = 0 04 rad R= 25mm
0 0
I 0.8
7 0.4
I 1.2
I 1.6
motor current I [A] Fig. 5. Shear stress-current
calibration
of the RDS.
The RHIOS V 3.01 software directly displays the actual shear stress z from the torque measured by the force rebalance transducer. Fig. 5 shows the calibration curves obtained for two different plate diameters. A value of Z, = 0.4 A was found for the friction current. During the calibration the cone-plate gap was filled with polyisobutylene OPPANOL@B 10 having a zero shear rate viscosity of 4000 Pa s. In these tests a positive first normal stress difference was obtained as expected for a homogeneous polymer melt. Measurements of the time-dependent shear stress and first normal stress difference of the shear thickening dispersion could be performed at controlled motor torque simply by (pro forma) programming a step shear rate test but in practice switching the constant current manually on and off. The true average shear rate was calculated from the independently measured rotational speed of the driving shaft.
6. First normal stress difference in cone-plate
geometry
6.1. General
In cone-plate geometry with radius R which imposes a constant shear rate through the sample, the shear stress z and first normal stress difference N, are readily available from the measured torque M and normal force F,respectively [ 161:
(3)
H.M.
Laun 1 J. Non-Newtonian
Fluid Mech.
54 (1994) 87-108
93
.V”
Dispersion RDS
(imposed
cone-plate
C5G5
a = 0.04
torque)
200
rad
R=25mm T = 27°C
190 Pa
-3 k
q = 10.9 Pa5
0
time Fig. 6. Negative
6.2. Cone-plate
first normal
2 0
100
stress difference
t [s]
for an imposed
average
shear stress of 190 Pa,
measurements of N, in the shear thickening regime (RDS)
Fig. 6 shows an experiment for an average shear stress of 190 Pa. The resulting shear rate is 17 s-l and the viscosity 10.9 Pa s. The viscosity level one power of ten above the low viscosity regime, as well as the value of the shear rate, indicate flow conditions valid for the high viscosity regime (compare Fig. 2(a)). The periodic change of the shear stress signal is due to the fact that we are operating at a motor current Z which is only slightly above the value IF necessary to overcome the bearing friction. Obviously, the friction of the ball bearing or the torque constant cM vary during one revolution of the shaft. In fact, it could be verified that the frequency of the periodic shear stress corresponds to the rotational speed. The time difference between two adjacent maxima is equal to the duration of one revolution. It is further necessary to note that the slow onset of the shear stress signal in Fig. 6 at the beginning of shear does not reflect a time constant of the sample but is simply due to a gradual imposition of the motor current in the experiment shown. The striking result is the occurrence of a negative average first normal stress difference, the absolute value of which is close to the average shear stress. The oscillations observed in the normal stress signal have the same frequency as those of the shear stress but are of opposite sign. One might guess that the following relation holds for the momentary values of shear and normal stress difference: N1 = -_IzI.
(4)
The absolute value of the shear stress is used in Eq. (4) since the value of the normal stress difference should not depend on the direction of shear. To check the validity of Eq. (4) for the transient behaviour the time-dependent first normal stress difference from Fig. 6 is plotted in Fig. 7 vs. the momentary shear stress. Within experimental scatter the two stresses are proportional to each other.
H.M. Lam 1 J. Non-Newtonian Flurd Mech. 54 (1994) 87-108
94
0
0
shear stress Fig. 7. Transient
first normal
300
200
100
‘c [Pa]
stress difference
vs. shear stress from Fig. 6.
However, the transient stress ratio of -N, /r z 0.9 calculated from the average slope is somewhat smaller than expected from the average stress values (compare Fig. 10 below). A negative N, of absolute value equal to the shear stress is also found at still higher shear stresses as shown by Fig. 8 for z = 1600 Pa. The resulting shear rate is 16.7 SC’ and the viscosity 96 Pa s. At the higher torque level the ripple of the shear stress due to the friction torque is small. As a consequence the average values of f and N, are well defined.
y = 16.7 s-1 q = 96 Pa+
Nl
-1750 Pa
-I U
100 time
Fig. 8. Negative
first normal
stress difference
200
t [s]
for an imposed
average shear stress of 1600 Pa.
H.M. Laun /J.
Non-Newtonian
Fluid Mech. 54 (1994) 87-108
95
5000 a = 0.04 rad R= z
25mm
0
.e T 3
80
time t [s] Fig. 9. Negative first normal stress difference for an Imposed average shear stress of 6700 Pa.
At still higher shear stresses as in Fig. 9 for 6700 Pa (shear rate 20.8 s-l, viscosity 322 Pa s) the normal stress signal shows strong noise which is correlated with deviations from the pure concentric flow of the sample. Here, part of the dispersion is sheared out of the gap. In addition, the period of the shear stress oscillation is shorter than the time required for one revolution, The average first normal stress values as measured by means of the RDS in constant torque mode are plotted in Fig. 10 vs. the average shear stress. The diagram contains data for two different cone-plate geometries. This plot yields a stress ratio of -N, /z = 1 and verifies Eq. (4).
I
0
T
I
I
Dispersion C5G5 RDS (imposed torque)
k
z 2 -2OOOg 3 .o z4~_
O.lrad R = 12.5 mm
a=
5 70)
& cone-plate a = 0.04 rad R= 25mm
E
8
E
-6OOO-
D $ 0
T=
27”~ 1 2000
I 4000
I 6000
8000
shear stress ‘c [Pa] Fig. 10. Average first normal stress difference vs. shear stress in the shear thickening regime for two different cone-plate geometries.
H.M.
96
6.3. Reproduction
Lam
1 J. Non-Newtonian
of the -N,
Fluid Mech. 54 (1994) 87-108
17 ratio by a second rheometer
(NFC)
In order to reproduce the negative first normal stress difference by an independent instrument our Normal Force Cell (NFC) was used. This instrument is described in Ref. [ 171. It uses a stiff two component piezo transducer to measure both the torque and the normal force. Torques as high as 0.5 N m could be applied. The motor was driven by an imposed current to control the torque as for the RDS described above. The first series of cone-plate measurements using a small cone angle of a = 0.017 rad (0.974”) yielded a negative normal force and a proportionality between normal force and torque in the full range investigated (unfilled squares in Fig. 11). However, the ratio -N, /r was only 0.625 in contrast to the RDS results. Since an axially very stiff bearing and a small cone angle were used it was suspected that this discrepancy might be due to a “pumping effect” in the gap: small fluctuations of the gap setting during rotation of the cone cause a radial flow. The necessary positive pressure reduces the negative thrust caused by the normal stress differences. To check this assumption the measurements were repeated using a large cone angle of CI= 0.069 rad (3.95”). The results from this cone geometry (full circles in Fig. 11) come close to -N, /r = 1, albeit only in the range of IF) < 1.5 N and M < 0.05 N m. At higher torques the normal force signal levels off to a value of zz -2.5 N. This levelling-off is attributed to the fact that part of the sample is sheared out of the gap. Whereas the smaller amount of sample in the gap may still produce the necessary torque (as in Fig. 2(a)) the decrease of the effective area that contributes to a normal force signal limits the negative thrust. If only part of the sample remains in contact with the wall and the true area of contact is unknown, an evaluation of the true normal stress difference is no longer possible in this regime.
8 z 8-4 ;;i E 8 c -6
a = 0.017 rad . a = 0.069 rad
q
I
I
0.1 torque Fig. Il. angles.
Average
normal
force
vs. torque
I
0.3
0.2
I 0.4
M [Nm]
in the Normal
Force
Cell (NFC)
at two different
cone
H.M. Lam / J. Non-Newtonian Fluid Mech. 54 (1994) 87-108
7. Determination
of the N,/N,
91
ratio
7.1. General
Following Meissner et al. [ 181 the ratio N2/N, of second to first normal stress differences can be determined in a cone-plate rheometer if the normal force Fi acting on a disk of radius Ri < R (Fig. 12) is measured in addition to the total normal force F acting on the full plate of radius R:
F,IRf -=
(5)
F/R2
For the special case of N2/N, = -l/2
ClR,2 -= FIR’
1
or
one obtains
(6)
s=N,, 1
which enables the determination of N, also from the partial normal force F, independent of the ratio R,/R. The shear stress can either be determined from the total torque M acting at the full plate or from the partial torque A4, acting on the smaller disk: 3lW, r=m=m.
3M
(7)
Measurements of F, and h4, could be performed with the NFC by using the modified plate mounting as depicted in Fig. 13. Here, only the inner disk of radius R, is supported by the quartz washer. Thus the outer part of the plate does not contribute to the torque and force signal.
drive
bearing sample RI = 16 mm rubber gasket
.2 component quartz washer
Y
Fi
MI
torque normal force
Fig. 12. Torque and normal force on the full plate and corresponding values on a disk of radius for the determination of the second normal stress difference (schematic). Fig. 13. Modified
NFC
geometry
to measure
torque
and normal
force on the smaller
R, < R
disk.
H.M. Lam / J. Non-Newtonian Fluid Mech. 54 (1994) 87-108
98
Dispersion NFC
C5G5
(imposed
torque)
. R = 25 mm cx = 0.017 rad A R, = 16 mm a = 0.017 rad q
Ri = 16 mm a = 0.017 rad
-6000
I
I
I
4ooa
2ooo
0
shear stress Fig. 14. Comparison smaller disk (unfilled
r
6000
8000
IOCQO
z [Pa]
of the average normal stress signals from the full plate (full circles) and from the symbols) for a cone angle of 0.017 rad in the NFC.
7.2. Experimental
results
Fig. 14 shows a plot of the first normal stress difference N, (full circles) and of the quantity 2FJrcRf (unifilled symbols) evaluted from the modified geometry vs. shear stress for the small cone angle of 0.017 rad. Within experimental error both values coincide on the same straight line (slope -0.625). From Eq. (6) it follows therefore that N2 = -+N,.
(8) Fig. 15 gives the corresponding data for the larger cone angle of 0.069 rad. Up to a shear stress of 2000 Pa the unfilled symbols (smaller disk) and full symbols
Dispersion NFC
C5G5
(imposed
torque)
. .^ “p: 5
z”
\
T = 27°C -6000
/ 0
_
o-
R = 25 mm a = 0.069 rad o R, = 16 mm a = 0.069 rad .
N II
--/I 4000
2ooo
shear stress Fig. 15. Comparison smaller disk (unfilled
.
q
CCW-plate
x00-
-
t
I &A
[Pa]
of the average normal stress signals from the full plate (full circles) and from the symbols) for a cone angle of 0.069 rad in the NFC.
H.M. Lmm 1 J. Non-Newtonian Fluid Mech. 54 (1994) 87-108
99
(total plate) agree within experimental scatter. This verifies the validity of Eq. (8). In addition, we find - N,lr z 1 (straight line has slope 1) in agreement with the RDS results and Eq. (4). At higher shear stresses a levelling-off of the normal stress signal is observed as discussed above. This seems to occur at higher shear stresses in the case of the smaller disk, leading to the conjecture that the loss of contact with the wall starts near the rim. Finally, it is interesting to note that Eq. (8) obviously holds independently of whether -N, /z is close to unity (Fig. 15) or significantly smaller (Fig. 14).
8. State of stress in the regime of extreme shear thickening
Using N, = - Iz\ and N,/N, = -0.5, the resulting stress tensor (r for simple shear (and hydrostatic pressure p) is
(9)
Unlike the situation with polymer melts and solutions there is a negative normal stress (positive pressure) acting in the direction of shear, whereas a positive normal stress of equal magnitude is acting in the shear gradient direction. The resulting principle normal stresses are (T, = + J5/2 and au = - J5/2 for a coordinate system that is rotated in the x-y plane by an angle of x = 58.5”.
9. Normal forces in plate-plate
geometry (RDS)
9.1. General The pertinent relations differences in plate-plate respectively, are [ 161
for the evaluation of shear stress and normal stress geometry from the torque M and normal force F,
(10) (11) Here jR stands for the rim shear rate,
(12) Sz being the angular frequency and h the gap height. As long as M(jR) and F(jR) are not explicitly known it is common practice to evaluate apparent values (valid
H.M. Law 1 J. Non-Newtonian Fluid Mech. 54 (1994) 87-108
100
for a second order fluid). The apprent shear stress ra is evaluated by assuming M proportional to jn:
L(itrd
=g$.
(13)
The apparent first normal stress difference N,, follows by assuming F proportional to yi and neglecting the second normal stress difference: (14) 9.2. Positive apparent normal stress in the low viscosity regime (RDS) Using plate-plate geometry and imposed torque it was possible to measure the apparent first normal stress difference in the low viscosity regime of the dispersion. In order to reach normal force values that could be measured by the force rebalance transducer it was necessary to apply rather high shear rates. In fact, it was necessary to operate in the metastable range above the equilibrium critical shear rate (compare Fig. 2(b)). Fig. 16 shows the result for an apparent shear stress of 42 Pa. From the rim shear rate of 40 s-i a viscosity of 1.05 Pa s is calculated, indicating the subcritical flow of the dispersion. The strong oscillation of the shear stress signal is again due to the imposition of a motor current close to the friction current IF as already discussed for Fig. 6. The important result for the average apparent first normal stress difference in the low viscosity regime is the positive sign and its value being much smaller than the apparent shear stress. The small drift of the normal force zero after stopping the shear does not affect this conclusion.
--._~--
80
Dispersion
7 P5 ol
C5G5
RDS (imposed torque) plate-plate h= I mm R= 25mm T= 27°C 42 Pa
sa
40-
z .^ T 2
#j###Jjpa
s
._ _
Nla 0-
1 jR = 40
S-1
0
time t [s] Fig. 16. Positive apparent first normal average shear stress of 42 Pa.
stress
difference
in the low viscosity
regime
for an imposed
H.M.
L.uun / J. Non-Newtonian
Dispersion
Fluid Mech.
C5G5
RDS
54 (1994) 87-108
(imposed
torque)
plate-plate
120
R= 77 5
101
25mm
80
2
t
+R = 19
U
s-1
80 time
Fig. 17. Positive apparent first normal average shear stress of 85 Pa.
’
stress
difference
160
t [s] in the low viscosity
regime
for an imposed
These findings are reproduced in Fig. 17 for an apparent shear stress of 85 Pa. Here, a rim shear rate of 79 s-‘, with apparent viscosity 1.08 Pa s still in the metastable regime, could be achieved. The average apparent first normal stress difference is positive, too, and much smaller than the apparent shear stress. 9.3. Negative apparent normal stress d@erence in the shear thickening
regime (RDS)
A further increase of the imposed torque causes the dispersion to switch to the high viscosity regime and the rim shear rate reaches the equilibrium critical shear rate for plate-plate geometry. A typical test is shown in Fig. 18 for an apparent shear stress of 7000 Pa. The resulting rim shear rate is 28 s-’ which corresponds to an apparent viscosity of 250 Pa s. Now the apparent first normal stress difference N,, is negative, its absolute value being close to the shear stress value. In summary, the plate-plate results for the apparent first normal stress difference in the shear thickening range look very similar to those measured for cone-plate geometry. Indeed, a plot of N,, vs. z, (Fig. 19) yields a straight line of slope - 1 which means
N,, = - /~a). 9.4. Interpretation
of the plate -plate
(15) result
In the region of strong viscosity increase (compare Fig. 2(b)) the rim shear rate remains constant while the apparent shear stress increases by powers of ten. Under such conditions it is questionable whether one can assume a linear shear rate profile imposed by the relative movement of both plates. Rather partial wall slip may occur. We therefore start from a radial shear stress profile r(r) in the gap. Let us for simplicity assume the power law relation
102
H.M.
Lam
1 J. Non-Newtonian
8ooo - Dispersion
Fluid Mech.
C5G5
54 (1994) 87-108
RDS (imposed torque)
Ta p
7000 Pa
plate-plate h= R= T=
0
40
Fig. 18. Negative apparent first normal average shear stress of 7000 Pa.
120
80 time
stress difference
Imm 25mm 27°C
160
t [s] in the shear thickening
I
I
I
Dispersion RDS
regime for an imposed
(imposed
C5G5 torque)
plate-plate lmm h= R= 25mm 27°C T=
20&l
I
I
4ooo
6000
shear stress Fig. 19. Average apparent first normal regime for plate-plate geometry.
stress difference
*do0
z [Pa] vs. apparent
shear stress in the shear thickening
(16) The power law index n governs the steepness of the shear stress profile, ~~ being the value at the rim (r = R). By using normal stress profiles according to Eqs. (4) and (8), N,(r) = -[z(r)1 = -2&(r),
(17)
H.M. Lam 1 J. Non-Newtonian
Fluid Mech. 54 (1994) 87-108
103
the ratio - N,,/z, follows as (see Appendix A) (18)
Obviously, for a positive value of n and 12>>1 the minimum value of the stress ratio becomes 1.5 which is significantly higher that the experimental value of 1. n = 0 yields the even higher ratio of 2.25. The experimentally observed stress ratio could only be obtained for the physically unrealistic power law exponent of n = -5. A better agreement with the experiment is achieved, however, with the additional assumption of negligible normal stresses for the central part of the sample where the shear stress is smaller than a critical value 7,. The latter should be close to the product of equilibrium critical shear rate and the subcritical plateau viscosity (7, z 30 Pa for C5G5 at 27°C). This is an approximation of the fact that the inner part of the sample-for a dimensional radius x = r/R < x,-remains in the low viscosity regime where the apparent first normal stress is positive but small compared to the shear stress. The stress ratio follows as (see Appendix A)
-N,, _=;
I4
5
(1
-;x;+‘).
(19)
The x, values that give a stress ratio of 1 are listed in Table 1 for power law exponents of 1 s n s 5. It turns out that only a rather small part of the sample at the rim needs to be in the high viscosity state to create the normal force observed. In the centre of the gap for 0 I r/R < x, the sample remains in the low viscosity state and the true shear may indeed increase proportional to r. At r/R = xc the equilibrium critical shear rate j, is reached. Now the shear stress may change considerably while the shear rate remains constant. The relative movement of both plates imposes a linear increase of the shear rate with r, however. This contradiction can be overcome by partial slip of the dispersion near the wall such that the true shear rate remains equal to f,. This implies that the shear stress profile is determined by the necessary wall slip velocity u,. Experimental investigations [ 191 show that a power law is pertinent to describe V,(Z). This conjecture explains why Table 1 Dimensionless radius x, from Eq. (A7) that separates the gap calculated for various exponents n n
XC
1 2 3 4 5
0.909 0.915 0.922 0.929 0.935
the low viscosity
and shear thickening
regimes
in
104
H.M.
Lam
/ J. Non-Newtonian
Fluid Mech.
the equilibrium critical shear rate jRc in plate-plate the value Ij, for the cone-plate gap:
54 (1994) 87-108
geometry
should be higher than
In fact, the ratio of the critical shear rates from Figs. 2(a) and 2(b) yields X, = 0.85 which comes close to the values given in Table 1.
10. Consequences for concentric cylinder geometry In concentric cylinder geometry as used in Sear1 or Couette rheometers it was found that small deviations in the concentricity cause lateral forces which tend to pull the central bob towards the cup at the location where the gap is most narrow. Obviously, this effect follows from the simple shear observations. In concentric cylinder geometry the radial shear stress profile is given by [ 161
(21) r’, being the shear stress at the bob. The radial normal stress profiles from Eq. (9), a,, = -]42 = - err (see Appendix B), yield for the total normal stress rr,, acting on the bob
(22) This creates a lateral force directed towards the cup which is proportional to the absolute value of the shear stress acting at the bob. As a consequence, the occurrence of a small deviation from concentricity will create a higher shear stress r, at the bob in the narrower gap. The magnitude of the normal force on the bob in the narrower gap will thus also be higher compared to its counterpart on the opposite side of the gap. This results in a force proportional to the difference of shear stress levels on both sides. The direction of this force is such that it pulls the bob towards the wall on the side of the narrower gap. Due to the high shear stresses that can be reached this force may be strong enough to twist the shaft of a Sear1 rheometer, which has indeed been observed.
11. Final comment The focus of this paper is on the experimental determination of the first and second normal stress differences of an extremely shear thickening dispersion at the critical shear rate suing cone-plate geometry. It is further shown how these findings can be used to interpret the normal force level measured by plate-plate geometry as well as the reason for a lateral force that tries to pull the bob towards the cup in concentric cylinder geometry. A discussion of the structural reason for the stress
H.M. Lam 1 .I. Non-Newtoman Fluid Mech. 54 (1994) 87-108
105
state given by Eq. (9) is outside the scope of this paper. In this respect the reader is referred to an interpretation discussed by Hess [20] on the basis of a liquid crystalline structure.
Acknowledgements
The author is indebted to M. Reuther for the modification of the Rheometrics RDS and for the normal force measurements performed on this instrument. I. Ulmerich is thanked for the construction of the Normal Force Cell as well as for the NFC experiments. Dr. Machtle is thanked for contributing the AUC diameter distribution. Helpful discussions with Professor S. Hess are gratefully acknowledged.
Appendix A. Normal force in plate-plate
geometry
A radial shear stress profile given by Eq. (16) is assumed (Fig. Al). We further assume for x, I r/R I 1 normal stresses as described by Eq. (9) with ~~~= r~,.~= 0. The normal stress in the tangential direction is CJ11 = CT@~ = - ITl/2 and in the axial direction a,, G a,, = + ]zj/2. For x/R < x, the normal stresses are set equal to zero. For a rim shear stress of zR the radial pressure profile p(r) follows from dp(r) = -o&r)
d In
r = H2
0
xr
n
d In r,
(AlI
and p(r)=jidp=$$[(iy-11.
The total normal stress q;(r)
(A9 acting on the upper plate is for f
n&)
= p(r) - a,,(r) =
”
Fig. Al. Coordinate system and shear stress profile for plate-plate geometry (schematic). normal force F is the integral of the total normal stress K,~ acting on the upper plate.
(A3)
The measured
H.M. Lam /J.
106
Non-Newtonian Fluid Mech. 54 (1994) 87-108
This yields a total normal force f on the upper plate of
s s R
F=
3 - 2x2 2 2(n + 2)
7r&)2711. dr = +&R2
r=O
(A4)
.
The torque M is given by R
M=
21rR~ r(r)r2rn dr = zR -.
(A5)
n+3
0
Using Eqs. (13) and ( 14) the ratio of the negative apparent first normal stress difference -N,, and the absolute value of the apparent shear stress Iz,I follows as (Ah) A stress ratio of -N,&[ xc =
n+5 [ 2(n + 3)
1
= 1 is obtained for
l/n+2
(A7)
.
Appendix B: Normal stress acting on the bob in concentric cylinder geometry The coordinate system used is depicted in Fig. Bl. The radial stress profile given by Eq. (21) is independent of the nature of the fluid. We assume c33 = (T,, = 0. From Eq. (9) the resulting tangential normal stress is oII = gBB= - (~(/2 and the radial normal stress 022 = err = + lr)/2. A radial pressure profile p(r) in the gap is created by dgs: dp(r) = -a,,(r)
d In r.
(Bl)
Its value p(Ri) at the bob (Y = Ri) follows as ~(4)
=
s
R1Tdlnr=!?!! I+91
“‘“f:’ -=-2 s RO
RO
k$-(!J]
VW
r 0 Fig. Bl. Coordinate system for concentric is due to the total normal stress x,,.
Ri cylinder
RO geometry
(schematic).
The lateral
force on the bob
H.M. Lam / J. Non-Newtonian Fluid Mech. 54 (1994) 87-108
107
Finally, the total normal force 7c,,(RJ acting on the bob is the sum of the negative pressure --p(R,) and the normal stress a,,(Ri) = + /r,(/2 at the bob (note the opposite direction of the total normal stress with respect to the sample, compared to Fig. Al):
TAR,) = -AR,)
+orr(R,)
=F [3-($71.
This normal stress is directed towards the wall, its magnitude being proportional to the absolute value of the shear stress zi at the bob. For small gaps with (R,/&)’ z 1 this simplifies to
The negative pressure according to Eq. (82) tends to suck the dispersion into the gap. To achieve oz, = 0 this pressure must be compensated by an elastic compression of the sample in the z direction. The magnitude of the pressure is highest at the bob. This explains the tendency of extremely shear thickening dispersions to produce a crack at the bob if sheared at high torque level.
References [II PI
I.M. Krieger, A dimensional approach to colloid rheology, Trans. Sot. Rheol., 7 (1963) lOl- 109. I.M. Krieger, Rheology of monodisperse latices, Adv. Colloid Interface Sci., 3 (1972) 11 l-136. effect in polymer latices, Trans. Sot. 131 I.M. Krieger and M. Eguiluz, The second electroviscous Rheol., 20 (1976) 29-45. Chem., [41 H.M. Laun, Rheological properties of aqueous polymer dispersions, Angew. Makromol. 123/124 (1984) 335-359. and dilatant viscosity behavior in concentrated suspensions. I. [51 R.L. Hoffman, Discontinuous Observation of flow instability, Trans. Sot. Rheol., 16 (1972) 155-173. and dilatant viscosity behavior in concentrated suspensions. II. [61 R.L. Hoffman, Discontinuous Theory and experimental data, J. Colloid Interface Sci., 46 (1974) 491-506. and dilatant viscosity behavior in concentrated suspensions. III. 171 R.L. Hoffman, Discontinuous Necessary conditions for their occurence in viscometric flows, Adv. Colloid Interface Sci., 17 (1982) 161-184. dispersion, PI W.H. Boersma, J. Laven and H.N. Stein, Shear thickening (dilatancy) in concentrated AIChE J., 36 (1990) 321-332. shear-thickening [91 W.H. Boersma, J. Laven and H.N. Stein, Viscoelastic properties of concentrated dispersions, J. Colloid Interface Sci., 149 (1992) 10-22. particle [lOI H.M. Laun, Rheological properties of polymer dispersions with respect to shear-induced structures, in H. Giesekus and M.F. Hibberd (Eds.). Progress and Trends in Rheology II, supplement to Rheol. Acta, 26 (1988) 287-290. polymer dispersions, in P.H.T. [Ill H.M. Laun, Rheology and particle structures of concentrated Uhlherr (Ed.), Xth International Congress on Rheology, Vol. 1, Australian Society of Rheology, Sydney, 1988, pp. 37-42. [I21 H.M. Laun, R. Bung and F. Schmidt, Rheology of extremely shear thickening polymer dispersions (passively viscosity switching fluids), J. Rheol., 35 (1991) 999-1034. by passive stiffness switching mounts, [I31 R. Helber, R. Bung and F. Doncker, Vibration attenuation J. Sound Vibration, 138 (1990) 47757.
108
H.M. Lam 1 J. Non-Newtonian Fluid Mech. 54 (1994) 87-108
[ 141 H.M. Laun, R. Bung, S. Hess, W. Loose, 0. Hess, K. Hahn, E. Hadicke, R. Hingmann, F. Schmidt and P. Lindner, Rheological and small angle neutron scattering investigation of shear-induced particle structures of concentrated polymer dispersions submitted to plane Poiseuille and Couette flow, J. Rheol., 36 (1992) 743-787. [15] W. Machtle, Future requirements for modem analytical ultracentrifuges., Progr. Colloid Polym. Sci., 86 (1991) 111-118. [16] R.B. Bird, R.C. Armstrong and 0. Hassager, Dynamics of Polymeric Liquids, Vol. 1. 2nd edn, J. Wiley, New York, 1987. [ 171 H.M. Laun and G.L. Hirsch, New laboratory tests to measure rheological properties of paper coatings in transient and steady-state flows, Rheol. Acta, 28 (1989) 267-280. [ 181 J. Meissner, R.W. Garbella and J. Hostettler, Measuring normal stress differences in polymer melt shear flow, J. Rheol., 33 (1989) 843-864. [ 191 E. Windhab, Untersuchungen zum rheologischen Verhalten konzentrierter Suspensionen, Fortschr.Ber. VDI Reihe 3 Nr. 118, VDI-Verlag, Dusseldorf 1986. [20] S. Hess Normal stress differences in liquid crystals and m extremely shear thickening dispersions, J. Non-Newtonian Fluid Mech., 54 (1994) 201.
J. Non-Newtonian Fluid Mech., 54 (1994) 87-108
Normal stresses in extremely shear thickening polymer dispersions * H.M. Laun BASF Aktiengesellschaft, Polymer Research Division, Polymer di Solid State Physics Department, ZKh4 - G201, D-67056 LudwigshafenlRhein, Germany
Received 22 March 1994
Abstract
Measurements of the first and second normal stress differences of an extremely shear thickening polymer dispersion have been performed at imposed shear stress both by means of a Rheometrics RDS and a home-made Normal Force Cell using cone-plate geometry. The sample is a 58.7 vol.% dispersion of styrenelethyl acrylate copolymer particles with 280 nm average diameter in glycol. The spheres are electrostatically stabilized by negative charges due to carboxylic groups. In the regime of strong shear thickening a negative first normal stresses difference N, is found, the absolute value of which is equal to the absolute value of the shear stress T: N, = - 171. Measurements of the normal force acting on a disk of smaller radius than the full plate radius yield for the second normal stress difference N2 the relation N2 = -N, /2. The experimentally observed relation of the apparent shear stress 7a and apparent first normal stress difference N,, in plate-plate geometry, N,, = -17a), can be explained by assuming that only the sample part near the rim is shear thickening whereas the inner part remains in the low viscosity regime. In the latter regime N,, is positive and smaller than 7,. It is further explained why small deviations of the concentricity in Searle geometry create a strong lateral force that pulls the bob towards the cup. Keywords:
Normal stresses; Polymer dispersions; Rotational
rheometry; Shear thickening
* Dedicated to Professor Ken Walters FRS on the occasion of his 60th birthday. 0377-0257/94/$07.00 0 1994 - Elsevier Science B.V. All rights reserved SSDZ 0377-0257( 94)01307-4
88
H.M.
Lam
1 J. Non-Newtoman
Fluid Mech.
54 (1994) 87-108
1. Introduction Concentrated suspensions of nearly monodisperse polymer particles dispersed in a low viscosity Newtonian fluid and stabilized by electrostatic forces exhibit distinct changes of the viscosity with increasing shear rate. Typically, a shearing thinning behaviour is observed at low shear rates. Whereas this low shear rate regime, dominated by Brownian forces, has been investigated widely (e.g. Refs. [l-4]) there exists a much smaller number of publications dealing with the high shear rate behaviour and the occurrence of shear thickening [4-121. A better understanding of the phenomenon of dilatancy is of importance for the design of passively switching mechanical elements like clutches and dampers [ 131. Extremely shear thickening dispersions exhibit a reversible step-like shear stress increase at a critical shear rate which reaches values higher than lo5 Pa! An investigation of the related particle structures in the flowing dispersions was performed by using neutron scattering [14]. The work on normal stresses of the dispersions described in this paper was motivated by observations during measurements using concentric cylinder geometry: it was found difficult to maintain the concentricity of the cylinders in the regime of shear thickening. Strong lateral forces on the bob caused a twist of the driving shaft of our Physica Rheolab MC 20 rheometer when the DIN 24 cup with 14 mm diameter was used.
2. Sample The sample investigated here is the glycol model dispersion C5G5 as described in Ref. [14]. It consists of styrene/ethyl acrylate copolymer particles prepared by emulsion polymerization, styrene being the dominant monomer. Small amounts of mono- and dicarboxylic acids were added to the monomers to incorporate carboxylic groups on the particle surface, a partial dissociation of which causes a negative particle charge in polar fluids. The average particle diameter of latex 18/157 as determined by quasi-elastic light scattering (QELS) is 295 nm [14]. For comparison Fig. 1 shows the integral and differential partical size (mass) distribution measured by means of the analytical ultracentrifuge (AUC) [ 151. The distribution is rather narrow. The peak of the differential distribution yields an average particle diameter of 280 nm. Sample C5G5 was prepared by redispersion of the freeze-dried latex powder in glycol at a solid concentration of 58.7 vol.%.
3. Viscosity function Fig. 2 shows the viscosity vs. shear rate as measured in a Rheometrics Dynamic Stress Rheometer (DSR) at 27°C in a shear stress ramp experiment. Both coneplate (cone angle CI= 0.1 rad, radius R = 12.5 mm) and plate-plate (gap h = 1 mm, R = 12.5 mm) geometries were used. A logarithmic sweep of the shear stress up to 5000 Pa within 3 min was applied. In the low shear rate range a weakly shear
H.M.
Lam
/J. Non-Newtonian
Fluid Mech.
54 (1994) 87-108
89
latex IS/157
0.0 100
150
200
250
diameter Fig. 1. Size distnbution
of the latex particles
300
350
400
[nm]
from the analytical
ultracentrifuge
(AUC)
thinning behaviour is found. At the beginning, with increasing shear stress, the curve remains on a quasi-Newtonian plateau of viscosity ( z 1.2 Pa s) up to a shear rate of about 55 SC’ (shear stress 7.2 Pa). Subsequently the shear rate decreases to the equilibrium critical shear rate j, = 22 s-’ (cone-plate) or rim shear rate compare Eq. (12), below, respectively. ?Lc = 26 SC’ (plate-plate), At r > 7.2 Pa the sample has switched to the high viscosity regime. Here, the average shear rate remains constant while the shear stress rises by about a factor of 40. The upper limit of the viscosity step is about an order of magnitude higher and cannot be measured in the DSR. The observation of a metastable range of low viscosity as well as fluctuations of the equilibrium critical shear rate for an extremely shear thickening dispersion have been described before [ 121. It has also been shown that the critical shear rate depends on the geometry of the gap [ 121. A repetition of the shear stress ramp for plate-plate geometry immediately after the first run reproduced the measurement shown in Fig. 2(b). In the case of cone-plate geometry, however, the low viscosity level was distinctly lower in the second run. An inspection of the sample showed that part of the material has been sheared out of the gap during the application of high torque levels. After refilling the gap the measurement represented in Fig. 2(a) could be reproduced.
4. Limitations
of imposed shear rate rheometers
For normal stress measurements on concentrated dispersions our shear strain or shear rate controlled Rheometrics Dynamic Spectrometer (RDS) equipped with a force rebalance transducer would in principle be the pertinent instrument. On these are only reasonable in the strongly shear thickening samples, however, subcritical regime since the possibility of a large and rapid change of the shear stress (torque) at shear rates 3 2 3, could damage the transducer.
90
H.M.
Lam / J. Non-Newtonian
- Dispersion DSR
Fluid Mech. 54 (1994) 87-108
C5G5
(shear stress ramp)
102 -
- cone-plate _ a= O.lrad
77 GJ
k
R = 12.5 mm T= 27°C
F h .% z 5: ‘5
101 7
100 -
I 100
10’ shear rate
L
- Dispersion DSR
7 5 $ B E? z ‘5
y [s-l]
C5G5
(shear stress ramp)
102:
_ plate-plate _ h= 1llllTi _
R = 12.5 mm 21°C T=
10’ -
5 9 B 100 7
I 10’
100 (b)
rim shear rate
Fig. 2. Viscosity function measured in the Rheometrics cone-plate geometry; (b) using plate-plate geometry.
YR [s-l]
Dynamic
Stress Rheometer
(DSR):
(4 using
Fig. 3 shows a constant shear rate test (time interval 30 s < t < 60 s) at 3 = 4 SC’ using cone-plate geometry. Although the shear rate is still subcritical, positive spikes higher than the average shear stress plateau are observed. These spikes indicate a partial switching to the high viscosity state, albeit for a short period only. The magnitude and frequency of spikes become higher the closer the imposed shear rate approaches j,. It should be mentioned that our instrument uses a ball bearing for the driving shaft. The bearing may cause small fluctuations of sample geometry during shear. It was not possible to measure normal stresses with cone-plate geometry in the subcritical regime.
H.M. Lam 1 J. Non-Newtonian Fluid Mech. 54 (1994) 87-108
60
r-
I
- Dispersion RDS
t 0
.
C5G5
(imposed
I
I
1’ cone-plate
shear rate 9
I
I
u=O.oQrad R= 25mm T= 21°C
.I
,I
*
50 time
Fig. 3. Constant
91
100
t [s]
shear rate test in a Rheometrics
Dynamic
Spectrometer
(RDS).
5. Poor man’s torque controlled rotational rheometer
To avoid difficulties due to the viscosity discontinuity, experiments should be performed at controlled shear stress. Therefore, the RDS drive motor was disconnected from the Rheometrics control board. Instead, a controlled motor current was imposed by means of an external current source (Fig. 4). The relation between the resulting torque M and the motor current I should be linear if the current IF necessary to overcome the friction torque remains independent of the speed and angle of rotation of the shaft (cM torque constant): M = c,(l
- IF).
(1)
A
z 3 s
3
e s
L!!M=cM.( I-IF)
IF
=’
motor current I [A]
Fig. 4. Poor man’s constant
stress rheometer
for the measurement
of normal
force.
92
H.M. Laun / J. Non-Newionian
Fluid Mech. 54 (1994) 87-108
16OCGcone-plate u= O.lrad
R= 125mm
/
Rheometrics
RDS t
/
D $
8000-
*m B 4OOC
a = 0 04 rad R= 25mm
0 0
I 0.8
7 0.4
I 1.2
I 1.6
motor current I [A] Fig. 5. Shear stress-current
calibration
of the RDS.
The RHIOS V 3.01 software directly displays the actual shear stress z from the torque measured by the force rebalance transducer. Fig. 5 shows the calibration curves obtained for two different plate diameters. A value of Z, = 0.4 A was found for the friction current. During the calibration the cone-plate gap was filled with polyisobutylene OPPANOL@B 10 having a zero shear rate viscosity of 4000 Pa s. In these tests a positive first normal stress difference was obtained as expected for a homogeneous polymer melt. Measurements of the time-dependent shear stress and first normal stress difference of the shear thickening dispersion could be performed at controlled motor torque simply by (pro forma) programming a step shear rate test but in practice switching the constant current manually on and off. The true average shear rate was calculated from the independently measured rotational speed of the driving shaft.
6. First normal stress difference in cone-plate
geometry
6.1. General
In cone-plate geometry with radius R which imposes a constant shear rate through the sample, the shear stress z and first normal stress difference N, are readily available from the measured torque M and normal force F,respectively [ 161:
(3)
H.M.
Laun 1 J. Non-Newtonian
Fluid Mech.
54 (1994) 87-108
93
.V”
Dispersion RDS
(imposed
cone-plate
C5G5
a = 0.04
torque)
200
rad
R=25mm T = 27°C
190 Pa
-3 k
q = 10.9 Pa5
0
time Fig. 6. Negative
6.2. Cone-plate
first normal
2 0
100
stress difference
t [s]
for an imposed
average
shear stress of 190 Pa,
measurements of N, in the shear thickening regime (RDS)
Fig. 6 shows an experiment for an average shear stress of 190 Pa. The resulting shear rate is 17 s-l and the viscosity 10.9 Pa s. The viscosity level one power of ten above the low viscosity regime, as well as the value of the shear rate, indicate flow conditions valid for the high viscosity regime (compare Fig. 2(a)). The periodic change of the shear stress signal is due to the fact that we are operating at a motor current Z which is only slightly above the value IF necessary to overcome the bearing friction. Obviously, the friction of the ball bearing or the torque constant cM vary during one revolution of the shaft. In fact, it could be verified that the frequency of the periodic shear stress corresponds to the rotational speed. The time difference between two adjacent maxima is equal to the duration of one revolution. It is further necessary to note that the slow onset of the shear stress signal in Fig. 6 at the beginning of shear does not reflect a time constant of the sample but is simply due to a gradual imposition of the motor current in the experiment shown. The striking result is the occurrence of a negative average first normal stress difference, the absolute value of which is close to the average shear stress. The oscillations observed in the normal stress signal have the same frequency as those of the shear stress but are of opposite sign. One might guess that the following relation holds for the momentary values of shear and normal stress difference: N1 = -_IzI.
(4)
The absolute value of the shear stress is used in Eq. (4) since the value of the normal stress difference should not depend on the direction of shear. To check the validity of Eq. (4) for the transient behaviour the time-dependent first normal stress difference from Fig. 6 is plotted in Fig. 7 vs. the momentary shear stress. Within experimental scatter the two stresses are proportional to each other.
H.M. Lam 1 J. Non-Newtonian Flurd Mech. 54 (1994) 87-108
94
0
0
shear stress Fig. 7. Transient
first normal
300
200
100
‘c [Pa]
stress difference
vs. shear stress from Fig. 6.
However, the transient stress ratio of -N, /r z 0.9 calculated from the average slope is somewhat smaller than expected from the average stress values (compare Fig. 10 below). A negative N, of absolute value equal to the shear stress is also found at still higher shear stresses as shown by Fig. 8 for z = 1600 Pa. The resulting shear rate is 16.7 SC’ and the viscosity 96 Pa s. At the higher torque level the ripple of the shear stress due to the friction torque is small. As a consequence the average values of f and N, are well defined.
y = 16.7 s-1 q = 96 Pa+
Nl
-1750 Pa
-I U
100 time
Fig. 8. Negative
first normal
stress difference
200
t [s]
for an imposed
average shear stress of 1600 Pa.
H.M. Laun /J.
Non-Newtonian
Fluid Mech. 54 (1994) 87-108
95
5000 a = 0.04 rad R= z
25mm
0
.e T 3
80
time t [s] Fig. 9. Negative first normal stress difference for an Imposed average shear stress of 6700 Pa.
At still higher shear stresses as in Fig. 9 for 6700 Pa (shear rate 20.8 s-l, viscosity 322 Pa s) the normal stress signal shows strong noise which is correlated with deviations from the pure concentric flow of the sample. Here, part of the dispersion is sheared out of the gap. In addition, the period of the shear stress oscillation is shorter than the time required for one revolution, The average first normal stress values as measured by means of the RDS in constant torque mode are plotted in Fig. 10 vs. the average shear stress. The diagram contains data for two different cone-plate geometries. This plot yields a stress ratio of -N, /z = 1 and verifies Eq. (4).
I
0
T
I
I
Dispersion C5G5 RDS (imposed torque)
k
z 2 -2OOOg 3 .o z4~_
O.lrad R = 12.5 mm
a=
5 70)
& cone-plate a = 0.04 rad R= 25mm
E
8
E
-6OOO-
D $ 0
T=
27”~ 1 2000
I 4000
I 6000
8000
shear stress ‘c [Pa] Fig. 10. Average first normal stress difference vs. shear stress in the shear thickening regime for two different cone-plate geometries.
H.M.
96
6.3. Reproduction
Lam
1 J. Non-Newtonian
of the -N,
Fluid Mech. 54 (1994) 87-108
17 ratio by a second rheometer
(NFC)
In order to reproduce the negative first normal stress difference by an independent instrument our Normal Force Cell (NFC) was used. This instrument is described in Ref. [ 171. It uses a stiff two component piezo transducer to measure both the torque and the normal force. Torques as high as 0.5 N m could be applied. The motor was driven by an imposed current to control the torque as for the RDS described above. The first series of cone-plate measurements using a small cone angle of a = 0.017 rad (0.974”) yielded a negative normal force and a proportionality between normal force and torque in the full range investigated (unfilled squares in Fig. 11). However, the ratio -N, /r was only 0.625 in contrast to the RDS results. Since an axially very stiff bearing and a small cone angle were used it was suspected that this discrepancy might be due to a “pumping effect” in the gap: small fluctuations of the gap setting during rotation of the cone cause a radial flow. The necessary positive pressure reduces the negative thrust caused by the normal stress differences. To check this assumption the measurements were repeated using a large cone angle of CI= 0.069 rad (3.95”). The results from this cone geometry (full circles in Fig. 11) come close to -N, /r = 1, albeit only in the range of IF) < 1.5 N and M < 0.05 N m. At higher torques the normal force signal levels off to a value of zz -2.5 N. This levelling-off is attributed to the fact that part of the sample is sheared out of the gap. Whereas the smaller amount of sample in the gap may still produce the necessary torque (as in Fig. 2(a)) the decrease of the effective area that contributes to a normal force signal limits the negative thrust. If only part of the sample remains in contact with the wall and the true area of contact is unknown, an evaluation of the true normal stress difference is no longer possible in this regime.
8 z 8-4 ;;i E 8 c -6
a = 0.017 rad . a = 0.069 rad
q
I
I
0.1 torque Fig. Il. angles.
Average
normal
force
vs. torque
I
0.3
0.2
I 0.4
M [Nm]
in the Normal
Force
Cell (NFC)
at two different
cone
H.M. Lam / J. Non-Newtonian Fluid Mech. 54 (1994) 87-108
7. Determination
of the N,/N,
91
ratio
7.1. General
Following Meissner et al. [ 181 the ratio N2/N, of second to first normal stress differences can be determined in a cone-plate rheometer if the normal force Fi acting on a disk of radius Ri < R (Fig. 12) is measured in addition to the total normal force F acting on the full plate of radius R:
F,IRf -=
(5)
F/R2
For the special case of N2/N, = -l/2
ClR,2 -= FIR’
1
or
one obtains
(6)
s=N,, 1
which enables the determination of N, also from the partial normal force F, independent of the ratio R,/R. The shear stress can either be determined from the total torque M acting at the full plate or from the partial torque A4, acting on the smaller disk: 3lW, r=m=m.
3M
(7)
Measurements of F, and h4, could be performed with the NFC by using the modified plate mounting as depicted in Fig. 13. Here, only the inner disk of radius R, is supported by the quartz washer. Thus the outer part of the plate does not contribute to the torque and force signal.
drive
bearing sample RI = 16 mm rubber gasket
.2 component quartz washer
Y
Fi
MI
torque normal force
Fig. 12. Torque and normal force on the full plate and corresponding values on a disk of radius for the determination of the second normal stress difference (schematic). Fig. 13. Modified
NFC
geometry
to measure
torque
and normal
force on the smaller
R, < R
disk.
H.M. Lam / J. Non-Newtonian Fluid Mech. 54 (1994) 87-108
98
Dispersion NFC
C5G5
(imposed
torque)
. R = 25 mm cx = 0.017 rad A R, = 16 mm a = 0.017 rad q
Ri = 16 mm a = 0.017 rad
-6000
I
I
I
4ooa
2ooo
0
shear stress Fig. 14. Comparison smaller disk (unfilled
r
6000
8000
IOCQO
z [Pa]
of the average normal stress signals from the full plate (full circles) and from the symbols) for a cone angle of 0.017 rad in the NFC.
7.2. Experimental
results
Fig. 14 shows a plot of the first normal stress difference N, (full circles) and of the quantity 2FJrcRf (unifilled symbols) evaluted from the modified geometry vs. shear stress for the small cone angle of 0.017 rad. Within experimental error both values coincide on the same straight line (slope -0.625). From Eq. (6) it follows therefore that N2 = -+N,.
(8) Fig. 15 gives the corresponding data for the larger cone angle of 0.069 rad. Up to a shear stress of 2000 Pa the unfilled symbols (smaller disk) and full symbols
Dispersion NFC
C5G5
(imposed
torque)
. .^ “p: 5
z”
\
T = 27°C -6000
/ 0
_
o-
R = 25 mm a = 0.069 rad o R, = 16 mm a = 0.069 rad .
N II
--/I 4000
2ooo
shear stress Fig. 15. Comparison smaller disk (unfilled
.
q
CCW-plate
x00-
-
t
I &A
[Pa]
of the average normal stress signals from the full plate (full circles) and from the symbols) for a cone angle of 0.069 rad in the NFC.
H.M. Lmm 1 J. Non-Newtonian Fluid Mech. 54 (1994) 87-108
99
(total plate) agree within experimental scatter. This verifies the validity of Eq. (8). In addition, we find - N,lr z 1 (straight line has slope 1) in agreement with the RDS results and Eq. (4). At higher shear stresses a levelling-off of the normal stress signal is observed as discussed above. This seems to occur at higher shear stresses in the case of the smaller disk, leading to the conjecture that the loss of contact with the wall starts near the rim. Finally, it is interesting to note that Eq. (8) obviously holds independently of whether -N, /z is close to unity (Fig. 15) or significantly smaller (Fig. 14).
8. State of stress in the regime of extreme shear thickening
Using N, = - Iz\ and N,/N, = -0.5, the resulting stress tensor (r for simple shear (and hydrostatic pressure p) is
(9)
Unlike the situation with polymer melts and solutions there is a negative normal stress (positive pressure) acting in the direction of shear, whereas a positive normal stress of equal magnitude is acting in the shear gradient direction. The resulting principle normal stresses are (T, = + J5/2 and au = - J5/2 for a coordinate system that is rotated in the x-y plane by an angle of x = 58.5”.
9. Normal forces in plate-plate
geometry (RDS)
9.1. General The pertinent relations differences in plate-plate respectively, are [ 161
for the evaluation of shear stress and normal stress geometry from the torque M and normal force F,
(10) (11) Here jR stands for the rim shear rate,
(12) Sz being the angular frequency and h the gap height. As long as M(jR) and F(jR) are not explicitly known it is common practice to evaluate apparent values (valid
H.M. Law 1 J. Non-Newtonian Fluid Mech. 54 (1994) 87-108
100
for a second order fluid). The apprent shear stress ra is evaluated by assuming M proportional to jn:
L(itrd
=g$.
(13)
The apparent first normal stress difference N,, follows by assuming F proportional to yi and neglecting the second normal stress difference: (14) 9.2. Positive apparent normal stress in the low viscosity regime (RDS) Using plate-plate geometry and imposed torque it was possible to measure the apparent first normal stress difference in the low viscosity regime of the dispersion. In order to reach normal force values that could be measured by the force rebalance transducer it was necessary to apply rather high shear rates. In fact, it was necessary to operate in the metastable range above the equilibrium critical shear rate (compare Fig. 2(b)). Fig. 16 shows the result for an apparent shear stress of 42 Pa. From the rim shear rate of 40 s-i a viscosity of 1.05 Pa s is calculated, indicating the subcritical flow of the dispersion. The strong oscillation of the shear stress signal is again due to the imposition of a motor current close to the friction current IF as already discussed for Fig. 6. The important result for the average apparent first normal stress difference in the low viscosity regime is the positive sign and its value being much smaller than the apparent shear stress. The small drift of the normal force zero after stopping the shear does not affect this conclusion.
--._~--
80
Dispersion
7 P5 ol
C5G5
RDS (imposed torque) plate-plate h= I mm R= 25mm T= 27°C 42 Pa
sa
40-
z .^ T 2
#j###Jjpa
s
._ _
Nla 0-
1 jR = 40
S-1
0
time t [s] Fig. 16. Positive apparent first normal average shear stress of 42 Pa.
stress
difference
in the low viscosity
regime
for an imposed
H.M.
L.uun / J. Non-Newtonian
Dispersion
Fluid Mech.
C5G5
RDS
54 (1994) 87-108
(imposed
torque)
plate-plate
120
R= 77 5
101
25mm
80
2
t
+R = 19
U
s-1
80 time
Fig. 17. Positive apparent first normal average shear stress of 85 Pa.
’
stress
difference
160
t [s] in the low viscosity
regime
for an imposed
These findings are reproduced in Fig. 17 for an apparent shear stress of 85 Pa. Here, a rim shear rate of 79 s-‘, with apparent viscosity 1.08 Pa s still in the metastable regime, could be achieved. The average apparent first normal stress difference is positive, too, and much smaller than the apparent shear stress. 9.3. Negative apparent normal stress d@erence in the shear thickening
regime (RDS)
A further increase of the imposed torque causes the dispersion to switch to the high viscosity regime and the rim shear rate reaches the equilibrium critical shear rate for plate-plate geometry. A typical test is shown in Fig. 18 for an apparent shear stress of 7000 Pa. The resulting rim shear rate is 28 s-’ which corresponds to an apparent viscosity of 250 Pa s. Now the apparent first normal stress difference N,, is negative, its absolute value being close to the shear stress value. In summary, the plate-plate results for the apparent first normal stress difference in the shear thickening range look very similar to those measured for cone-plate geometry. Indeed, a plot of N,, vs. z, (Fig. 19) yields a straight line of slope - 1 which means
N,, = - /~a). 9.4. Interpretation
of the plate -plate
(15) result
In the region of strong viscosity increase (compare Fig. 2(b)) the rim shear rate remains constant while the apparent shear stress increases by powers of ten. Under such conditions it is questionable whether one can assume a linear shear rate profile imposed by the relative movement of both plates. Rather partial wall slip may occur. We therefore start from a radial shear stress profile r(r) in the gap. Let us for simplicity assume the power law relation
102
H.M.
Lam
1 J. Non-Newtonian
8ooo - Dispersion
Fluid Mech.
C5G5
54 (1994) 87-108
RDS (imposed torque)
Ta p
7000 Pa
plate-plate h= R= T=
0
40
Fig. 18. Negative apparent first normal average shear stress of 7000 Pa.
120
80 time
stress difference
Imm 25mm 27°C
160
t [s] in the shear thickening
I
I
I
Dispersion RDS
regime for an imposed
(imposed
C5G5 torque)
plate-plate lmm h= R= 25mm 27°C T=
20&l
I
I
4ooo
6000
shear stress Fig. 19. Average apparent first normal regime for plate-plate geometry.
stress difference
*do0
z [Pa] vs. apparent
shear stress in the shear thickening
(16) The power law index n governs the steepness of the shear stress profile, ~~ being the value at the rim (r = R). By using normal stress profiles according to Eqs. (4) and (8), N,(r) = -[z(r)1 = -2&(r),
(17)
H.M. Lam 1 J. Non-Newtonian
Fluid Mech. 54 (1994) 87-108
103
the ratio - N,,/z, follows as (see Appendix A) (18)
Obviously, for a positive value of n and 12>>1 the minimum value of the stress ratio becomes 1.5 which is significantly higher that the experimental value of 1. n = 0 yields the even higher ratio of 2.25. The experimentally observed stress ratio could only be obtained for the physically unrealistic power law exponent of n = -5. A better agreement with the experiment is achieved, however, with the additional assumption of negligible normal stresses for the central part of the sample where the shear stress is smaller than a critical value 7,. The latter should be close to the product of equilibrium critical shear rate and the subcritical plateau viscosity (7, z 30 Pa for C5G5 at 27°C). This is an approximation of the fact that the inner part of the sample-for a dimensional radius x = r/R < x,-remains in the low viscosity regime where the apparent first normal stress is positive but small compared to the shear stress. The stress ratio follows as (see Appendix A)
-N,, _=;
I4
5
(1
-;x;+‘).
(19)
The x, values that give a stress ratio of 1 are listed in Table 1 for power law exponents of 1 s n s 5. It turns out that only a rather small part of the sample at the rim needs to be in the high viscosity state to create the normal force observed. In the centre of the gap for 0 I r/R < x, the sample remains in the low viscosity state and the true shear may indeed increase proportional to r. At r/R = xc the equilibrium critical shear rate j, is reached. Now the shear stress may change considerably while the shear rate remains constant. The relative movement of both plates imposes a linear increase of the shear rate with r, however. This contradiction can be overcome by partial slip of the dispersion near the wall such that the true shear rate remains equal to f,. This implies that the shear stress profile is determined by the necessary wall slip velocity u,. Experimental investigations [ 191 show that a power law is pertinent to describe V,(Z). This conjecture explains why Table 1 Dimensionless radius x, from Eq. (A7) that separates the gap calculated for various exponents n n
XC
1 2 3 4 5
0.909 0.915 0.922 0.929 0.935
the low viscosity
and shear thickening
regimes
in
104
H.M.
Lam
/ J. Non-Newtonian
Fluid Mech.
the equilibrium critical shear rate jRc in plate-plate the value Ij, for the cone-plate gap:
54 (1994) 87-108
geometry
should be higher than
In fact, the ratio of the critical shear rates from Figs. 2(a) and 2(b) yields X, = 0.85 which comes close to the values given in Table 1.
10. Consequences for concentric cylinder geometry In concentric cylinder geometry as used in Sear1 or Couette rheometers it was found that small deviations in the concentricity cause lateral forces which tend to pull the central bob towards the cup at the location where the gap is most narrow. Obviously, this effect follows from the simple shear observations. In concentric cylinder geometry the radial shear stress profile is given by [ 161
(21) r’, being the shear stress at the bob. The radial normal stress profiles from Eq. (9), a,, = -]42 = - err (see Appendix B), yield for the total normal stress rr,, acting on the bob
(22) This creates a lateral force directed towards the cup which is proportional to the absolute value of the shear stress acting at the bob. As a consequence, the occurrence of a small deviation from concentricity will create a higher shear stress r, at the bob in the narrower gap. The magnitude of the normal force on the bob in the narrower gap will thus also be higher compared to its counterpart on the opposite side of the gap. This results in a force proportional to the difference of shear stress levels on both sides. The direction of this force is such that it pulls the bob towards the wall on the side of the narrower gap. Due to the high shear stresses that can be reached this force may be strong enough to twist the shaft of a Sear1 rheometer, which has indeed been observed.
11. Final comment The focus of this paper is on the experimental determination of the first and second normal stress differences of an extremely shear thickening dispersion at the critical shear rate suing cone-plate geometry. It is further shown how these findings can be used to interpret the normal force level measured by plate-plate geometry as well as the reason for a lateral force that tries to pull the bob towards the cup in concentric cylinder geometry. A discussion of the structural reason for the stress
H.M. Lam 1 .I. Non-Newtoman Fluid Mech. 54 (1994) 87-108
105
state given by Eq. (9) is outside the scope of this paper. In this respect the reader is referred to an interpretation discussed by Hess [20] on the basis of a liquid crystalline structure.
Acknowledgements
The author is indebted to M. Reuther for the modification of the Rheometrics RDS and for the normal force measurements performed on this instrument. I. Ulmerich is thanked for the construction of the Normal Force Cell as well as for the NFC experiments. Dr. Machtle is thanked for contributing the AUC diameter distribution. Helpful discussions with Professor S. Hess are gratefully acknowledged.
Appendix A. Normal force in plate-plate
geometry
A radial shear stress profile given by Eq. (16) is assumed (Fig. Al). We further assume for x, I r/R I 1 normal stresses as described by Eq. (9) with ~~~= r~,.~= 0. The normal stress in the tangential direction is CJ11 = CT@~ = - ITl/2 and in the axial direction a,, G a,, = + ]zj/2. For x/R < x, the normal stresses are set equal to zero. For a rim shear stress of zR the radial pressure profile p(r) follows from dp(r) = -o&r)
d In
r = H2
0
xr
n
d In r,
(AlI
and p(r)=jidp=$$[(iy-11.
The total normal stress q;(r)
(A9 acting on the upper plate is for f
n&)
= p(r) - a,,(r) =
”
Fig. Al. Coordinate system and shear stress profile for plate-plate geometry (schematic). normal force F is the integral of the total normal stress K,~ acting on the upper plate.
(A3)
The measured
H.M. Lam /J.
106
Non-Newtonian Fluid Mech. 54 (1994) 87-108
This yields a total normal force f on the upper plate of
s s R
F=
3 - 2x2 2 2(n + 2)
7r&)2711. dr = +&R2
r=O
(A4)
.
The torque M is given by R
M=
21rR~ r(r)r2rn dr = zR -.
(A5)
n+3
0
Using Eqs. (13) and ( 14) the ratio of the negative apparent first normal stress difference -N,, and the absolute value of the apparent shear stress Iz,I follows as (Ah) A stress ratio of -N,&[ xc =
n+5 [ 2(n + 3)
1
= 1 is obtained for
l/n+2
(A7)
.
Appendix B: Normal stress acting on the bob in concentric cylinder geometry The coordinate system used is depicted in Fig. Bl. The radial stress profile given by Eq. (21) is independent of the nature of the fluid. We assume c33 = (T,, = 0. From Eq. (9) the resulting tangential normal stress is oII = gBB= - (~(/2 and the radial normal stress 022 = err = + lr)/2. A radial pressure profile p(r) in the gap is created by dgs: dp(r) = -a,,(r)
d In r.
(Bl)
Its value p(Ri) at the bob (Y = Ri) follows as ~(4)
=
s
R1Tdlnr=!?!! I+91
“‘“f:’ -=-2 s RO
RO
k$-(!J]
VW
r 0 Fig. Bl. Coordinate system for concentric is due to the total normal stress x,,.
Ri cylinder
RO geometry
(schematic).
The lateral
force on the bob
H.M. Lam / J. Non-Newtonian Fluid Mech. 54 (1994) 87-108
107
Finally, the total normal force 7c,,(RJ acting on the bob is the sum of the negative pressure --p(R,) and the normal stress a,,(Ri) = + /r,(/2 at the bob (note the opposite direction of the total normal stress with respect to the sample, compared to Fig. Al):
TAR,) = -AR,)
+orr(R,)
=F [3-($71.
This normal stress is directed towards the wall, its magnitude being proportional to the absolute value of the shear stress zi at the bob. For small gaps with (R,/&)’ z 1 this simplifies to
The negative pressure according to Eq. (82) tends to suck the dispersion into the gap. To achieve oz, = 0 this pressure must be compensated by an elastic compression of the sample in the z direction. The magnitude of the pressure is highest at the bob. This explains the tendency of extremely shear thickening dispersions to produce a crack at the bob if sheared at high torque level.
References [II PI
I.M. Krieger, A dimensional approach to colloid rheology, Trans. Sot. Rheol., 7 (1963) lOl- 109. I.M. Krieger, Rheology of monodisperse latices, Adv. Colloid Interface Sci., 3 (1972) 11 l-136. effect in polymer latices, Trans. Sot. 131 I.M. Krieger and M. Eguiluz, The second electroviscous Rheol., 20 (1976) 29-45. Chem., [41 H.M. Laun, Rheological properties of aqueous polymer dispersions, Angew. Makromol. 123/124 (1984) 335-359. and dilatant viscosity behavior in concentrated suspensions. I. [51 R.L. Hoffman, Discontinuous Observation of flow instability, Trans. Sot. Rheol., 16 (1972) 155-173. and dilatant viscosity behavior in concentrated suspensions. II. [61 R.L. Hoffman, Discontinuous Theory and experimental data, J. Colloid Interface Sci., 46 (1974) 491-506. and dilatant viscosity behavior in concentrated suspensions. III. 171 R.L. Hoffman, Discontinuous Necessary conditions for their occurence in viscometric flows, Adv. Colloid Interface Sci., 17 (1982) 161-184. dispersion, PI W.H. Boersma, J. Laven and H.N. Stein, Shear thickening (dilatancy) in concentrated AIChE J., 36 (1990) 321-332. shear-thickening [91 W.H. Boersma, J. Laven and H.N. Stein, Viscoelastic properties of concentrated dispersions, J. Colloid Interface Sci., 149 (1992) 10-22. particle [lOI H.M. Laun, Rheological properties of polymer dispersions with respect to shear-induced structures, in H. Giesekus and M.F. Hibberd (Eds.). Progress and Trends in Rheology II, supplement to Rheol. Acta, 26 (1988) 287-290. polymer dispersions, in P.H.T. [Ill H.M. Laun, Rheology and particle structures of concentrated Uhlherr (Ed.), Xth International Congress on Rheology, Vol. 1, Australian Society of Rheology, Sydney, 1988, pp. 37-42. [I21 H.M. Laun, R. Bung and F. Schmidt, Rheology of extremely shear thickening polymer dispersions (passively viscosity switching fluids), J. Rheol., 35 (1991) 999-1034. by passive stiffness switching mounts, [I31 R. Helber, R. Bung and F. Doncker, Vibration attenuation J. Sound Vibration, 138 (1990) 47757.
108
H.M. Lam 1 J. Non-Newtonian Fluid Mech. 54 (1994) 87-108
[ 141 H.M. Laun, R. Bung, S. Hess, W. Loose, 0. Hess, K. Hahn, E. Hadicke, R. Hingmann, F. Schmidt and P. Lindner, Rheological and small angle neutron scattering investigation of shear-induced particle structures of concentrated polymer dispersions submitted to plane Poiseuille and Couette flow, J. Rheol., 36 (1992) 743-787. [15] W. Machtle, Future requirements for modem analytical ultracentrifuges., Progr. Colloid Polym. Sci., 86 (1991) 111-118. [16] R.B. Bird, R.C. Armstrong and 0. Hassager, Dynamics of Polymeric Liquids, Vol. 1. 2nd edn, J. Wiley, New York, 1987. [ 171 H.M. Laun and G.L. Hirsch, New laboratory tests to measure rheological properties of paper coatings in transient and steady-state flows, Rheol. Acta, 28 (1989) 267-280. [ 181 J. Meissner, R.W. Garbella and J. Hostettler, Measuring normal stress differences in polymer melt shear flow, J. Rheol., 33 (1989) 843-864. [ 191 E. Windhab, Untersuchungen zum rheologischen Verhalten konzentrierter Suspensionen, Fortschr.Ber. VDI Reihe 3 Nr. 118, VDI-Verlag, Dusseldorf 1986. [20] S. Hess Normal stress differences in liquid crystals and m extremely shear thickening dispersions, J. Non-Newtonian Fluid Mech., 54 (1994) 201.