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A simple empirical model describing the steady-state shear and extensional viscosities of polymer melts
Journal of Non-Newtonian Fluid Mechanics, 44 (1992) 113-126
113
Elsevier Science Publishers B.V., Amsterdam
A simple empirical model describing the steady-state and extensional viscosities of polymer melts
shear
H.A. Barnes and G.P. Roberts Unileuer Research, Port Sunlight Laboratory, Merseyside, L63 3JW (UK)
(Received January 13, 1992)
Abstract
The purpose of the work was to find a simple equation that predicts simultaneously both shear and extensional viscosity as a function of deformation rate, as well as giving a reasonable estimate of the steady-state elastic properties. The equation chosen to work with was the White-Metzner model. The special - and highly specific - forms of the deformation rate-dependent viscosity and relaxation time we chose were: $1,)
= Q/D
+ (K,II,)“l
and A(II,) = A,/[1 + K,II,l
These particular forms - with a careful choice of constants - allow us to keep the value of the extensional viscosity finite, but give enough freedom to predict the expected forms of the extensional-viscosity flow curve. A collection of literature data for concurrent extensional and shear melt viscosities was assembled to test the equation. The best values for the model parameters for the data were obtained using a Simplex multivariable data-fitting method. The model gives an excellent fit to the shear viscosity rate data and a reasonable fit to extensional viscosity data, particularly in the high deformation rate range. It should be noted that the data show that at high deformation rates the experimental curves are indeed parallel when plotted logarithmically - as predicted, a fact that as far as the authors are aware has not been pointed out before. The theory prediction here is very good, suggesting that the Trouton ratio does become constant. For simple shear experiments, the predictions are that the viscosity function can be fitted by the Cross model; the relaxation time should Correspondence to: H.A. Barnes, Unilever Research, Port Sunlight Laboratory,
Merseyside,
L63 3JW, UK. 0377-0257/92/$05.00
0 1992 - Elsevier Science Publishers B.V. Ail rights reserved
114
H.A. Barnes and G.P. Roberts/J.
Non-Newtonian Fluid Mech. 44 (1992) 113-126
decrease with shear rate, eventually ending up being inversely proportional to shear rate and that at high enough shear stress there should be a linear relationship between the first normal-stress difference and the shear stress or, put another way, their ratio (twice the Recoverable Shear Strain) should become constant. All these predictions are shown to be reasonable for a range of polymer melts. Keywords: extensional
viscosity; polymer melts; steady-state
shear; shear viscosity; White-
Metzner model
1. Introduction
The usefulness of relatively simple equations - for instance the Cross and Sisko models - in describing the shear viscosity/shear rate curves of a wide range of non-Newtonian liquids has previously been shown [l]. The challenge that remains is to find an equally simple equation that gives the simultaneous prediction of both shear and extensional viscosity as a function of shear or extensional deformation rate respectively, as well as to make a reasonable estimate of the steady-state elastic properties, e.g. the first normal stress difference. As polymer melts have yielded the most interesting and reliable data in this area as well as shown the most problems with extensional flow, these will be our main target for prediction We shall also include particle-filled melts - concentrated dispersions - where data are available. 2. Theory
The equation chosen to work with is one of the simplest non-linear, single relaxation-time models available - the so-called White-Metzner model (see Ref. 1, Chapter 8). It will predict both shear and extensional viscosities, as well as normal force and even transient behaviour, although the latter would actually be the weakest part of the model because it only has one relaxation time. However, we shall restrict ourselves to steady-state flow, where transient effects are ignored. Written in tensorial form, the White-Metzner model is: T + A(&,)?
where the second invariant of the deformation II,=@&
(1)
= rls(II,)D, tensor is
(2)
H.A. Barnes and G.P. Roberts/J.
Non-Newtonian Fluid Mech. 44 (1992) 113-126
115
In shear flow the second invariant is the shear rate (q) and in simple uniaxial extensional flow is equal to root-three times the extension rate (a;). T is the stress tensor; f its time derivative and A(I1,) the deformation rate-dependent relaxation time; &I,,) the deformation rate-dependent viscosity and D the rate of deformation. The special forms of the deformation rate-dependent viscosity and relaxation time we choose are: %(IM
= %/[ 1+ (WJ]
A&)
= A,/[1 + &II,]
7
(3)
;
(4)
i.e. a Cross-model type of representation of the functions where K,, K, are parameters with the dimensions of time and n is a numerical constant. The special form of the viscosity expression (3) was chosen following our successful use of it in the past [l]. The reason why the particular form of the relaxation time was chosen becomes clear when we write down the expression for the extensional viscosity (see ref. 1, Chapter 8): TjE= 2?7,(II,)/
1 - $A(rrn)Ir (II )/ 1 + ~(II,)II n] + ?JSD[ [ If we examine the first term in the extensional viscosity, i.e.
n].
(5)
2 TE = 277,(11D)/
1 -
~A(ll,)llD
(6)
7 I
we see the cause of many problems that arise in the use of this model; that is when (2/~)A(II,)II, > 1, the viscosity eventually becomes negative after having first become infinite at a value of (2/ fi) A(II,)II, of unity. In order to overcome this limitation, the expression given in (4) was chosen, for this allows the expression to approach a (large) constant value rather than become infinite because
1
1 - ~A(II,)II, 2
= 1 - ~A&/[1 2
+&II,]
1
)
(7)
then, eventually, ;AJD/ll
+&II,]
--) 2A,/&
9
hence 2 E~(~~,))~I,
+ 1 - 2A,/fiK,. I So if we keep the value of 2A,/fiK, less than 1, this keeps the value of the extensional viscosity finite, and, as we shall see, it also gives the I-
HA
116
Barnes and G.P. Roberts/J.
Non-Newtonian Fluid Mech. 44 (1992) 113-126
= 1.500 = 1.300 = 1.100 = 0.700
000001
00001
0001
001
0 1
1
10
100
1,wo
10,cm
1w.ooo1,wo.ow
Deformation Rate (II,),? ,i /s’ Fig. 1. Predicted form of eqns. (3) and (5) for a variety of A, and n = 0.5, with the other parameters set to unity.
expected form of the extensional viscosity flow curve. The factor [l 2A/ ~)(II,)II,l, being a divisor and becoming progressively smaller, can have an overwhelming influence on the extensional viscosity, because it can become very small, indeed it will dominate the extensional viscosity predicted by eqn. (5). The forms of the shear and extensional viscosities predicted by the equation is illustrated in Fig. 1, where they are plotted as functions of the appropriate value of the second invariant II,. Using A, as a variable - and keeping the other parameters constant - various shapes possible with the model are displayed. At low deformation rates we see the expected three times higher extensional viscosity for uniaxial extensional flow compared with its shear equivalent. (The proportionality would of course be different for other kinds of extensional flow, see ref. 1, Chapter 5.) We see that the extensional viscosity can either thereafter continuously decrease or otherwise show the often-seen increase followed by a decrease. This is encouraging as all experimental flow curves follow one of these kinds of behaviour. We may note that the eventual extensional viscosity line is parallel (in logarithmic terms) to the shear viscosity, although their ratio - called the Trouton ratio - can be very different, varying upwards from 3. This suggests that if there is a power-law behaviour in shear, then we also expect the same power-law behaviour at high extension rates. It is possible to define a form of Trouton ratio by dividing the extensional-viscosity function by the shear-viscosity function ~&IIn) =
HA. Barnes and G.P. R~b~rts~J. N~~-Newt~~~n Flak deco.
~/[l + (K,II,)“]. final value of 2/(1-
44 tl99~~ 113-126
117
This expression begins at a value of 3 and increases to a
2A,/\l?;K,)
+ l/(1
+ b”/\/;jK,)
at very high deformation rate. In cases where the value of 2hO/6K, is often nearly one, the final Trouton ratio is dominated by 2/(12h,/ \27K,>, which can be considerably greater than 3. Although our main interest is the simultaneous prediction of shear and extensional viscosity using this simple model, we must also note the form of the normal-force unction predicted. In the White-meaner general formulation we have (in simple-shear flow) Ni = 271(+)A(+)+~, which in our case becomes 1
*0
‘lo
N, -2
+ (K,j)”
1+ (K2j) $2*
(8)
(9)
At very high shear rates this expression becomes N
_ 1-
~~,Ao?‘-” rc;K-,
-
This might be compared with the shear stress (01 at high shear rate: *l-n Y a=770-$
(11) 1
This suggests an asymptotic linear relationship between the normal force and the shear stress at high enough shear rate. The constant of proportionality between the two functions is 2A,/K,. Simple inspection of the formula shows a quadratic behaviour at very low shear rates; thus, we expect a gradual movement in the relationship from quadratic to linear. This relationship also predicts that at high shear rates/shear stresses, the Recoverable Shear Strain (RSS) becomes constant - here the RSS is the first normal stress difference divided by twice the shear stress, i.e. N,/2a = A,/&. It should be noted that these ultimate forms couid also be derived by using an equivalent Carreau-type formulation; see ref. 1, p. 18. 3. Comparison with experimental results 3.1. Concomitant shear and extensional viscosities A collection of literature data for &oncurrent extensional and shear viscosities was assembled to test eqns. (3) and (sf; each set contained both
118
HA. Barnes and G.P. Roberts/J. Non-NewtonianFluid Mech. 44 (1992) 113-126
shear-viscosity and extensional-~s~si~ values as a function of deformation rate for the same material at the same temperature. The best values for the model parameters for the data were obtained using a Simplex multivariable data-fitting method [2]. The results are shown for each data set in Figs. 2-6 where the model-fitting parameters are also listed. The shear viscosity was fitted first, then, using the parameters so obtained, the extensional viscosity data was fitted; the parameters are collected together in Table 1. As will be seen, the model gives an excellent fit to the shear viscosityshear rate data and a reasonable fit to extensional viscosity data, particularly in the high deformation rate range. It is unable, however, to show the very fast rises in extensional viscosity often seen for materials like low-density polyethylene (LPDE). Even then, however, the fit near and past the maximum in the extensional viscosity is very good. It should be noted that the data show that at high deformation rates the experimental curves are parallel as predicted, a fact that as far as the authors are aware has not been pointed out before. The theory prediction here is very good, suggesting that the ultimate Trouton ratio - see Ref. 1, chapter 5 - becomes constant. Given the simple nature of the model and the limited number of parameters it contains, it has done remarkably well at fitting the data which we have assumed to be both accurate and steady state; the latter is often disputed for data derived from certain kinds of experiment (see ref. 1, Chapter 5). Lastly we consider the data of Munstedt and Laun [4] on a particle-filled melt. They made measurements of both shear and extensional viscosities of an LDPE melt at 125°C with 0, 35 and 50% glass beads. Surprisingly, although the shear viscosity increased with increase in filler level in the expected fashion, the extensional viscosity showed a rise only at very low extension rates - giving the expected three times increase - while thereafter the extensional viscosity decreased with increase in filler level! This data is fitted to the above theory and plotted in Fig. 3. This phenomenon is equivalent to a decrease in elasticity and has been seen in shear flow where, although the viscosity is increased as a simple filler is added, the normal stresses fall; a number of examples of this are shown in Chapter 3 of Han’s book on multiphase flow [lo]. By a simple filler we mean particles that are essentially spherical; when very asymmetric fibre-like particles are added, both viscosity and normal stresses are increased.
For simple-shear experiments, the predictions are that: (i) the viscosity function can be fitted by the Cross model; this will be shown later in comparison with the extensional prediction;
H.A. Barnes and G.P. Roberts/J.
Non-Newtonian Fluid Mech. 44 (1992) 113-126
119
120
HA. Barnes and G.P. Roberts/J.
Non-Newtonian F&d Me&. 44 fB9211113-_126
Fig. 4. Shear and extensional viscosities vs. deformation rate for (a) LDPE, (b) LDPE + 35% glass beads; Cc) LDPE + 50% glass beads. Symbols are data of Munstedt and Law [l]; lines are best fit to eqns. (3) and (5). E
CcaEfW
CcoEtcd
-
(4
Deformation Rate UI,,.+,&r
1
(t-4
10 -
) (4
-
Trouton Ratio
Rate (II,),+&-’
\
150°C; (c) PMMA,
Deformation Fate &,,,Q,(/d
Deformation
Fig. 5. Shear and extensional viscosities vs. deformation rate for (a) 6.6 nylon; (b) branched polyethylene, polypropylene, 190°C. Symbols are data of Cogswell [5]; lines are best fit to eqns. (3) and (5).
,
t
1 MOE+07
3 moE+M
190°C; (d)
H.A. Barnes and G.P. Roberts/J.
Non-Newtonian Fluid Mech. 44 (1992) 113-126
123
1
I
1
I Polystvrene
6d
6.7E+02
4.9E+02
I 3.9E+Oi
I LDPE 9
6c
6
1 LDPE
6b
1 I lE+03
I 4.9E+Oi
5.9E+02
9.97E+Ol
lE+03
8.2E+02
6.47E+Ol
Polystyrene
HDPE
6a
1.6E+02
1 .OE+02
Butyl
5c
5d
7.4E+02
6.1E+02
7.4E+02
6.1E+02
LDPE
LDPE
1.605E+OO
1.384E+OO
5.19E+Ol
I .955E-03
l.lE+02
5a
1
1
(
5b
i.l91E+OO
l.O39E+OO
Polypropylene
PMMA
I
1.662E-03
4.40E+Oi
Nylon
7.38E+Ol
1 Polyethylene
4d
4c
4b
4a
1
1 LDPE/Glass
3c
5.4E+02
2E+03
3.8E+02
2E+03
LDPE
3a
1
2.2E+02
6.60E+oi
HDPE
2d
1 LDPE/Glass
3.36E+Ol
2.22E+Ol
LDPE/HDPE
ac
3b
5.92E+Ol
4.6E+Ol
LDPEIHDPE
2b
Parameters used to fit data to eqns. (3) and (5)
TABLE 1
1.259E+05
2.922E+O4
1.2E+02
8.079E+06
6E+03
4.663E+04
3.614E+04
1.255E+05
1.807E+05
1.8E+02
3.832E+05
2.9E+05
1
4.79lE+05
I 2.954E+04
1 4.748E+04
1
1
1
/
2E+04
3E+04
3.300Ei03
9.801E+OO
1.239E-03
2.52E+Ol
5.3E+OO
I 1.47E+Ol
I 2.314E+OO
1
4.606E+OO
4.6E+02
3.09E+Ol
8.75tE+OO
6.074E+OO
1.733E+OO
1 9,0880E-01
(
1
1 2.34E+Ol
1.6E+ol
6.2660E-01
2.412E+OO
3.682E+OO
0.6335
0.5546
0.6437
0.6179
0.5748
0.5717
0.6483
0.5776
0.7217
0.6777
0.6310
0.6286
0.5946
1 0.799
I 0.6408
1
1
1
1
1
0.5623
0.5163
0.5290
H.A. Barnes and G.P. Roberts/J.
Non-Newtonian Fluid Mech. 44 (1992) 113-126
125
(ii) the relaxation time should decrease with shear rate, eventually ending up being inversely proportional to shear rate, and (iii) there should eventually - at high enough shear stress - be linearity between the first normal stress difference and the shear stress or, put another way, their ratio (twice the Recoverable Shear Strain) should become constant. The first prediction has been adequately proved elsewhere [l]. For the second prediction we can cite examples from the extensive work of Han [lo]. His data on a high-density polyethylene at 24O”C, polypropylene at 200°C and polystyrene at 200°C nylon at 280°C and polyethylene terephthalate at 300°C show that the terminal (logarithmic) values of the relax-1.03, -1.08 and ation time/shear rate curves are -1.14, -0.824, - 1.075 respectively. Chan et al. [ll] studied the same relationship for various samples of polystyrene at 180°C filled with glass fibre (0, 20, and 40 %wt.); they found that the terminal values were -0.808, -0.967 and -0.864 respectively. Taken together, these data show that a predicted value of - 1 for the terminal logarithmic slope is quite reasonable. The model also predicts that the relationship between the first normal stress difference to shear stress should start as quadratic and then decrease to an eventual linear relationship. This is indeed usually seen; perusal of the collected data of Han [lo] for instance shows this often to be the case. All polymer melts do give values of the logarithmic relationship that lie between 2 and 1, and a number actually are near 1 at high enough shear stress. Examples of this eventual linear relationship are high-density polyethylene (1.06), polypropylene at 200°C (1.15), polypropylene at 200°C with 50% wt. glass beads (l), and polyethylene (1.02). Han also produced curves of the Recoverable Shear Strain vs. shear stress for low-density polyethylene at 200°C; polypropylene at 200°C; polystyrene at 200°C; nylon at 280°C and polyethylene terephthalate at 300°C. Above a shear stress of about lo4 Pa, the values did tend to a constant value as predicted, with asymptotic values of about 6, 2, 1.5, 0.11 and 0.025 respectively. Overall then, the model presented seems quite reasonable at describing the steady-state shear properties of polymer melts. 4. Conclusions
and future work
The model gives better-than-expected fits to the data given its simplicity and small number of parameters. The real test of any such model, however, is not just the reduction of many data points to a simple equation; the real test would come in applying the equation to flow in complicated geometries, e.g. flow through a contraction. This will be tested soon, and then we
126
H.A. Barnes and G.P. Roberts/J.
Non-Newtonian Fluid Mech. 44 (1992) 113-126
can, for instance, test the predicted pressure drop/flow rate data against such theories as that of Binding [12]. These seek to solve the inverse problem of predicting the model from the data for simple contraction flow. In order to carry out such a prediction, computer simulations will be attempted using the theory. We shall also collect data on polymer/particle mixtures and see how the theory presented above describes these. Such mixed systems often show that elasticity is reduced as particles are added to polymeric systems. This has important consequences for real products such as toothpaste and the filled polymer melts used in packaging. References 1 H.A. Barnes, J.F. Hutton and K. Walters, An Introduction to Rheology, Chapter 2, Elsevier, Amsterdam, 1989. 2 A. Valenza, F.P. La Mantia and D. Acierno, J. Rheol., 30 (1986) 1085. 3 H.M. Laun and H. Schuch, J. Rheol., 33 (1989) 119. 4 H. Munstedt and H.M. Laun, in G. Astarita et al. (Eds.), Rheology, Naples, 1980, p. 415. 5 F.N. Cogswell, Polymer Melt Rheology, George Godwin, London, 1981. 6 J. Meisner, Trans. Sot. Rheol., 16 (1972) 59. 7 J.V. Aleman, Proc. Xth Int. Congress on Rheology, Sydney, Vol. 1, Australian Society of Rheology, 1988, p. 143. 8 J.F. Stevenson, Ph.D. Thesis, University of Wisconsin, 1970. 9 R.L. Ballman, Rheol. Acta, 4 (1965) 2. 10 C.D. Han, Rheology of Polymer Processing, Academic Press, New York, 1976; C.D. Han, Multiphase Flow in Polymer Processing, Academic Press, New York, 1981. 11 Y. Chan et al., J. Rheol., 22 (1978) 507. 12 D. Binding, J. Non-Newtonian Fluid Mech., 27 (1988) 173.
113
Elsevier Science Publishers B.V., Amsterdam
A simple empirical model describing the steady-state and extensional viscosities of polymer melts
shear
H.A. Barnes and G.P. Roberts Unileuer Research, Port Sunlight Laboratory, Merseyside, L63 3JW (UK)
(Received January 13, 1992)
Abstract
The purpose of the work was to find a simple equation that predicts simultaneously both shear and extensional viscosity as a function of deformation rate, as well as giving a reasonable estimate of the steady-state elastic properties. The equation chosen to work with was the White-Metzner model. The special - and highly specific - forms of the deformation rate-dependent viscosity and relaxation time we chose were: $1,)
= Q/D
+ (K,II,)“l
and A(II,) = A,/[1 + K,II,l
These particular forms - with a careful choice of constants - allow us to keep the value of the extensional viscosity finite, but give enough freedom to predict the expected forms of the extensional-viscosity flow curve. A collection of literature data for concurrent extensional and shear melt viscosities was assembled to test the equation. The best values for the model parameters for the data were obtained using a Simplex multivariable data-fitting method. The model gives an excellent fit to the shear viscosity rate data and a reasonable fit to extensional viscosity data, particularly in the high deformation rate range. It should be noted that the data show that at high deformation rates the experimental curves are indeed parallel when plotted logarithmically - as predicted, a fact that as far as the authors are aware has not been pointed out before. The theory prediction here is very good, suggesting that the Trouton ratio does become constant. For simple shear experiments, the predictions are that the viscosity function can be fitted by the Cross model; the relaxation time should Correspondence to: H.A. Barnes, Unilever Research, Port Sunlight Laboratory,
Merseyside,
L63 3JW, UK. 0377-0257/92/$05.00
0 1992 - Elsevier Science Publishers B.V. Ail rights reserved
114
H.A. Barnes and G.P. Roberts/J.
Non-Newtonian Fluid Mech. 44 (1992) 113-126
decrease with shear rate, eventually ending up being inversely proportional to shear rate and that at high enough shear stress there should be a linear relationship between the first normal-stress difference and the shear stress or, put another way, their ratio (twice the Recoverable Shear Strain) should become constant. All these predictions are shown to be reasonable for a range of polymer melts. Keywords: extensional
viscosity; polymer melts; steady-state
shear; shear viscosity; White-
Metzner model
1. Introduction
The usefulness of relatively simple equations - for instance the Cross and Sisko models - in describing the shear viscosity/shear rate curves of a wide range of non-Newtonian liquids has previously been shown [l]. The challenge that remains is to find an equally simple equation that gives the simultaneous prediction of both shear and extensional viscosity as a function of shear or extensional deformation rate respectively, as well as to make a reasonable estimate of the steady-state elastic properties, e.g. the first normal stress difference. As polymer melts have yielded the most interesting and reliable data in this area as well as shown the most problems with extensional flow, these will be our main target for prediction We shall also include particle-filled melts - concentrated dispersions - where data are available. 2. Theory
The equation chosen to work with is one of the simplest non-linear, single relaxation-time models available - the so-called White-Metzner model (see Ref. 1, Chapter 8). It will predict both shear and extensional viscosities, as well as normal force and even transient behaviour, although the latter would actually be the weakest part of the model because it only has one relaxation time. However, we shall restrict ourselves to steady-state flow, where transient effects are ignored. Written in tensorial form, the White-Metzner model is: T + A(&,)?
where the second invariant of the deformation II,=@&
(1)
= rls(II,)D, tensor is
(2)
H.A. Barnes and G.P. Roberts/J.
Non-Newtonian Fluid Mech. 44 (1992) 113-126
115
In shear flow the second invariant is the shear rate (q) and in simple uniaxial extensional flow is equal to root-three times the extension rate (a;). T is the stress tensor; f its time derivative and A(I1,) the deformation rate-dependent relaxation time; &I,,) the deformation rate-dependent viscosity and D the rate of deformation. The special forms of the deformation rate-dependent viscosity and relaxation time we choose are: %(IM
= %/[ 1+ (WJ]
A&)
= A,/[1 + &II,]
7
(3)
;
(4)
i.e. a Cross-model type of representation of the functions where K,, K, are parameters with the dimensions of time and n is a numerical constant. The special form of the viscosity expression (3) was chosen following our successful use of it in the past [l]. The reason why the particular form of the relaxation time was chosen becomes clear when we write down the expression for the extensional viscosity (see ref. 1, Chapter 8): TjE= 2?7,(II,)/
1 - $A(rrn)Ir (II )/ 1 + ~(II,)II n] + ?JSD[ [ If we examine the first term in the extensional viscosity, i.e.
n].
(5)
2 TE = 277,(11D)/
1 -
~A(ll,)llD
(6)
7 I
we see the cause of many problems that arise in the use of this model; that is when (2/~)A(II,)II, > 1, the viscosity eventually becomes negative after having first become infinite at a value of (2/ fi) A(II,)II, of unity. In order to overcome this limitation, the expression given in (4) was chosen, for this allows the expression to approach a (large) constant value rather than become infinite because
1
1 - ~A(II,)II, 2
= 1 - ~A&/[1 2
+&II,]
1
)
(7)
then, eventually, ;AJD/ll
+&II,]
--) 2A,/&
9
hence 2 E~(~~,))~I,
+ 1 - 2A,/fiK,. I So if we keep the value of 2A,/fiK, less than 1, this keeps the value of the extensional viscosity finite, and, as we shall see, it also gives the I-
HA
116
Barnes and G.P. Roberts/J.
Non-Newtonian Fluid Mech. 44 (1992) 113-126
= 1.500 = 1.300 = 1.100 = 0.700
000001
00001
0001
001
0 1
1
10
100
1,wo
10,cm
1w.ooo1,wo.ow
Deformation Rate (II,),? ,i /s’ Fig. 1. Predicted form of eqns. (3) and (5) for a variety of A, and n = 0.5, with the other parameters set to unity.
expected form of the extensional viscosity flow curve. The factor [l 2A/ ~)(II,)II,l, being a divisor and becoming progressively smaller, can have an overwhelming influence on the extensional viscosity, because it can become very small, indeed it will dominate the extensional viscosity predicted by eqn. (5). The forms of the shear and extensional viscosities predicted by the equation is illustrated in Fig. 1, where they are plotted as functions of the appropriate value of the second invariant II,. Using A, as a variable - and keeping the other parameters constant - various shapes possible with the model are displayed. At low deformation rates we see the expected three times higher extensional viscosity for uniaxial extensional flow compared with its shear equivalent. (The proportionality would of course be different for other kinds of extensional flow, see ref. 1, Chapter 5.) We see that the extensional viscosity can either thereafter continuously decrease or otherwise show the often-seen increase followed by a decrease. This is encouraging as all experimental flow curves follow one of these kinds of behaviour. We may note that the eventual extensional viscosity line is parallel (in logarithmic terms) to the shear viscosity, although their ratio - called the Trouton ratio - can be very different, varying upwards from 3. This suggests that if there is a power-law behaviour in shear, then we also expect the same power-law behaviour at high extension rates. It is possible to define a form of Trouton ratio by dividing the extensional-viscosity function by the shear-viscosity function ~&IIn) =
HA. Barnes and G.P. R~b~rts~J. N~~-Newt~~~n Flak deco.
~/[l + (K,II,)“]. final value of 2/(1-
44 tl99~~ 113-126
117
This expression begins at a value of 3 and increases to a
2A,/\l?;K,)
+ l/(1
+ b”/\/;jK,)
at very high deformation rate. In cases where the value of 2hO/6K, is often nearly one, the final Trouton ratio is dominated by 2/(12h,/ \27K,>, which can be considerably greater than 3. Although our main interest is the simultaneous prediction of shear and extensional viscosity using this simple model, we must also note the form of the normal-force unction predicted. In the White-meaner general formulation we have (in simple-shear flow) Ni = 271(+)A(+)+~, which in our case becomes 1
*0
‘lo
N, -2
+ (K,j)”
1+ (K2j) $2*
(8)
(9)
At very high shear rates this expression becomes N
_ 1-
~~,Ao?‘-” rc;K-,
-
This might be compared with the shear stress (01 at high shear rate: *l-n Y a=770-$
(11) 1
This suggests an asymptotic linear relationship between the normal force and the shear stress at high enough shear rate. The constant of proportionality between the two functions is 2A,/K,. Simple inspection of the formula shows a quadratic behaviour at very low shear rates; thus, we expect a gradual movement in the relationship from quadratic to linear. This relationship also predicts that at high shear rates/shear stresses, the Recoverable Shear Strain (RSS) becomes constant - here the RSS is the first normal stress difference divided by twice the shear stress, i.e. N,/2a = A,/&. It should be noted that these ultimate forms couid also be derived by using an equivalent Carreau-type formulation; see ref. 1, p. 18. 3. Comparison with experimental results 3.1. Concomitant shear and extensional viscosities A collection of literature data for &oncurrent extensional and shear viscosities was assembled to test eqns. (3) and (sf; each set contained both
118
HA. Barnes and G.P. Roberts/J. Non-NewtonianFluid Mech. 44 (1992) 113-126
shear-viscosity and extensional-~s~si~ values as a function of deformation rate for the same material at the same temperature. The best values for the model parameters for the data were obtained using a Simplex multivariable data-fitting method [2]. The results are shown for each data set in Figs. 2-6 where the model-fitting parameters are also listed. The shear viscosity was fitted first, then, using the parameters so obtained, the extensional viscosity data was fitted; the parameters are collected together in Table 1. As will be seen, the model gives an excellent fit to the shear viscosityshear rate data and a reasonable fit to extensional viscosity data, particularly in the high deformation rate range. It is unable, however, to show the very fast rises in extensional viscosity often seen for materials like low-density polyethylene (LPDE). Even then, however, the fit near and past the maximum in the extensional viscosity is very good. It should be noted that the data show that at high deformation rates the experimental curves are parallel as predicted, a fact that as far as the authors are aware has not been pointed out before. The theory prediction here is very good, suggesting that the ultimate Trouton ratio - see Ref. 1, chapter 5 - becomes constant. Given the simple nature of the model and the limited number of parameters it contains, it has done remarkably well at fitting the data which we have assumed to be both accurate and steady state; the latter is often disputed for data derived from certain kinds of experiment (see ref. 1, Chapter 5). Lastly we consider the data of Munstedt and Laun [4] on a particle-filled melt. They made measurements of both shear and extensional viscosities of an LDPE melt at 125°C with 0, 35 and 50% glass beads. Surprisingly, although the shear viscosity increased with increase in filler level in the expected fashion, the extensional viscosity showed a rise only at very low extension rates - giving the expected three times increase - while thereafter the extensional viscosity decreased with increase in filler level! This data is fitted to the above theory and plotted in Fig. 3. This phenomenon is equivalent to a decrease in elasticity and has been seen in shear flow where, although the viscosity is increased as a simple filler is added, the normal stresses fall; a number of examples of this are shown in Chapter 3 of Han’s book on multiphase flow [lo]. By a simple filler we mean particles that are essentially spherical; when very asymmetric fibre-like particles are added, both viscosity and normal stresses are increased.
For simple-shear experiments, the predictions are that: (i) the viscosity function can be fitted by the Cross model; this will be shown later in comparison with the extensional prediction;
H.A. Barnes and G.P. Roberts/J.
Non-Newtonian Fluid Mech. 44 (1992) 113-126
119
120
HA. Barnes and G.P. Roberts/J.
Non-Newtonian F&d Me&. 44 fB9211113-_126
Fig. 4. Shear and extensional viscosities vs. deformation rate for (a) LDPE, (b) LDPE + 35% glass beads; Cc) LDPE + 50% glass beads. Symbols are data of Munstedt and Law [l]; lines are best fit to eqns. (3) and (5). E
CcaEfW
CcoEtcd
-
(4
Deformation Rate UI,,.+,&r
1
(t-4
10 -
) (4
-
Trouton Ratio
Rate (II,),+&-’
\
150°C; (c) PMMA,
Deformation Fate &,,,Q,(/d
Deformation
Fig. 5. Shear and extensional viscosities vs. deformation rate for (a) 6.6 nylon; (b) branched polyethylene, polypropylene, 190°C. Symbols are data of Cogswell [5]; lines are best fit to eqns. (3) and (5).
,
t
1 MOE+07
3 moE+M
190°C; (d)
H.A. Barnes and G.P. Roberts/J.
Non-Newtonian Fluid Mech. 44 (1992) 113-126
123
1
I
1
I Polystvrene
6d
6.7E+02
4.9E+02
I 3.9E+Oi
I LDPE 9
6c
6
1 LDPE
6b
1 I lE+03
I 4.9E+Oi
5.9E+02
9.97E+Ol
lE+03
8.2E+02
6.47E+Ol
Polystyrene
HDPE
6a
1.6E+02
1 .OE+02
Butyl
5c
5d
7.4E+02
6.1E+02
7.4E+02
6.1E+02
LDPE
LDPE
1.605E+OO
1.384E+OO
5.19E+Ol
I .955E-03
l.lE+02
5a
1
1
(
5b
i.l91E+OO
l.O39E+OO
Polypropylene
PMMA
I
1.662E-03
4.40E+Oi
Nylon
7.38E+Ol
1 Polyethylene
4d
4c
4b
4a
1
1 LDPE/Glass
3c
5.4E+02
2E+03
3.8E+02
2E+03
LDPE
3a
1
2.2E+02
6.60E+oi
HDPE
2d
1 LDPE/Glass
3.36E+Ol
2.22E+Ol
LDPE/HDPE
ac
3b
5.92E+Ol
4.6E+Ol
LDPEIHDPE
2b
Parameters used to fit data to eqns. (3) and (5)
TABLE 1
1.259E+05
2.922E+O4
1.2E+02
8.079E+06
6E+03
4.663E+04
3.614E+04
1.255E+05
1.807E+05
1.8E+02
3.832E+05
2.9E+05
1
4.79lE+05
I 2.954E+04
1 4.748E+04
1
1
1
/
2E+04
3E+04
3.300Ei03
9.801E+OO
1.239E-03
2.52E+Ol
5.3E+OO
I 1.47E+Ol
I 2.314E+OO
1
4.606E+OO
4.6E+02
3.09E+Ol
8.75tE+OO
6.074E+OO
1.733E+OO
1 9,0880E-01
(
1
1 2.34E+Ol
1.6E+ol
6.2660E-01
2.412E+OO
3.682E+OO
0.6335
0.5546
0.6437
0.6179
0.5748
0.5717
0.6483
0.5776
0.7217
0.6777
0.6310
0.6286
0.5946
1 0.799
I 0.6408
1
1
1
1
1
0.5623
0.5163
0.5290
H.A. Barnes and G.P. Roberts/J.
Non-Newtonian Fluid Mech. 44 (1992) 113-126
125
(ii) the relaxation time should decrease with shear rate, eventually ending up being inversely proportional to shear rate, and (iii) there should eventually - at high enough shear stress - be linearity between the first normal stress difference and the shear stress or, put another way, their ratio (twice the Recoverable Shear Strain) should become constant. The first prediction has been adequately proved elsewhere [l]. For the second prediction we can cite examples from the extensive work of Han [lo]. His data on a high-density polyethylene at 24O”C, polypropylene at 200°C and polystyrene at 200°C nylon at 280°C and polyethylene terephthalate at 300°C show that the terminal (logarithmic) values of the relax-1.03, -1.08 and ation time/shear rate curves are -1.14, -0.824, - 1.075 respectively. Chan et al. [ll] studied the same relationship for various samples of polystyrene at 180°C filled with glass fibre (0, 20, and 40 %wt.); they found that the terminal values were -0.808, -0.967 and -0.864 respectively. Taken together, these data show that a predicted value of - 1 for the terminal logarithmic slope is quite reasonable. The model also predicts that the relationship between the first normal stress difference to shear stress should start as quadratic and then decrease to an eventual linear relationship. This is indeed usually seen; perusal of the collected data of Han [lo] for instance shows this often to be the case. All polymer melts do give values of the logarithmic relationship that lie between 2 and 1, and a number actually are near 1 at high enough shear stress. Examples of this eventual linear relationship are high-density polyethylene (1.06), polypropylene at 200°C (1.15), polypropylene at 200°C with 50% wt. glass beads (l), and polyethylene (1.02). Han also produced curves of the Recoverable Shear Strain vs. shear stress for low-density polyethylene at 200°C; polypropylene at 200°C; polystyrene at 200°C; nylon at 280°C and polyethylene terephthalate at 300°C. Above a shear stress of about lo4 Pa, the values did tend to a constant value as predicted, with asymptotic values of about 6, 2, 1.5, 0.11 and 0.025 respectively. Overall then, the model presented seems quite reasonable at describing the steady-state shear properties of polymer melts. 4. Conclusions
and future work
The model gives better-than-expected fits to the data given its simplicity and small number of parameters. The real test of any such model, however, is not just the reduction of many data points to a simple equation; the real test would come in applying the equation to flow in complicated geometries, e.g. flow through a contraction. This will be tested soon, and then we
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H.A. Barnes and G.P. Roberts/J.
Non-Newtonian Fluid Mech. 44 (1992) 113-126
can, for instance, test the predicted pressure drop/flow rate data against such theories as that of Binding [12]. These seek to solve the inverse problem of predicting the model from the data for simple contraction flow. In order to carry out such a prediction, computer simulations will be attempted using the theory. We shall also collect data on polymer/particle mixtures and see how the theory presented above describes these. Such mixed systems often show that elasticity is reduced as particles are added to polymeric systems. This has important consequences for real products such as toothpaste and the filled polymer melts used in packaging. References 1 H.A. Barnes, J.F. Hutton and K. Walters, An Introduction to Rheology, Chapter 2, Elsevier, Amsterdam, 1989. 2 A. Valenza, F.P. La Mantia and D. Acierno, J. Rheol., 30 (1986) 1085. 3 H.M. Laun and H. Schuch, J. Rheol., 33 (1989) 119. 4 H. Munstedt and H.M. Laun, in G. Astarita et al. (Eds.), Rheology, Naples, 1980, p. 415. 5 F.N. Cogswell, Polymer Melt Rheology, George Godwin, London, 1981. 6 J. Meisner, Trans. Sot. Rheol., 16 (1972) 59. 7 J.V. Aleman, Proc. Xth Int. Congress on Rheology, Sydney, Vol. 1, Australian Society of Rheology, 1988, p. 143. 8 J.F. Stevenson, Ph.D. Thesis, University of Wisconsin, 1970. 9 R.L. Ballman, Rheol. Acta, 4 (1965) 2. 10 C.D. Han, Rheology of Polymer Processing, Academic Press, New York, 1976; C.D. Han, Multiphase Flow in Polymer Processing, Academic Press, New York, 1981. 11 Y. Chan et al., J. Rheol., 22 (1978) 507. 12 D. Binding, J. Non-Newtonian Fluid Mech., 27 (1988) 173.