The Yield Stress Myth
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Article CC:CCL Rheologica acta Rheologica acta Rheologica acta. The yield stress myth? Barnes, 24 4 1985 323-326 0035-4511 [TN:627955][ODYSSEY:206.107.42.72/PUL]
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Rheologica Acta
Rheol Acta 24:323-326 (1985)
ORIGINAL C O N T R I B U T I O N S
The yield stress myth? *)
H. A. Barnes and K. Walters Unilever Research Laboratory, Port Sunlight (U.K.) and Department of Applied Mathematics, University College of Wales, Aberystwyth (U.K.) Abstract: New experimental data obtained from constant stress rheometers are used to show that the yield stress concept is an idealization, and that, given accurate measurements, no yield stress exists• The simple Cross model is shown to be a useful empiricism for many non-Newtonian fluids, including those which have hitherto been thought to possess a yield stress. Key words:Yield stress, viscosity, Cross model
1. Introduction
This paper is concerned with practical and industrial rheology. In these areas, it is generally conceded that the apparent viscosity function is of paramount importance and other non-Newtonian effects are of secondary influence. Of course, it is certainly possible to quote situations, in polymer processing for example, where the extensional viscosity is more important than the shear viscosity, and, further, that viscoelasticity as manifested through normal-stress differences can also be of significant moment in such operations as mixing. Nevertheless, the shear viscosity still maintains its position as the most important material function in an industrial context and this is certainly borne out by industrial practice. In this paper, we therefore confine attention to the shear stress r/shear rate q behaviour of non-Newtonian materials, representative examples of which are given in figure 1. The "yield stress" hypothesis associated with Bingham (and non-Bingham) l) plastic materials has long been widely accepted and considered useful if not indispensable. Indeed, the yield stress itself is now a British Standards term (1975) defined as "that stress *) Paper presented at the 9th International Congress on Rheology, Acapulco, Mexico, October 1984. 1) Non-Bingham behaviour is associated with a non-linear behaviour between z and q above the yield stress ry.
49
Fig. 1. Representative (z, q) rheograms below which the substance is an elastic solid and above it a liquid with a plastic viscosity r/p, such that
r = ry + qr]p.
i
,.q
(1)
One of the purposes of the present paper is to show that the yield stress concept, whether presented in the above form or any of its variants, is an idealization and that, given accurate measurements, no yield stress exists.
2. The yield stress myth
The concept of a yield stress was standard until about 30 years ago. Then improvements in measuring
324 techniques, particularly in the low shear rate range, showed that eq. (1) was too simplistic and that Bingham-type behaviour was an approximation, essentially valid only at high shear rates. Scott Blair [5] anticipated this state of affairs when he proposed that the yield stress was "that stress below which no flow can be observed under the conditions of experimentation". A new range of constant stress instruments have recently been developed (like the Deer Rheometer) which allow accurate stress measurements to be made at shear rates as low as 1 0 - 6 s -1. In our opinion, such instruments are exploding the yield stress myth and we assert that no one has ever measured a yield stress, they have only extrapolated to it. In effect, yield stress only defines what cannot be measured. If a material "flows" at high stresses it will also flow, however slowly, at low stresses. We conjecture that the viscosity is always finite and, if measured over a wide enough shear rate range, shows the type o f q/q behaviour shown in figure 2. At low enough shear rates, the viscosity reaches a Newtonian plateau. Similarly, at high enough shear rates there is a so-called "second Newtonian" region. Between the two extremes, the viscosity falls monotonically and a "power law" behaviour is often observed with i/ varying as qm, where m is a negative exponent 2). To develop our arguments, we consider two test cases; many others could be quoted. The first concerns a commercially available PVA latex adhesive (Sample A). Using a conventional rheometer, like the Weissenberg Rheogoniometer, it is difficult to obtain accurate results at shear rates below about 10 -2 s -1 and certainly not below 10 -3 s -1. Testing the sample on such an instrument yields the type of graphs shown in figure 3 a - c , from which one might be tempted to conclude that the material has a yield stress, the value of which depends on the shear rate range used. However, the availability of results at very low shear rates now shows a Newtonian plateau, characteristic of standard shearthinning fluids. The definition of a yield stress would be reasonable on the basis of figure 3 a - c but it would be unacceptable with the added data provided in figure 3 d - e . A further example of our general argument is provided by a 0.5% solution of Carbopol (Sample B), a well known thickening agent for aqueous systems3).
Rheologica Acta, Vol. 24, No. 4 (1985) Figure 4 shows (r/, q) data for the solution with a clear indication of a high Newtonian plateau at low shear rates. In both cases, we note in passing that the power law model, which is appropriate over a limited shear rate range, eventually breaks down and that the viscosity does not tend to an infinite values as q ~ 0 as the power law model dictates. To develop further our general argument, we give closer consideration to the test samples A and B studied in figures 3 and 4; many others could be quoted. Both liquids were tested on the Deer Rheometer Mk II, which is a constant stress rheometer capable of measuring over a wide shear rate range and, for the kind of samples under consideration, the instrument can give results at shear rates as low as 10 -6 s-~. If we consider the results over various portions of the shear rate range, we can virtually produce a historical development of the yield stress concept (table 1). The range 10-150 s- l represents that typical of older laboratory viscometers and figure 3a shows that the material exhibits Bingham-type behaviour. If one now looks at data a decade lower in shear rate (representing the better type of laboratory viscometer) then again Bingham behaviour is seen, but the intercept is now lower (fig. 3b). Going down even further to the 1 10-3s -j range (as one can just manage to do on a Weissenberg Rheogoniometer), and plotting the results this time as stress/log shear rate, then a yield stress still appears to be present (fig. 3c). However, when the full range of the constant stress rheometer is utilized, we see that eventually a Newtonian plateau is reached (fig. 3 e).
Table 1 Shear rate range (s-1) 150 -10
10 1 - 1 -10 -3
"yield stress" (Pa) 9.5 5.0 1.0 0
10-3-10 -5
2) As is clear from figure 2 we limit ourselves in the present paper to shear-thinning non-Newtonian materials. 3) Details of the preparation of the sample and its complete rheological characterization will be published separately.
)]l
Fig. 2. Schematic representation of shear-thinning behaviour
20
T c~
14
15
~
10
o_ tO
~ lC
if) uJ I ffl
UJ
~ 6
< I.U -r 2
I
I I I I
Z,
8 SHEAR RATE ×10 -1
(s"1)
12
15 SHEAR RATE (s -1)
101
/
c)
5
Q_ tO
L~ 100 3
rv
~¢o "~'°'
n,. < uJ ..r to
1 = = "--'-" . . . . . . . . . . . .
7I I I
I
10-4
10-3 SHEAR RATE [s -1}
e)
10o
10-
/,
I
I
i0-~ SHEAR
i0-2 RATE (s-I)
L I 0 ~ ~
Fig. 3. Viscometric data for sample A
10°10.s
'
'0.3 ' 1'0_ 1 ' SHEAR RATE (s -1)
1'01
10 3
0.5°1o CARBOPOL 934 105
u
A
tn
D_ >i.--
10~
m 103
Fig. 4. Viscometric data for sample B
10 2
I
I
10 " 5
I RATE ( s -1)
IC
I 10 -3
I
~,
I
10-I
SHEAR
326 Table 2 System Reference Approximate limit of Newtonian behaviour r/0 (Pa s) K (s)
Rheologica Acta, Vol. 24, No. 4 (1985)
n -
(s-l)
Sample B Sample A Synovial fluid Polyacrylamide solution Anionic polyacrylamide Aqueous PVA
This work This work Zeidler and Altman [6] Boger [1] Lenge and Rehage [4] Cross [3]
10-s
10 . 4
10-2 10-l 10° 101
1.4 x l0 s 7.6 x 103 13.2 1.82 0.1 11.1
7.5 x 10 4 5 x 103 19 1.5 2 0.013
0.845 0.80 0.685 0.6 0.85 0.66
When we plot the data over the whole range of shear rates, figure 3e demonstrates our postulate that all liquids show Newtonian behaviour at low enough shear rates. Further, the behaviour in the middle shear rate range is of the power-law type and a second Newtonian plateau is approached at very high shear rates. Our low shear rate results for sample B show similar behaviour (fig. 4). This time, the plateau is higher and is not attained until even lower shear rates are reached.
3. Choice of models to describe low shear rate behaviour
4. Conclusion
With the availability of the new generation of constant stress rheometers, it is possible to show that the yield stress hypothesis, which has hitherto been a useful empiricism, is no longer necessary, and that fluids which flow at high stresses will flow at all lower stresses; i.e. the viscosity, although large, is always finite and there is no yield stress. Numerous simple models exist to simulate such departure from Newtonian behaviour. We have used the Cross model as an illustrative example.
References
If one restricts attention to low shear rates, it is of interest to seek simple flow models which show first a Newtonian plateau and then a power-law type behaviour at higher shear rates. Equations of the form 1 1 + (Kq) n (Form 1) (2) r/0 (1 + (Kq)2) m (Form 2)
170 or
1. Boger DV (1977) Nature 265:127 2. British Standard 5168 (1975) Glossary of Rheological Terms 3. Cross MM (1965) J Coil Sci 20:417 4. Lenge W, Rehage G (1980) In: Astarita G, Marrucci G, Nicolais L (eds) Rheology. vol 2, Plenum Press, p 354 5. Scott Blair GW (1933) J Applied Phys 4:113 6. Zeidler H, Altman S (1980) In: Astarita G, Marrucci G, Nicolais L (eds) Rheology, vol 3, Plenum Press, p 512 (Received October 10, 1984) Authors' addresses: Dr. H. A. Barnes Unilever Research Laboratory Port Sunlight L63 3JW (U.K.) Prof. K. Walters Department of Applied Mathematics The University College of Wales Penglais, Aberystwyth SY23 3BZ (U.K.)
satisfy these requirements with n > 0, m > 0. The Cross model [3] is an example of Form 1 and we take this model as an illustrative example. We show in table 2 the parameters necessary to fit the data for samples A and B and also other non-Newtonian fluids available in the literature. We do this to show that the same general type of flow curve is found, irrespective of the shear rate at which departure from Newtonian behaviour is first observed.
CALL #: LOCATION:
TYPE: JOURNAL TITLE: USER JOURNAL TITLE: CSL CATALOG TITLE: ARTICLE TITLE: ARTICLE AUTHOR: VOLUME: ISSUE: MONTH: YEAR: PAGES: ISSN: OCLC #: CROSS REFERENCE ID: VERIFIED:
Odyssey IP: 206.107.42.72/PUL
QC189 .R45 CSL :: Grand :: GRANDDEPOS BOOKSTACKS
Article CC:CCL Rheologica acta Rheologica acta Rheologica acta. The yield stress myth? Barnes, 24 4 1985 323-326 0035-4511 [TN:627955][ODYSSEY:206.107.42.72/PUL]
BORROWER: PATRON:
NTE ::
Main Library
This material may be protected by copyright law (Title 17 U.S. Code) System Date/Time: 2/21/2013 3:20:22 PM MST
Rheologica Acta
Rheol Acta 24:323-326 (1985)
ORIGINAL C O N T R I B U T I O N S
The yield stress myth? *)
H. A. Barnes and K. Walters Unilever Research Laboratory, Port Sunlight (U.K.) and Department of Applied Mathematics, University College of Wales, Aberystwyth (U.K.) Abstract: New experimental data obtained from constant stress rheometers are used to show that the yield stress concept is an idealization, and that, given accurate measurements, no yield stress exists• The simple Cross model is shown to be a useful empiricism for many non-Newtonian fluids, including those which have hitherto been thought to possess a yield stress. Key words:Yield stress, viscosity, Cross model
1. Introduction
This paper is concerned with practical and industrial rheology. In these areas, it is generally conceded that the apparent viscosity function is of paramount importance and other non-Newtonian effects are of secondary influence. Of course, it is certainly possible to quote situations, in polymer processing for example, where the extensional viscosity is more important than the shear viscosity, and, further, that viscoelasticity as manifested through normal-stress differences can also be of significant moment in such operations as mixing. Nevertheless, the shear viscosity still maintains its position as the most important material function in an industrial context and this is certainly borne out by industrial practice. In this paper, we therefore confine attention to the shear stress r/shear rate q behaviour of non-Newtonian materials, representative examples of which are given in figure 1. The "yield stress" hypothesis associated with Bingham (and non-Bingham) l) plastic materials has long been widely accepted and considered useful if not indispensable. Indeed, the yield stress itself is now a British Standards term (1975) defined as "that stress *) Paper presented at the 9th International Congress on Rheology, Acapulco, Mexico, October 1984. 1) Non-Bingham behaviour is associated with a non-linear behaviour between z and q above the yield stress ry.
49
Fig. 1. Representative (z, q) rheograms below which the substance is an elastic solid and above it a liquid with a plastic viscosity r/p, such that
r = ry + qr]p.
i
,.q
(1)
One of the purposes of the present paper is to show that the yield stress concept, whether presented in the above form or any of its variants, is an idealization and that, given accurate measurements, no yield stress exists.
2. The yield stress myth
The concept of a yield stress was standard until about 30 years ago. Then improvements in measuring
324 techniques, particularly in the low shear rate range, showed that eq. (1) was too simplistic and that Bingham-type behaviour was an approximation, essentially valid only at high shear rates. Scott Blair [5] anticipated this state of affairs when he proposed that the yield stress was "that stress below which no flow can be observed under the conditions of experimentation". A new range of constant stress instruments have recently been developed (like the Deer Rheometer) which allow accurate stress measurements to be made at shear rates as low as 1 0 - 6 s -1. In our opinion, such instruments are exploding the yield stress myth and we assert that no one has ever measured a yield stress, they have only extrapolated to it. In effect, yield stress only defines what cannot be measured. If a material "flows" at high stresses it will also flow, however slowly, at low stresses. We conjecture that the viscosity is always finite and, if measured over a wide enough shear rate range, shows the type o f q/q behaviour shown in figure 2. At low enough shear rates, the viscosity reaches a Newtonian plateau. Similarly, at high enough shear rates there is a so-called "second Newtonian" region. Between the two extremes, the viscosity falls monotonically and a "power law" behaviour is often observed with i/ varying as qm, where m is a negative exponent 2). To develop our arguments, we consider two test cases; many others could be quoted. The first concerns a commercially available PVA latex adhesive (Sample A). Using a conventional rheometer, like the Weissenberg Rheogoniometer, it is difficult to obtain accurate results at shear rates below about 10 -2 s -1 and certainly not below 10 -3 s -1. Testing the sample on such an instrument yields the type of graphs shown in figure 3 a - c , from which one might be tempted to conclude that the material has a yield stress, the value of which depends on the shear rate range used. However, the availability of results at very low shear rates now shows a Newtonian plateau, characteristic of standard shearthinning fluids. The definition of a yield stress would be reasonable on the basis of figure 3 a - c but it would be unacceptable with the added data provided in figure 3 d - e . A further example of our general argument is provided by a 0.5% solution of Carbopol (Sample B), a well known thickening agent for aqueous systems3).
Rheologica Acta, Vol. 24, No. 4 (1985) Figure 4 shows (r/, q) data for the solution with a clear indication of a high Newtonian plateau at low shear rates. In both cases, we note in passing that the power law model, which is appropriate over a limited shear rate range, eventually breaks down and that the viscosity does not tend to an infinite values as q ~ 0 as the power law model dictates. To develop further our general argument, we give closer consideration to the test samples A and B studied in figures 3 and 4; many others could be quoted. Both liquids were tested on the Deer Rheometer Mk II, which is a constant stress rheometer capable of measuring over a wide shear rate range and, for the kind of samples under consideration, the instrument can give results at shear rates as low as 10 -6 s-~. If we consider the results over various portions of the shear rate range, we can virtually produce a historical development of the yield stress concept (table 1). The range 10-150 s- l represents that typical of older laboratory viscometers and figure 3a shows that the material exhibits Bingham-type behaviour. If one now looks at data a decade lower in shear rate (representing the better type of laboratory viscometer) then again Bingham behaviour is seen, but the intercept is now lower (fig. 3b). Going down even further to the 1 10-3s -j range (as one can just manage to do on a Weissenberg Rheogoniometer), and plotting the results this time as stress/log shear rate, then a yield stress still appears to be present (fig. 3c). However, when the full range of the constant stress rheometer is utilized, we see that eventually a Newtonian plateau is reached (fig. 3 e).
Table 1 Shear rate range (s-1) 150 -10
10 1 - 1 -10 -3
"yield stress" (Pa) 9.5 5.0 1.0 0
10-3-10 -5
2) As is clear from figure 2 we limit ourselves in the present paper to shear-thinning non-Newtonian materials. 3) Details of the preparation of the sample and its complete rheological characterization will be published separately.
)]l
Fig. 2. Schematic representation of shear-thinning behaviour
20
T c~
14
15
~
10
o_ tO
~ lC
if) uJ I ffl
UJ
~ 6
< I.U -r 2
I
I I I I
Z,
8 SHEAR RATE ×10 -1
(s"1)
12
15 SHEAR RATE (s -1)
101
/
c)
5
Q_ tO
L~ 100 3
rv
~¢o "~'°'
n,. < uJ ..r to
1 = = "--'-" . . . . . . . . . . . .
7I I I
I
10-4
10-3 SHEAR RATE [s -1}
e)
10o
10-
/,
I
I
i0-~ SHEAR
i0-2 RATE (s-I)
L I 0 ~ ~
Fig. 3. Viscometric data for sample A
10°10.s
'
'0.3 ' 1'0_ 1 ' SHEAR RATE (s -1)
1'01
10 3
0.5°1o CARBOPOL 934 105
u
A
tn
D_ >i.--
10~
m 103
Fig. 4. Viscometric data for sample B
10 2
I
I
10 " 5
I RATE ( s -1)
IC
I 10 -3
I
~,
I
10-I
SHEAR
326 Table 2 System Reference Approximate limit of Newtonian behaviour r/0 (Pa s) K (s)
Rheologica Acta, Vol. 24, No. 4 (1985)
n -
(s-l)
Sample B Sample A Synovial fluid Polyacrylamide solution Anionic polyacrylamide Aqueous PVA
This work This work Zeidler and Altman [6] Boger [1] Lenge and Rehage [4] Cross [3]
10-s
10 . 4
10-2 10-l 10° 101
1.4 x l0 s 7.6 x 103 13.2 1.82 0.1 11.1
7.5 x 10 4 5 x 103 19 1.5 2 0.013
0.845 0.80 0.685 0.6 0.85 0.66
When we plot the data over the whole range of shear rates, figure 3e demonstrates our postulate that all liquids show Newtonian behaviour at low enough shear rates. Further, the behaviour in the middle shear rate range is of the power-law type and a second Newtonian plateau is approached at very high shear rates. Our low shear rate results for sample B show similar behaviour (fig. 4). This time, the plateau is higher and is not attained until even lower shear rates are reached.
3. Choice of models to describe low shear rate behaviour
4. Conclusion
With the availability of the new generation of constant stress rheometers, it is possible to show that the yield stress hypothesis, which has hitherto been a useful empiricism, is no longer necessary, and that fluids which flow at high stresses will flow at all lower stresses; i.e. the viscosity, although large, is always finite and there is no yield stress. Numerous simple models exist to simulate such departure from Newtonian behaviour. We have used the Cross model as an illustrative example.
References
If one restricts attention to low shear rates, it is of interest to seek simple flow models which show first a Newtonian plateau and then a power-law type behaviour at higher shear rates. Equations of the form 1 1 + (Kq) n (Form 1) (2) r/0 (1 + (Kq)2) m (Form 2)
170 or
1. Boger DV (1977) Nature 265:127 2. British Standard 5168 (1975) Glossary of Rheological Terms 3. Cross MM (1965) J Coil Sci 20:417 4. Lenge W, Rehage G (1980) In: Astarita G, Marrucci G, Nicolais L (eds) Rheology. vol 2, Plenum Press, p 354 5. Scott Blair GW (1933) J Applied Phys 4:113 6. Zeidler H, Altman S (1980) In: Astarita G, Marrucci G, Nicolais L (eds) Rheology, vol 3, Plenum Press, p 512 (Received October 10, 1984) Authors' addresses: Dr. H. A. Barnes Unilever Research Laboratory Port Sunlight L63 3JW (U.K.) Prof. K. Walters Department of Applied Mathematics The University College of Wales Penglais, Aberystwyth SY23 3BZ (U.K.)
satisfy these requirements with n > 0, m > 0. The Cross model [3] is an example of Form 1 and we take this model as an illustrative example. We show in table 2 the parameters necessary to fit the data for samples A and B and also other non-Newtonian fluids available in the literature. We do this to show that the same general type of flow curve is found, irrespective of the shear rate at which departure from Newtonian behaviour is first observed.
