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Opposing-jet viscometry of fluids with viscosity approaching that of water
Nea-~Imeiaa 1~ Medmics ELSEVIER
J. Non-Newtonian Fluid Mech., 56 (1995) 1- 14
Opposing-jet viscometry of fluids with viscosity approaching that of water Clarence G. Hermansky a, David V. Boger b DuPont Agricultural Products, Wilmington, DE 19880-0402, USA b The Advanced Mineral Products Research Centre, The University of Melbourne, Parkville 3052, Vic., Australia
Received 15 December 1994; in revised form 3 June 1994
Abstract Flows which were predominantly uniaxial in extension were generated through 0.5 mm diameter aligned jets. The torque required to keep the jets at a fixed distance apart, and thus the apparent stress generated by the fluid, was measured by a torque rebalance transducer attached to one of the moveable jet arms. Strain rates from 1000 to 20 000 s-t were spanned. Data are presented for 1 to ~150 centipoise (cP) Newtonian water and glycerol/water mixtures. The trends found were consistent with a contribution to the measured torque by the inertia of the fluid, becoming negligibly small as the shear viscosity of the fluid approached ~ 75 cP. A method for correcting opposing-jet measurements made on fluids of viscosity less than 75 cP is described and the extent of the correction illustrated on a series of solutions with identical shear viscosities prepared from polyethylene glycols of molecular weight 8 × 103 to 1 x 10 6. The validity of the correction method for use on non-Newtonian fluids is discussed. The corrected results for the polyethylene glycol solutions of identical shear viscosity clearly illustrate the influence of molecular weight on extensional flow properties. .Keywords: Flow viscosity; Opposing-jet viscometry; Uniaxial flow
1. Introduction The study of rigid rods/polymers a n d flexible polymers in extensional flow fields has received significant a t t e n t i o n in recent years. This focus is, in part, due to a n 0377-0257/95/$09.50 © 1995 - Elsevier Science B.V. All rights reserved SSDI 0377-0257(94)01265-J
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C.G. Hermansky, D.V. Boger / J. Non-Newtonian Fluid Mech. 56 (1995) 1 14
appreciation that extensional flows occur and can dominate over shear flows in industrial processes which operate at high rates, e.g. coatings, extrusion, lamination, drag-reduction and atomization. While many techniques have been used to generate data in extensional flows [1], it is generally acknowledged that no one technique measures a true extensional viscosity [2]. All suffer from an inability to reach a steady rate of strain as the fluid element under study passes through the measurement volume. Nevertheless, these measurements still remain attractive given their ability to approximate the strain rates developed in real processes. Techniques for measuring the extensional properties of fluids normally fall into two categories: flow through and stagnation point devices [3]. Most of the former rely on the fluid being spinnable, like the tubeless siphon [4], or various spinning techniques [5,6]. Although significant strides have now been made in the measurement of elongational properties [7,8], such devices are still limited to low rates of strain and generally to highly viscous or elastic fluids, since a column of at least a few centimeters of the fluid under study must be self-sustaining over a range of strain rates. Alternatively, orifice flow techniques which measure the pressure drop across a contraction offer an option which overcomes the need to have a spinnable fluid [9]. However, interpretation of the orifice data even in Newtonian liquids is not straightforward and can be further compounded by the presence of recirculating vortices and flow instabilities in viscoelastic fluids [10,11]. Conversely, stagnation point devices such as the roll mill [12], lubricated-die converging flow rheometer [13], cross-slot cell [14,15] and the opposing-jet device of this study can be used to collect extensional data in low viscosity fluids. With the exception of the lubricated-die converging flow rheometer which requires large volumes (100 liters) to operate, any of these devices can be used to make measurements on low viscosity fluids provided the limitation of short residence time in the measurement volume is acknowledged. Recent publications using the opposing-jet technique to measure rheological properties have focused on a critique of the technique [ 16] and measurement of radial flow velocity distributions and pressure drop across the jets [17], its utility in measuring the extensional properties of dilute and semi-dilute solutions of rigid collagen rods [ 18] and various concentrations of Xanthan and flexible polyacrylamides in high viscosity Newtonian glycerol/water media [3]. In this study we have selected the Rheometrics RFX opposing jet device since it is commercially available, has the potential to span a wide range of strain rates, and because we are not aware of any attempt to address its use on fluids whose shear viscosities are less than 100 centipoise. What follows are the results of extensional measurements made on Newtonian glycerol/water mixtures and on elastic polymer solutions whose shear viscosities range from 1 to ~ 150 centipoise (cP). For the Newtonian fluids, contrary to what should be expected, Trouton's rule was not obeyed when the shear viscosity of the fluid was below ~ 75 cP, due to the fluid inertia associated with high rates of flow. Using data on the Newtonian fluids tested, a data correction technique was developed which is usable over a strain rate range of at least 1000 to 20 000 s -~. Adherence to Trouton's rule was demonstrated for Newtonian fluids and the utility
C.G. Hermansky, D.V. Boger [ J. Non-Newton&n Fluid Mech. 56 (1995) 1-14
3
of the correction technique illustrated by application to a series of elastic polyethylene glycol solutions of matched shear viscosities.
2. Experimental apparatus and materials
2. I. Materials and methods The water used in this study was taken from a Millipore "Milli-Q" water purification unit. Reagent grade glycerol purchased from BDH Chemicals was used. A series of polyethylene glycols (PEG) ranging from 8.9 x 103 to 1.0 x 10 6 in molecular weight was purchased from Aldrich Chemical Co. and used as received. All compositions were prepared by weight and are reported as weight percents. All mixtures were used within a week of preparation. The aqueous P E G solutions were allowed to turn end over end for several hours to days depending on the concentration and molecular weight. Shear viscosity was measured at 25.0°C (_+0.1°C) using a C a r r i - M e d constant stress rheometer equipped with a 6 cm, 2 ° cone and plate geometry with a 56 micron cone truncation. A broad enough range of stress was applied during each measurement to insure that a shear rate range of at least 1 to 800 s-l was spanned. Extensional viscosity measurements were determined as a function of strain rate using a Rheometrics RFX, at 25.0°C (_+0.5°C). The Rheometrics R F X is the commercial version of the opposing-jet device described by other authors [3,18-20]. A schematic of the opposed-nozzle device (Fig. 1) has been reproduced from Schunk et al. [ 16] to illustrate the experimental set up. Unless otherwise noted, all data were taken using a set of 0.5 mm diameter nozzles fixed at a nozzle-to-nozzle separation of 0.5 mm. The software supplied with the commercial instrument was used to automate data collection. However, the output of the torque transducer was independently monitored on a strip chart recorder to insure adequate time was allowed for torque (M) to come to "equilibrium" and to detect the presence of flow instabilities if present. The instrument operates by stepping through a series of pre-programmed strain rates and taking an average of the torque over the last few seconds of the measurement. Fluid flow through the nozzles is accomplished by a stepper motor attached to a pair of 20 cc syringes which draw the fluid of interest through the nozzles via the torque arms at a fixed volumetric flow rate (Q). During this time the torque (M) produced at the torque rebalance transducer is measured. In the instrument of this study, the jets were attached to a pivot arm which was 7.62 centimeters in length (L). All calculations were performed by the software. The force developed at the face of each nozzle was determined by dividing the torque on the moment arm by its length (M/L). This force was then converted to an apparent stress at the face of each nozzle on division by the area of the nozzle bore (re, R2), where R is the radius of the nozzle bore. The strain rate ( 0 is similarly derived by dividing each flow rate by the nozzle bore area and half distance of separation between the nozzles (2Q/r~R2d), where d is the distance to the midplane in Fig. 1. Thus the apparent
4
C.G. Hermansky, D.V. Boger/J. Non-Newtonian Fluid Mech. 56 (1995) 1-14
TORQUE M ~...~f
PIVOTAXIS TIP ~- FACEINCLINATION THICKNESS ~
PIVOTARML B O R E ~ MIDPtANE
-.~ ! ¢
CENTERLINE
SEPARATIONd Fig. 1. Schematic diagram of opposing-jet rheometer.
extensional viscosity (r/e) is given by the quotient of the stress and strain rate (Eq. (1)), provided the stepper motor and torque transducer are properly calibrated and care is taken to assure that the transducer is nulled before each measurement. Md ~ - 2QL"
(1)
3. Results and discussion 3. I. Newtonian fluids
Fig. 2 shows the ratio of the extensional viscosity (qe) to three times the shear viscosity (3r/s) for two glycerol/water mixtures whose shear viscosities were 156 cP (90% glycerol/water) and 110 cP (88% glycerol/water). The results are consistent with a Trouton ratio of 3 to within ~ 15% over a strain rate range of 500 to 8000 s -]. This range represents both the upper and lower limits attainable by the instrument for fluids having the above shear viscosity. Fluids of lesser viscosity allow greater flexibility expanding the high end of the strain rates attainable. Fig. 3 demonstrates the apparent discrepancy which results in the Trouton ratio (r/e/r/s) when the shear viscosity is reduced to below ~ 100 cP. The Trouton ratio increases with increasing strain rate and decreases with increasing viscosity in all cases. The data shown are for Newtonian, glycerol/water mixtures with the compositions, measured shear viscosities and linear regression slopes summarized in Table 1. The slopes given in Table 1 were determined by assuming (a) the best fit intercept; and
C.G. Herrnansky, D.V. Boger / J. Non-Newtonian Fluid Mech. 56 (1995) 1 14
5
2.0 ,~ 4.m (D 0 0 W ~>
1.8
II .C if) X
1.2
~
0.8
1.6 1.4
tll r,-
0.6
~-
0.4
:< LU
0.2
.2ffl
0
0
[] [] []
1.0
0.0 0
n
I~
U
[]
2000
r7
~
4000
[]
[]
tO'
6ooo
sooo
10000
Strain Rate (1/Sec.)
Fig. 2. Ratio of the extensional to three times the shear viscosity as a function of strain rate for: C), 90% glycerol/10% water (156 cP) and [~, 88% glycerol/12% water (110 cP) mixtures. i •
~ . 150
1 Cp&
0 0
._w > 100 e.(/) n C 0 :~ 50 m O~ C
o
UJ
~ 5000
10000
3.6 cps 5.5 cps. 10.5 cps
15000
20000
25000
Strain Rate (1/Sec.)
Fig. 3. Trouton ratio for the Newtonian glycerol/water mixture of Table 1 which have shear viscosities below 100 cP. (b) an intercept o f 3, the latter being consistent with the T r o u t o n ratio which would be expected in the limit o f low strain rates. The data show that this assumption had little effect on the fitted slope. I f such departure f r o m the expected T r o u t o n ratio o f three were attributed to fluid inertia an additional contribution to the measured torque which would be p r o p o r t i o n a l to the density o f the fluid (p) and to the square o f its velocity (U 2) might be expected as follows: M m = M c + a ' ( R ) p U z.
(2)
6
C.G. Hermansky, D.V. Boger /J. Non-Newtonian Fluid Mech. 56 (1995) 1-14
Table 1 Newtonian compositions, shear viscosities and slope from Trouton ratio-strain rate plots of Fig, 3 Composition % Glycerol in water
Shear viscosity (cP)
Slope a
slope b
90.0 88.0 85.0 75.0 60.0 50.0 40.0 0.0
156 110 77 30 10.5 5.5 3.6 1.3
0.0000 0.0000 0.0001 0.0003 0.0008 0.0015 0.0026 0.0074
0.0000 0.0000 0.0000 0.0003 0.0007 0.0017 0.0028 0.0078
1.3 1.0
0.0074 0.0115
Water at 25°C Carbon tetrachloride at 18°C
a Slope of the linear regression of the Trouton ratio-strain rate data of Fig. 3. b Slope of the linear regression of the Trouton ratio-strain rate data of Fig. 3, holding the intercept fixed at a value of 3.0.
In Eq. (2), the value o f a'(R) is anticipated to be a function of nozzle radius but not o f nozzle separation, for reasons discussed below. M m is the torque measured by the instrument and M c is the corrected torque, i.e. the torque which would have been measured in the absence o f fluid inertial effects. While the general f o r m o f a'(R) might include the consideration that it be a function o f separation [i.e. a'(R,d)], measurements by ourselves and others [3,16] indicate that it is not. In the present study, the slope o f the measured extensional viscosity versus strain rate plots o f Fig. 3 did not vary from that reported in Table 1 by m o r e than +_ 10% over a range o f separation (d) o f 0.35 to 1.0 mm. At higher separations, while the measured extensional viscosity was still linear in strain rate, a decreasing trend in b o t h the values o f qe and slope was found. This finding is consistent with existing literature indicating that the strain rate remains constant across the gap, provided the gap between nozzles is less than 2 - 3 times their diameter [19]. As noted earlier, the strain rate (~) is given by 2Q ~R2d,
(3)
where Q is the flow rate, R is the radius o f the nozzles bore and d is half the distance o f separation between the nozzles. Given the mean velocity O o f the fluid flow t h r o u g h the jets has been shown to be given by Q/ztR 2 to within ~ 15% for 0.5 m m jets [19], to a g o o d approximation the m e a n velocity o f the fluid can be rewritten in terms o f the strain rate on c o m b i n a t i o n with Eq. (3), to give d~ 2
O = --.
(4)
C.G. Hermansky, D.V. Boger / J. Non-Newtonian Fluid Mech. 56 (1995) 1-14
103
I
O
1.0 cps.
V 3o.o~,,. 10 =
A
E
Z
7
•. "
~
~jp
~
101
0 I--
10 0
i/ 10 s
I 10 ~
Strain Rate {1/Sec.)
Fig. 4. Measured torque versus strain rate for Newtonian glycerol/watersolutions of composition and shear viscosity selected from Table 1. Substituting Eq. (4) for the fluid velocity given in Eq. (2) results in the relationship to be expected between the measured (M,,) and corrected (Me) torques in terms of the contribution made by the inertia of the fluid. a'(R)d2p42
M m = M c q-
(5)
4
Eq. (5) suggests that the measured torque (Mm) should show quadratic behavior if plotted against strain rate, if a significant contribution from fluid inertia is present. Such behavior is demonstrated in Fig. 4 for a selection of fluids in Table 1. The data for the 1.0, 5.5, 30 and 77 cP Newtonian, glycerol/water fluids also serve to show that the quadratic behavior is all but diminished in the 77 cP (85% glycerol/ water) fluid, over the range of operation of the instrument. In the case of this particular fluid, cavitation occurs in the syringes at ~ 12 000 s- 1, thus defining the upper limit of the instruments capability. To demonstrate that the linear trend in the Trouton ratio-strain rate data of Fig. 3 directly follows from the quadratic behavior given by Eq. (5), the following substitutions and rearrangements were made. Eq. (3) was rearranged to give the flow rate in terms of the strain rate and inserted into Eq. (1), to eliminate Q and arrive at the generic relationship between the torque and the strain rate given by (6)
M = 7"cR2Lrle 4.
Substitution of this into Eq. (5) and subsequent rearrangement, to give both the measured and corrected extensional viscosity in terms of the contributions to be expected from the inertia of the fluid, gives a'(R)pd 2 . qc = qe
~4 ~ L R
~.
(7)
8
C.G. Hermansky, D.V. Boger/J. Non-Newtonian Fluid Mech. 56 (1995) 1-14
In Eq. (7) r/e is the extensional viscosity calculated from the measured torque and r/c is the extensional viscosity corrected for the inertia of the fluid under study. Division of Eq. (7) by the shear viscosity (r/s) results in a relationship between the measured and corrected Trouton ratio as a function of strain rate, as shown below. r/c
r/e
a'(R)pd 2 .
r/s
r/s
4gLR2r/s
-
-
E.
(8)
From Eq. (8) it is clear that the differences in the slopes of the Trouton ratio-strain plots of Fig. 3 will only be consistent with the proposed dominance of fluid inertia to the meaured torque if the value of a'(R) is found to be constant for a given jet radius. To test this, the densities of the glycerol/water solutions of Table 1 were assumed to be constant and a plot of the slopes from Table 1 versus the reciprocal of the Newtonian shear viscosity (1/r/s) was constructed. The result is shown in Fig. 5. The reciprocal dependence found clearly illustrates that the value of a'(R) is constant and has a value of 0.708 cm 3. While maintenance of a constant strain across the gap limits the working range of the instrument to 2 - 3 times the diameter of the nozzles in use, the value of a'(R) has been shown to be constant, i.e. the numerical value of a'(R) calculated from Fig. 5 can be said to be applicable for all 0.5 mm jets. However, Eq. (2) acknowledges the contribution of total mass per unit time to the torque, so a dependence of a'(R) on the area of the jet (i.e. its radius) should be anticipated. The nature of this dependence was explored by measuring the extensional viscosity versus strain rate for water with 0.5, 1.0 and 2.0 mm jets in place. The value of d in each experiment was again matched to the radius of the jets being used. The resulting data is shown in Fig. 6.
0.008
I
I
I
I
I
0 Q.
o
t/) 0.006 n'C t,/)
0.004
|
o
I s,op,- 0.00741
0 0.002 > x I.[.I 0,~0 0.0
0.2
0.4
0.6
o.a
1.0
1.2
Reciprocal of Shear Viscosity Fig. 5. Slope of the Trouton ratio-strain rate plots from Table 1 versus the reciprocal of the fluid shear viscosity in cP.
C.G. Hermansky, D.F. Boger / J. Non-Newtonian FluM Mech. 56 (1995) 1-14
9
°°IJ Z
'~ 0
Slope: 8.176 (2.8 m m J e t s )
300
~
ua "o Ip
76
200
L
¢u
100
Slope: 8.8874 (8.5 m m Jets)
~-0
o
I
I
sooo
~oooo
J
~sooo
I
zoooo
Strain Rate (I/Sec.}
Fig. 6. Measured extensional viscosity of water over a range of strain rates using 0.5, 1.0 and 2.0 mm nozzles.
In all three cases, the linearity of the measured extensional viscosity-strain rate plot was retained over the range of strain rates accessible to the instrument. And, as expected, a considerable difference in the slope of each plot was seen. The numerical value of the slope for each plot and the respective nozzle used is also shown in Fig. 6. From the slopes of Fig. 6 and Eq. (7) the values of a'(R) were calculated for each of the nozzles and were found to be 0.708, 2.64 and 16.8 cm 3 for the 0.5, 1.0 and 2.0 mm nozzles, respectively. On considering recent work which suggested that inertial contributions would be a function of the area swept out by the measurement volume [ 16], it seemed prudent to test these data by plotting the values of a'(R) against the area of the nozzle bore being used. The result is shown in Fig. 7. While the data is limited it does not support an inertial contribution which scales with area and suggests that the radial and axial velocity profiles are not appreciably changed by the choice of nozzle. This is consistent with previously reported photon-correlation velocimetry scans which have shown the radial velocity distribution to be constant for pure solvent and dilute high molecular weight polymer solutions [17]. Also consistent with the relationship given by Eq. (8) is limited, but significant, data which shows that the slope of Fig. 3 are directly proportionality to the density of the fluid. Specifically, Table 1 gives the measured slopes for water at 25°C and for carbon tetrachloride at 18°C. Both fluids have nearly identical shear viscosities at these temperatures, with the ratio of their densities being 1.59. A ratio of 1.55 (within 5%) was calculated from the carbon tetrachloride/water slopes reported in Table 1. This result is consistent with the expected proportionality of fluid inertia to fluid density as found by ourselves and described by others for the opposing-jet method of measurement [16].
10
C.G. Hermansky, D.V. Boger /J. Non-Newtonian Fluid Mech. 56 (1995) 1-14
E w m
0.00
I
I
0.05
0.10
0.15
A r e a o f J e t ( c m 2)
Fig. 7. Proportionality constant a'(R) in c m
3 versus
the area of nozzle core in c m
2.
From the foregoing it is suggested that extensional viscosity data on low viscosity fluids, inelastic or elastic, can be corrected from a knowledge of the shear viscosity and density of the fluid. This generality, of course, assumes that the radial and axial velocity distributions between the jets is not appreciably different for elastic and inelastic fluids. While such an assumption is known not to be true for viscous, highly elastic, fluid flow through an abrupt contraction [11], it may be acceptable for opposing-jet rheometry of low viscosity, elastic fluids and for higher viscosity elastic fluids in the limit of low strain rates.
3.2. Low viscosity elastic polymer solutions Application of this technique of correcting opposing-jet data and its significance to the interpretation of such data was demonstrated by testing a series of elastic polyethylene glycol solutions of matched shear viscosites. Specific information on these solutions is given in Table 2. Shear viscosities of 85 and 5 cP were chosen so that data could be compared in regions where fluid inertial corrections were (5 cP) and were not (85 cP) necessary. Fig. 8 shows typical shear viscosity-shear rate data for select mixtures from Table 2. All solutions exhibited a constant viscosity over this range and were well matched in shear viscosity. Data for corresponding glycerol/water mixtures of like viscosity are also shown for reference. Extensional data for the 85 cP P E G solutions are shown in Fig. 9. Over the range of strain rates shown all but the lowest molecular weight P E G solutions were elastic and exhibited a maximum. The 8000 molecular weight P E G solution was inelastic even at a concentration of 34 percent by weight, giving a Trouton ratio of ~ 3. All of the other solutions trended towards a Trouton ratio of 3 h in the limit of low strain rate, in agreement with what would be expected from continuum mechanics.
C.G. Hermansky, D.V. Boger / J. Non-Newtonian Fluid Mech. 56 (1995) 1 14
Table 2 Polyethylene glycol solutions and their properties Molecular weight 8000 100 000 300 000 600 000
Composition % PEG in water
Shear viscosity cP
34.0 8.3 2.6 1.8
87 85 81 84 88
1 000 000
1.45
100 000 300 000 600 000 1 000 000
2.1 0.76 0.48 0.36
5.4 6.0 5.0 5.2
i 11111
:0~0-
~ 0 Glycerol/ Water [ ] PEG 300,000 L~ PEG 600,000 4~ PEG 1,000,000
o
o 0 ._=
¢-
l0
1
I
100
1000
Shear Rate (1/Sec.)
Fig. 8. Viscosity-shear rate plot of aqueous polyethylene glycol solutions selected from Table 2. Maxima in extensional data have been reported by others, e.g. low-viscosity polyacrylamide solutions [21] and in highly elastic, constant viscosity, Boger fluids containing a small quantity of polyacrylamide (0.1%) in a highly viscous corn syrup/water solution [ 13]. However, the polymer solutions in these studies could be considered as dilute, whereas those used to produce the data shown in Fig. 9 are not. Given the concentration of polyethylene glycol used in the present study and the lack of additional data, it is not possible to determine the cause of these maxima at this time. However, possibilities include: the transient nature of the experiment, such that the time required to traverse the measurement volume may be too rapid for the stress to reach equilibrium even though a time independent force is recorded during the measurement; constraints due to molecular "entanglements"; transitions
12
C.G. Hermansky, D.V. Boger / J. Non-Newtonian Fluid Mech. 56 (1995) 1-14 30
1
I"
25
~>
T - - ' I ~
O
A •
AA <>
AA O 20 .=
0 0 •
Q
n-
o
~- 15
•
'~
~
'~"
I
A
¢
A
•
I- 111 5 0 0
S000
10000
15000
Strain Rate
(1/S~.)
20000
Fig. 9. Trouton ratio as a function of strain rate for the 85 cP polyethylene glycol solutions of Table 2: O, 8000; [~, 100 000; A, 300 000; ~ , 600 000 and O, 1 000 000 molecular weight.
from flexible to fully extended rigid rods or even cleavage of the polymer. Of all of the above, the data collected on the more dilute 5 cP polyethylene glycol solutions suggest that the major contribution to the maximum may arise from molecular entanglements. This possibility will be discussed in more detail below. Fig. 10 graphically summarizes the uncorrected extensional data for the polyethylene glycols found to be elastic at the concentrations required to achieve 5 cP, i.e. those with molecular weight of 105 and above. At first glance the uncorrected data would suggest that all four polyethylene glycols are again elastic, having 120
o ¢l cc to
2
I-qD
I
I
I
I
5000
~oooo
lsooo
2oom
100
80
60
m ¢n
40;
=Z 2O
0
Strain Rate (1/Se¢.) Fig. 10. Uncorrected Trouton ratio strain rate data for the 5 cP polyethylene glycol solutions of Table 2: O, 100 000; [3, 300 000; A, 600 000 and ~ , 1 000 000 molecular weight.
C.G. Hermansky, D.V. Boger /J. Non-Newtonian Fluid Mech. 56 (1995) 1-14
13
0 M C
2
I-"0
8O
60
¢~ 40 O ¢J
20 0
v
0
5000
v
10000
15000
20000
Strain Rate (1/Sec.) Fig. 11. Corrected T r o u t o n r a t i o - s t r a i n rate data for the 5 cP polyethylene glycol solutions of Table 2: O, 100000; El, 300000; A, 600000 and ©, 1 000000 molecular weight.
Trouton ratios in excess of 3. However, on application of the fluid inertia correction previously developed the Trouton ratios of all of the P E G solutions are significantly lowered, as shown in Fig. 11. In fact, without the application of the correction, in more extreme cases were the concentration or molecular weight of the polymer to be lowered it would be very likely that a fluid could be mistakenly considered to be elastic when it is not. In addition, the possibility that a maximum may exist in the 10 6 molecular weight solution is accentuated and the data for the 1 × 105 molecular weight P E G becomes nearly rate independent, similar to that which would be expected for dilute rigid rods or semi-dilute rods at high strain rates.
4. Conclusion
An opposed-jet appartus is used to measure an extensional viscosity for low viscosity fluids. For shear viscosities of ~ 75 cP or greater a Trouton ratio of 3 is obtained for inelastic Newtonian fluids, as expected. Results obtained for a series of constant and equal viscosity fluids at 85 cP, constructed with increasing molecular weights for polyethylene glycol, also demonstrated trends which would be expected for polymeric fluids in extensional flows. For shear viscosities significantly lower than 75 cP inertia-like corrections are necessary to force the data to a Trouton ration of 3. Application of this correction to a series of equal and constant viscosity fluids at 5 cP, constructed with a series of increasing molecular weight polyethylene glycols was used to illustrate the significance of the correction and its potential to greatly aid the interpretation of the low shear viscosity extensional property data. It is concluded that the opposed-jet technique is a valuable technique for measuring elasticity in extremely low viscosity fluids especially since other measurements, like
14
C.G. Hermansky, D.V. Boger / J. Non-Newtonian Fluid Mech. 56 (1995) 1-14
the normal stresses and/or the storage modulus, are not possible on such fluids. Such indication of elasticity in dilute solutions would be most useful in phenomena associated with jet breakup and in drag reduction, to quote but two examples.
Acknowledgment C.H. Hermansky would like to acknowledge the support received from DuPont Agricultural Products for this and other related studies done during his stay at the AMPC. D.V. Boger would like to acknowledge the support received from the ARC, both for his work in the AMPC and for support received for work on elastic liquids.
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21]
"J. Ferguson, K. Waiters and C. Wolff, Rheol. Acta., 29 (1990) 571. D.M. Binding, D.M. Jones and K. Walters, J. Non-Newtonian Fluid Mech., 35 (1990) 121. G.G. Fuller, C.A. Cathey, B. Hubbard and B.E. Zebrowski, J. Rheol., 31(3) (1987) 235. W.C. Macsporran, J. Non-Newtonian Fluid Mech., 8 (1981) 119. R.B. Secor, C.W. Macosko and L.E. Scriven, J. Non-Newtonian Fluid Mech., 23 (1987) 355. J. Mewis and A.B. Metzner, J. Fluid Mech., 62 (1974) 593. T. Sridhar, D.A. Nguyen and R.K. Gupta, J. Non-Newtonian Fluid Mech., 40 (1991) 271. V. Tirtaamadja and T. Sridhar, in P. Moldenaers and R. Keunings (Eds.), Theoretical & Applied Rheology, Proc. Xlth Int. Congr. on Rheology, 1992, p. 495. D.F. James and J.H. Saringer, J. Non-Newtonian Fluid Mech., 11 (1982) 317. D.M. Binding and K. Waiters, J. Non-Newtonian Fluid Mech., 30 (1988) 233. D.V. Boger, Ann. Rev. Fluid Mech., 19 (1987) 157. G.I. Taylor, Proc. R. Soc. (London), 146 (1934) 510. P.R. Williams and R.W. Williams, J. Non-Newtonian Fluid Mech., 19 (1985) 53. C.J. Farrell, A. Keller, M.J. Miles and D.P. Pope, Polymer, 21 (1980) 1292. M.J. Miles and A. Keller, Polymer, 21 (1980) 1295. P.R. Schunk, J.M. DeSantos and L.E. Scriven, J. Rheol., 34(3) (1990) 387. A. Chow, A. Keller, A.J. Muller and J.A. Odell, Macromolecules, 21 (1988) 250. C.A. Cathey and G.G. Fuller, J. Non-Newtonian Fluid Mech., 30 (1988) 303. M.R. Mackley and A. Keller, Phil. Trans. R. Soc. London, A278 (1975) 29. F.C. Frank, A. Keller and M.R. Mackley, Polymer, 12 (1971) 467. D.M. Jones, K. Walters and P.R. Williams, Rheol. Acta., 26 (1987) 20.
J. Non-Newtonian Fluid Mech., 56 (1995) 1- 14
Opposing-jet viscometry of fluids with viscosity approaching that of water Clarence G. Hermansky a, David V. Boger b DuPont Agricultural Products, Wilmington, DE 19880-0402, USA b The Advanced Mineral Products Research Centre, The University of Melbourne, Parkville 3052, Vic., Australia
Received 15 December 1994; in revised form 3 June 1994
Abstract Flows which were predominantly uniaxial in extension were generated through 0.5 mm diameter aligned jets. The torque required to keep the jets at a fixed distance apart, and thus the apparent stress generated by the fluid, was measured by a torque rebalance transducer attached to one of the moveable jet arms. Strain rates from 1000 to 20 000 s-t were spanned. Data are presented for 1 to ~150 centipoise (cP) Newtonian water and glycerol/water mixtures. The trends found were consistent with a contribution to the measured torque by the inertia of the fluid, becoming negligibly small as the shear viscosity of the fluid approached ~ 75 cP. A method for correcting opposing-jet measurements made on fluids of viscosity less than 75 cP is described and the extent of the correction illustrated on a series of solutions with identical shear viscosities prepared from polyethylene glycols of molecular weight 8 × 103 to 1 x 10 6. The validity of the correction method for use on non-Newtonian fluids is discussed. The corrected results for the polyethylene glycol solutions of identical shear viscosity clearly illustrate the influence of molecular weight on extensional flow properties. .Keywords: Flow viscosity; Opposing-jet viscometry; Uniaxial flow
1. Introduction The study of rigid rods/polymers a n d flexible polymers in extensional flow fields has received significant a t t e n t i o n in recent years. This focus is, in part, due to a n 0377-0257/95/$09.50 © 1995 - Elsevier Science B.V. All rights reserved SSDI 0377-0257(94)01265-J
2
C.G. Hermansky, D.V. Boger / J. Non-Newtonian Fluid Mech. 56 (1995) 1 14
appreciation that extensional flows occur and can dominate over shear flows in industrial processes which operate at high rates, e.g. coatings, extrusion, lamination, drag-reduction and atomization. While many techniques have been used to generate data in extensional flows [1], it is generally acknowledged that no one technique measures a true extensional viscosity [2]. All suffer from an inability to reach a steady rate of strain as the fluid element under study passes through the measurement volume. Nevertheless, these measurements still remain attractive given their ability to approximate the strain rates developed in real processes. Techniques for measuring the extensional properties of fluids normally fall into two categories: flow through and stagnation point devices [3]. Most of the former rely on the fluid being spinnable, like the tubeless siphon [4], or various spinning techniques [5,6]. Although significant strides have now been made in the measurement of elongational properties [7,8], such devices are still limited to low rates of strain and generally to highly viscous or elastic fluids, since a column of at least a few centimeters of the fluid under study must be self-sustaining over a range of strain rates. Alternatively, orifice flow techniques which measure the pressure drop across a contraction offer an option which overcomes the need to have a spinnable fluid [9]. However, interpretation of the orifice data even in Newtonian liquids is not straightforward and can be further compounded by the presence of recirculating vortices and flow instabilities in viscoelastic fluids [10,11]. Conversely, stagnation point devices such as the roll mill [12], lubricated-die converging flow rheometer [13], cross-slot cell [14,15] and the opposing-jet device of this study can be used to collect extensional data in low viscosity fluids. With the exception of the lubricated-die converging flow rheometer which requires large volumes (100 liters) to operate, any of these devices can be used to make measurements on low viscosity fluids provided the limitation of short residence time in the measurement volume is acknowledged. Recent publications using the opposing-jet technique to measure rheological properties have focused on a critique of the technique [ 16] and measurement of radial flow velocity distributions and pressure drop across the jets [17], its utility in measuring the extensional properties of dilute and semi-dilute solutions of rigid collagen rods [ 18] and various concentrations of Xanthan and flexible polyacrylamides in high viscosity Newtonian glycerol/water media [3]. In this study we have selected the Rheometrics RFX opposing jet device since it is commercially available, has the potential to span a wide range of strain rates, and because we are not aware of any attempt to address its use on fluids whose shear viscosities are less than 100 centipoise. What follows are the results of extensional measurements made on Newtonian glycerol/water mixtures and on elastic polymer solutions whose shear viscosities range from 1 to ~ 150 centipoise (cP). For the Newtonian fluids, contrary to what should be expected, Trouton's rule was not obeyed when the shear viscosity of the fluid was below ~ 75 cP, due to the fluid inertia associated with high rates of flow. Using data on the Newtonian fluids tested, a data correction technique was developed which is usable over a strain rate range of at least 1000 to 20 000 s -~. Adherence to Trouton's rule was demonstrated for Newtonian fluids and the utility
C.G. Hermansky, D.V. Boger [ J. Non-Newton&n Fluid Mech. 56 (1995) 1-14
3
of the correction technique illustrated by application to a series of elastic polyethylene glycol solutions of matched shear viscosities.
2. Experimental apparatus and materials
2. I. Materials and methods The water used in this study was taken from a Millipore "Milli-Q" water purification unit. Reagent grade glycerol purchased from BDH Chemicals was used. A series of polyethylene glycols (PEG) ranging from 8.9 x 103 to 1.0 x 10 6 in molecular weight was purchased from Aldrich Chemical Co. and used as received. All compositions were prepared by weight and are reported as weight percents. All mixtures were used within a week of preparation. The aqueous P E G solutions were allowed to turn end over end for several hours to days depending on the concentration and molecular weight. Shear viscosity was measured at 25.0°C (_+0.1°C) using a C a r r i - M e d constant stress rheometer equipped with a 6 cm, 2 ° cone and plate geometry with a 56 micron cone truncation. A broad enough range of stress was applied during each measurement to insure that a shear rate range of at least 1 to 800 s-l was spanned. Extensional viscosity measurements were determined as a function of strain rate using a Rheometrics RFX, at 25.0°C (_+0.5°C). The Rheometrics R F X is the commercial version of the opposing-jet device described by other authors [3,18-20]. A schematic of the opposed-nozzle device (Fig. 1) has been reproduced from Schunk et al. [ 16] to illustrate the experimental set up. Unless otherwise noted, all data were taken using a set of 0.5 mm diameter nozzles fixed at a nozzle-to-nozzle separation of 0.5 mm. The software supplied with the commercial instrument was used to automate data collection. However, the output of the torque transducer was independently monitored on a strip chart recorder to insure adequate time was allowed for torque (M) to come to "equilibrium" and to detect the presence of flow instabilities if present. The instrument operates by stepping through a series of pre-programmed strain rates and taking an average of the torque over the last few seconds of the measurement. Fluid flow through the nozzles is accomplished by a stepper motor attached to a pair of 20 cc syringes which draw the fluid of interest through the nozzles via the torque arms at a fixed volumetric flow rate (Q). During this time the torque (M) produced at the torque rebalance transducer is measured. In the instrument of this study, the jets were attached to a pivot arm which was 7.62 centimeters in length (L). All calculations were performed by the software. The force developed at the face of each nozzle was determined by dividing the torque on the moment arm by its length (M/L). This force was then converted to an apparent stress at the face of each nozzle on division by the area of the nozzle bore (re, R2), where R is the radius of the nozzle bore. The strain rate ( 0 is similarly derived by dividing each flow rate by the nozzle bore area and half distance of separation between the nozzles (2Q/r~R2d), where d is the distance to the midplane in Fig. 1. Thus the apparent
4
C.G. Hermansky, D.V. Boger/J. Non-Newtonian Fluid Mech. 56 (1995) 1-14
TORQUE M ~...~f
PIVOTAXIS TIP ~- FACEINCLINATION THICKNESS ~
PIVOTARML B O R E ~ MIDPtANE
-.~ ! ¢
CENTERLINE
SEPARATIONd Fig. 1. Schematic diagram of opposing-jet rheometer.
extensional viscosity (r/e) is given by the quotient of the stress and strain rate (Eq. (1)), provided the stepper motor and torque transducer are properly calibrated and care is taken to assure that the transducer is nulled before each measurement. Md ~ - 2QL"
(1)
3. Results and discussion 3. I. Newtonian fluids
Fig. 2 shows the ratio of the extensional viscosity (qe) to three times the shear viscosity (3r/s) for two glycerol/water mixtures whose shear viscosities were 156 cP (90% glycerol/water) and 110 cP (88% glycerol/water). The results are consistent with a Trouton ratio of 3 to within ~ 15% over a strain rate range of 500 to 8000 s -]. This range represents both the upper and lower limits attainable by the instrument for fluids having the above shear viscosity. Fluids of lesser viscosity allow greater flexibility expanding the high end of the strain rates attainable. Fig. 3 demonstrates the apparent discrepancy which results in the Trouton ratio (r/e/r/s) when the shear viscosity is reduced to below ~ 100 cP. The Trouton ratio increases with increasing strain rate and decreases with increasing viscosity in all cases. The data shown are for Newtonian, glycerol/water mixtures with the compositions, measured shear viscosities and linear regression slopes summarized in Table 1. The slopes given in Table 1 were determined by assuming (a) the best fit intercept; and
C.G. Herrnansky, D.V. Boger / J. Non-Newtonian Fluid Mech. 56 (1995) 1 14
5
2.0 ,~ 4.m (D 0 0 W ~>
1.8
II .C if) X
1.2
~
0.8
1.6 1.4
tll r,-
0.6
~-
0.4
:< LU
0.2
.2ffl
0
0
[] [] []
1.0
0.0 0
n
I~
U
[]
2000
r7
~
4000
[]
[]
tO'
6ooo
sooo
10000
Strain Rate (1/Sec.)
Fig. 2. Ratio of the extensional to three times the shear viscosity as a function of strain rate for: C), 90% glycerol/10% water (156 cP) and [~, 88% glycerol/12% water (110 cP) mixtures. i •
~ . 150
1 Cp&
0 0
._w > 100 e.(/) n C 0 :~ 50 m O~ C
o
UJ
~ 5000
10000
3.6 cps 5.5 cps. 10.5 cps
15000
20000
25000
Strain Rate (1/Sec.)
Fig. 3. Trouton ratio for the Newtonian glycerol/water mixture of Table 1 which have shear viscosities below 100 cP. (b) an intercept o f 3, the latter being consistent with the T r o u t o n ratio which would be expected in the limit o f low strain rates. The data show that this assumption had little effect on the fitted slope. I f such departure f r o m the expected T r o u t o n ratio o f three were attributed to fluid inertia an additional contribution to the measured torque which would be p r o p o r t i o n a l to the density o f the fluid (p) and to the square o f its velocity (U 2) might be expected as follows: M m = M c + a ' ( R ) p U z.
(2)
6
C.G. Hermansky, D.V. Boger /J. Non-Newtonian Fluid Mech. 56 (1995) 1-14
Table 1 Newtonian compositions, shear viscosities and slope from Trouton ratio-strain rate plots of Fig, 3 Composition % Glycerol in water
Shear viscosity (cP)
Slope a
slope b
90.0 88.0 85.0 75.0 60.0 50.0 40.0 0.0
156 110 77 30 10.5 5.5 3.6 1.3
0.0000 0.0000 0.0001 0.0003 0.0008 0.0015 0.0026 0.0074
0.0000 0.0000 0.0000 0.0003 0.0007 0.0017 0.0028 0.0078
1.3 1.0
0.0074 0.0115
Water at 25°C Carbon tetrachloride at 18°C
a Slope of the linear regression of the Trouton ratio-strain rate data of Fig. 3. b Slope of the linear regression of the Trouton ratio-strain rate data of Fig. 3, holding the intercept fixed at a value of 3.0.
In Eq. (2), the value o f a'(R) is anticipated to be a function of nozzle radius but not o f nozzle separation, for reasons discussed below. M m is the torque measured by the instrument and M c is the corrected torque, i.e. the torque which would have been measured in the absence o f fluid inertial effects. While the general f o r m o f a'(R) might include the consideration that it be a function o f separation [i.e. a'(R,d)], measurements by ourselves and others [3,16] indicate that it is not. In the present study, the slope o f the measured extensional viscosity versus strain rate plots o f Fig. 3 did not vary from that reported in Table 1 by m o r e than +_ 10% over a range o f separation (d) o f 0.35 to 1.0 mm. At higher separations, while the measured extensional viscosity was still linear in strain rate, a decreasing trend in b o t h the values o f qe and slope was found. This finding is consistent with existing literature indicating that the strain rate remains constant across the gap, provided the gap between nozzles is less than 2 - 3 times their diameter [19]. As noted earlier, the strain rate (~) is given by 2Q ~R2d,
(3)
where Q is the flow rate, R is the radius o f the nozzles bore and d is half the distance o f separation between the nozzles. Given the mean velocity O o f the fluid flow t h r o u g h the jets has been shown to be given by Q/ztR 2 to within ~ 15% for 0.5 m m jets [19], to a g o o d approximation the m e a n velocity o f the fluid can be rewritten in terms o f the strain rate on c o m b i n a t i o n with Eq. (3), to give d~ 2
O = --.
(4)
C.G. Hermansky, D.V. Boger / J. Non-Newtonian Fluid Mech. 56 (1995) 1-14
103
I
O
1.0 cps.
V 3o.o~,,. 10 =
A
E
Z
7
•. "
~
~jp
~
101
0 I--
10 0
i/ 10 s
I 10 ~
Strain Rate {1/Sec.)
Fig. 4. Measured torque versus strain rate for Newtonian glycerol/watersolutions of composition and shear viscosity selected from Table 1. Substituting Eq. (4) for the fluid velocity given in Eq. (2) results in the relationship to be expected between the measured (M,,) and corrected (Me) torques in terms of the contribution made by the inertia of the fluid. a'(R)d2p42
M m = M c q-
(5)
4
Eq. (5) suggests that the measured torque (Mm) should show quadratic behavior if plotted against strain rate, if a significant contribution from fluid inertia is present. Such behavior is demonstrated in Fig. 4 for a selection of fluids in Table 1. The data for the 1.0, 5.5, 30 and 77 cP Newtonian, glycerol/water fluids also serve to show that the quadratic behavior is all but diminished in the 77 cP (85% glycerol/ water) fluid, over the range of operation of the instrument. In the case of this particular fluid, cavitation occurs in the syringes at ~ 12 000 s- 1, thus defining the upper limit of the instruments capability. To demonstrate that the linear trend in the Trouton ratio-strain rate data of Fig. 3 directly follows from the quadratic behavior given by Eq. (5), the following substitutions and rearrangements were made. Eq. (3) was rearranged to give the flow rate in terms of the strain rate and inserted into Eq. (1), to eliminate Q and arrive at the generic relationship between the torque and the strain rate given by (6)
M = 7"cR2Lrle 4.
Substitution of this into Eq. (5) and subsequent rearrangement, to give both the measured and corrected extensional viscosity in terms of the contributions to be expected from the inertia of the fluid, gives a'(R)pd 2 . qc = qe
~4 ~ L R
~.
(7)
8
C.G. Hermansky, D.V. Boger/J. Non-Newtonian Fluid Mech. 56 (1995) 1-14
In Eq. (7) r/e is the extensional viscosity calculated from the measured torque and r/c is the extensional viscosity corrected for the inertia of the fluid under study. Division of Eq. (7) by the shear viscosity (r/s) results in a relationship between the measured and corrected Trouton ratio as a function of strain rate, as shown below. r/c
r/e
a'(R)pd 2 .
r/s
r/s
4gLR2r/s
-
-
E.
(8)
From Eq. (8) it is clear that the differences in the slopes of the Trouton ratio-strain plots of Fig. 3 will only be consistent with the proposed dominance of fluid inertia to the meaured torque if the value of a'(R) is found to be constant for a given jet radius. To test this, the densities of the glycerol/water solutions of Table 1 were assumed to be constant and a plot of the slopes from Table 1 versus the reciprocal of the Newtonian shear viscosity (1/r/s) was constructed. The result is shown in Fig. 5. The reciprocal dependence found clearly illustrates that the value of a'(R) is constant and has a value of 0.708 cm 3. While maintenance of a constant strain across the gap limits the working range of the instrument to 2 - 3 times the diameter of the nozzles in use, the value of a'(R) has been shown to be constant, i.e. the numerical value of a'(R) calculated from Fig. 5 can be said to be applicable for all 0.5 mm jets. However, Eq. (2) acknowledges the contribution of total mass per unit time to the torque, so a dependence of a'(R) on the area of the jet (i.e. its radius) should be anticipated. The nature of this dependence was explored by measuring the extensional viscosity versus strain rate for water with 0.5, 1.0 and 2.0 mm jets in place. The value of d in each experiment was again matched to the radius of the jets being used. The resulting data is shown in Fig. 6.
0.008
I
I
I
I
I
0 Q.
o
t/) 0.006 n'C t,/)
0.004
|
o
I s,op,- 0.00741
0 0.002 > x I.[.I 0,~0 0.0
0.2
0.4
0.6
o.a
1.0
1.2
Reciprocal of Shear Viscosity Fig. 5. Slope of the Trouton ratio-strain rate plots from Table 1 versus the reciprocal of the fluid shear viscosity in cP.
C.G. Hermansky, D.F. Boger / J. Non-Newtonian FluM Mech. 56 (1995) 1-14
9
°°IJ Z
'~ 0
Slope: 8.176 (2.8 m m J e t s )
300
~
ua "o Ip
76
200
L
¢u
100
Slope: 8.8874 (8.5 m m Jets)
~-0
o
I
I
sooo
~oooo
J
~sooo
I
zoooo
Strain Rate (I/Sec.}
Fig. 6. Measured extensional viscosity of water over a range of strain rates using 0.5, 1.0 and 2.0 mm nozzles.
In all three cases, the linearity of the measured extensional viscosity-strain rate plot was retained over the range of strain rates accessible to the instrument. And, as expected, a considerable difference in the slope of each plot was seen. The numerical value of the slope for each plot and the respective nozzle used is also shown in Fig. 6. From the slopes of Fig. 6 and Eq. (7) the values of a'(R) were calculated for each of the nozzles and were found to be 0.708, 2.64 and 16.8 cm 3 for the 0.5, 1.0 and 2.0 mm nozzles, respectively. On considering recent work which suggested that inertial contributions would be a function of the area swept out by the measurement volume [ 16], it seemed prudent to test these data by plotting the values of a'(R) against the area of the nozzle bore being used. The result is shown in Fig. 7. While the data is limited it does not support an inertial contribution which scales with area and suggests that the radial and axial velocity profiles are not appreciably changed by the choice of nozzle. This is consistent with previously reported photon-correlation velocimetry scans which have shown the radial velocity distribution to be constant for pure solvent and dilute high molecular weight polymer solutions [17]. Also consistent with the relationship given by Eq. (8) is limited, but significant, data which shows that the slope of Fig. 3 are directly proportionality to the density of the fluid. Specifically, Table 1 gives the measured slopes for water at 25°C and for carbon tetrachloride at 18°C. Both fluids have nearly identical shear viscosities at these temperatures, with the ratio of their densities being 1.59. A ratio of 1.55 (within 5%) was calculated from the carbon tetrachloride/water slopes reported in Table 1. This result is consistent with the expected proportionality of fluid inertia to fluid density as found by ourselves and described by others for the opposing-jet method of measurement [16].
10
C.G. Hermansky, D.V. Boger /J. Non-Newtonian Fluid Mech. 56 (1995) 1-14
E w m
0.00
I
I
0.05
0.10
0.15
A r e a o f J e t ( c m 2)
Fig. 7. Proportionality constant a'(R) in c m
3 versus
the area of nozzle core in c m
2.
From the foregoing it is suggested that extensional viscosity data on low viscosity fluids, inelastic or elastic, can be corrected from a knowledge of the shear viscosity and density of the fluid. This generality, of course, assumes that the radial and axial velocity distributions between the jets is not appreciably different for elastic and inelastic fluids. While such an assumption is known not to be true for viscous, highly elastic, fluid flow through an abrupt contraction [11], it may be acceptable for opposing-jet rheometry of low viscosity, elastic fluids and for higher viscosity elastic fluids in the limit of low strain rates.
3.2. Low viscosity elastic polymer solutions Application of this technique of correcting opposing-jet data and its significance to the interpretation of such data was demonstrated by testing a series of elastic polyethylene glycol solutions of matched shear viscosites. Specific information on these solutions is given in Table 2. Shear viscosities of 85 and 5 cP were chosen so that data could be compared in regions where fluid inertial corrections were (5 cP) and were not (85 cP) necessary. Fig. 8 shows typical shear viscosity-shear rate data for select mixtures from Table 2. All solutions exhibited a constant viscosity over this range and were well matched in shear viscosity. Data for corresponding glycerol/water mixtures of like viscosity are also shown for reference. Extensional data for the 85 cP P E G solutions are shown in Fig. 9. Over the range of strain rates shown all but the lowest molecular weight P E G solutions were elastic and exhibited a maximum. The 8000 molecular weight P E G solution was inelastic even at a concentration of 34 percent by weight, giving a Trouton ratio of ~ 3. All of the other solutions trended towards a Trouton ratio of 3 h in the limit of low strain rate, in agreement with what would be expected from continuum mechanics.
C.G. Hermansky, D.V. Boger / J. Non-Newtonian Fluid Mech. 56 (1995) 1 14
Table 2 Polyethylene glycol solutions and their properties Molecular weight 8000 100 000 300 000 600 000
Composition % PEG in water
Shear viscosity cP
34.0 8.3 2.6 1.8
87 85 81 84 88
1 000 000
1.45
100 000 300 000 600 000 1 000 000
2.1 0.76 0.48 0.36
5.4 6.0 5.0 5.2
i 11111
:0~0-
~ 0 Glycerol/ Water [ ] PEG 300,000 L~ PEG 600,000 4~ PEG 1,000,000
o
o 0 ._=
¢-
l0
1
I
100
1000
Shear Rate (1/Sec.)
Fig. 8. Viscosity-shear rate plot of aqueous polyethylene glycol solutions selected from Table 2. Maxima in extensional data have been reported by others, e.g. low-viscosity polyacrylamide solutions [21] and in highly elastic, constant viscosity, Boger fluids containing a small quantity of polyacrylamide (0.1%) in a highly viscous corn syrup/water solution [ 13]. However, the polymer solutions in these studies could be considered as dilute, whereas those used to produce the data shown in Fig. 9 are not. Given the concentration of polyethylene glycol used in the present study and the lack of additional data, it is not possible to determine the cause of these maxima at this time. However, possibilities include: the transient nature of the experiment, such that the time required to traverse the measurement volume may be too rapid for the stress to reach equilibrium even though a time independent force is recorded during the measurement; constraints due to molecular "entanglements"; transitions
12
C.G. Hermansky, D.V. Boger / J. Non-Newtonian Fluid Mech. 56 (1995) 1-14 30
1
I"
25
~>
T - - ' I ~
O
A •
AA <>
AA O 20 .=
0 0 •
Q
n-
o
~- 15
•
'~
~
'~"
I
A
¢
A
•
I- 111 5 0 0
S000
10000
15000
Strain Rate
(1/S~.)
20000
Fig. 9. Trouton ratio as a function of strain rate for the 85 cP polyethylene glycol solutions of Table 2: O, 8000; [~, 100 000; A, 300 000; ~ , 600 000 and O, 1 000 000 molecular weight.
from flexible to fully extended rigid rods or even cleavage of the polymer. Of all of the above, the data collected on the more dilute 5 cP polyethylene glycol solutions suggest that the major contribution to the maximum may arise from molecular entanglements. This possibility will be discussed in more detail below. Fig. 10 graphically summarizes the uncorrected extensional data for the polyethylene glycols found to be elastic at the concentrations required to achieve 5 cP, i.e. those with molecular weight of 105 and above. At first glance the uncorrected data would suggest that all four polyethylene glycols are again elastic, having 120
o ¢l cc to
2
I-qD
I
I
I
I
5000
~oooo
lsooo
2oom
100
80
60
m ¢n
40;
=Z 2O
0
Strain Rate (1/Se¢.) Fig. 10. Uncorrected Trouton ratio strain rate data for the 5 cP polyethylene glycol solutions of Table 2: O, 100 000; [3, 300 000; A, 600 000 and ~ , 1 000 000 molecular weight.
C.G. Hermansky, D.V. Boger /J. Non-Newtonian Fluid Mech. 56 (1995) 1-14
13
0 M C
2
I-"0
8O
60
¢~ 40 O ¢J
20 0
v
0
5000
v
10000
15000
20000
Strain Rate (1/Sec.) Fig. 11. Corrected T r o u t o n r a t i o - s t r a i n rate data for the 5 cP polyethylene glycol solutions of Table 2: O, 100000; El, 300000; A, 600000 and ©, 1 000000 molecular weight.
Trouton ratios in excess of 3. However, on application of the fluid inertia correction previously developed the Trouton ratios of all of the P E G solutions are significantly lowered, as shown in Fig. 11. In fact, without the application of the correction, in more extreme cases were the concentration or molecular weight of the polymer to be lowered it would be very likely that a fluid could be mistakenly considered to be elastic when it is not. In addition, the possibility that a maximum may exist in the 10 6 molecular weight solution is accentuated and the data for the 1 × 105 molecular weight P E G becomes nearly rate independent, similar to that which would be expected for dilute rigid rods or semi-dilute rods at high strain rates.
4. Conclusion
An opposed-jet appartus is used to measure an extensional viscosity for low viscosity fluids. For shear viscosities of ~ 75 cP or greater a Trouton ratio of 3 is obtained for inelastic Newtonian fluids, as expected. Results obtained for a series of constant and equal viscosity fluids at 85 cP, constructed with increasing molecular weights for polyethylene glycol, also demonstrated trends which would be expected for polymeric fluids in extensional flows. For shear viscosities significantly lower than 75 cP inertia-like corrections are necessary to force the data to a Trouton ration of 3. Application of this correction to a series of equal and constant viscosity fluids at 5 cP, constructed with a series of increasing molecular weight polyethylene glycols was used to illustrate the significance of the correction and its potential to greatly aid the interpretation of the low shear viscosity extensional property data. It is concluded that the opposed-jet technique is a valuable technique for measuring elasticity in extremely low viscosity fluids especially since other measurements, like
14
C.G. Hermansky, D.V. Boger / J. Non-Newtonian Fluid Mech. 56 (1995) 1-14
the normal stresses and/or the storage modulus, are not possible on such fluids. Such indication of elasticity in dilute solutions would be most useful in phenomena associated with jet breakup and in drag reduction, to quote but two examples.
Acknowledgment C.H. Hermansky would like to acknowledge the support received from DuPont Agricultural Products for this and other related studies done during his stay at the AMPC. D.V. Boger would like to acknowledge the support received from the ARC, both for his work in the AMPC and for support received for work on elastic liquids.
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