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Analysis of entrance pressure drop techniques for extensional viscosity determination
Polymer Testing 28 (2009) 843–853
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Test Method
Analysis of entrance pressure drop techniques for extensional viscosity determination Martin Zatloukal*, Jan Musil Polymer Centre, Faculty of Technology, Tomas Bata University in Zlı´n, TGM 275, Zlı´n 76272, Czech Republic
a r t i c l e i n f o
a b s t r a c t
Article history: Received 6 June 2009 Accepted 23 July 2009
In this work, a novel (patent pending) orifice die design for precise extensional viscosity data determination from entrance pressure drop measurements has been developed and tested both theoretically (through Finite Element Analysis) and experimentally. It has been demonstrated that the proposed novel orifice die allows much more precise extensional viscosity measurements for polymer melts in comparison with conventional orifice dies. Moreover, it has been found that, for extensional strain hardening and extensional strain thinning polymer melts, the corrected Cogswell model and Binding/Gibson model should be preferred, respectively. Otherwise, the extensional viscosity determination can be rather erroneous. Ó 2009 Elsevier Ltd. All rights reserved.
Keywords: Extensional viscosity Entrance pressure drop Orifice die Cogswell model Binding model Gibson model Polymer melts
1. Introduction Extensional viscosity is an important rheological property which expresses the resistance of the polymer melt to extensional flow (stretching). The knowledge of extensional viscosity is crucially important for better understanding of polymer molecular structure, and it significantly helps to optimize the polymer processing conditions and equipment design to achieve stable flow conditions, especially in technologies with predominantly extensional flow (the flow through converging channels in extrusion/coextrusion dies, injection moulding, calendering and post die processes such as fibre spinning, tubular film blowing, cast film). It should be mentioned that the extensional viscosity of the polymer melt is much more difficult to measure in comparison with the shear viscosity [1]. In the open literature, several types of extensional rheometers and experimental techniques have been developed [1–9] to measure this very important property. However, each of them is applicable for only * Corresponding author. E-mail address: [email protected] (M. Zatloukal). 0142-9418/$ – see front matter Ó 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.polymertesting.2009.07.007
a limited range of extensional rates or stresses and a low experimental error can be observed only in the case when the elongation flow is stable for a sufficient time for large enough volume of the sample to pass. Interestingly, the only available experimental technique to determine extensional viscosity at high extensional strain rates is based on entrance pressure drop measurements [1,5–9] by using capillary rheometers (see Fig. 1). The use of an orifice die for entrance pressure drop measurements is a very attractive methodology because it minimizes errors due to pressure dependence of viscosity, wall slip and viscous heating, which can occur during alternative extrapolation based on the Bagley plot approach. On the other hand, Kim and Dealy [8] recently pointed out that entrance pressure drop measurements can be highly erroneous by using orifice dies for the cases when the orifice die exit section is wetted by the polymer melt. In order to understand more deeply advantages and disadvantages of the entrance pressure drop techniques for extensional viscosity determination, the effect of orifice die design on the entrance pressure drop determination as well as applicability of the different models relating entrance pressure drop and extensional viscosity will be investigated theoretically and
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M. Zatloukal, J. Musil / Polymer Testing 28 (2009) 843–853
Fig. 1. Transparent section view of barrel part of twin-bore capillary rheometer.
experimentally for polyolefins having different level of extensional strain hardening. 2. Theoretical analysis In the theoretical part of this work, isothermal viscoelastic steady state two-dimensional finite element simulations were performed by solving the well known mass and momentum conservation equations using the commercially available Compuplast software VEL 6.3 together with the modified White-Metzner (mWM) model according to Barnes and Roberts [10]. The numerical scheme used for this model is a mixed scheme with the stresses as unknowns (u-v-p-s scheme). The main advantage of this phenomenological model is, firstly, the existence of analytical expressions for both shear and uniaxial extensional viscosities, which simplifies the model parameters identification process from the measured steady shear and steady uniaxial extensional viscosity data by a direct last square minimization method. Secondly, it has very good capability to describe steady shear and uniaxial extensional viscosities for various polymeric materials such as unfilled polymers, polymers filled with
glass beads and polymer blends [10]. Finally, the numerical solution remains stable even for strong mixed shear and extensional flows, mainly due to the fact that recoverable shear, N1/(2sxy), is predicted by the model to be constant at very high shear rates [10]. The isothermal mWM model is given by Eqs. (1)–(3) V
s þ lðIId Þs ¼ hðIId Þd ho
hðIId Þ ¼
(1)
a
1 þ ðK1 IId Þ
lðIId Þ ¼
lo 1 þ K2 IId
1n a
(2)
(3)
where d is the rate of deformation tensor, IId is the second invariant of the rate of deformation tensor, s is the stress V tensor, s is the upper-convected time derivative of the stress tensor. l(IId) is the deformation rate-dependent relaxation time and h(IId) is the deformation rate-dependent viscosity, h0 is zero shear-rate viscosity and l0, K1, K2,
M. Zatloukal, J. Musil / Polymer Testing 28 (2009) 843–853 10 7
10 7
a
LDPE Escorene LD 165 BW1 at 200°C
Shear and uniaxial extensional viscosities (Pa.s)
Shear and uniaxial extensional viscosities (Pa.s)
845
10 6
10 5
10 4
10 3 Uniaxial extensional viscosity (SER-HV-A01) Shear viscosity (rotational rheometer) Shear viscosity (capillary rheometer) modified White-Metzner model prediction
10 2
10 1 10 -5
10 -4
10 -3
10 -2
10 -1
10 0
10 1
10 2
10 3
10 4
10 5
b
LDPE Escorene LD 165 BW1 at 200°C and virtual materials M1, M2, M3 and M4 at 200°C
10 6
10 5
10 4
10 3 Extensional viscosity for LDPE Escorene LD 165BW1 Extensional viscosity for virtual material M1 Extensional viscosity for virtual material M2 Extensional viscosity for virtual material M3 Extensional viscosity for virtual material M4 Shear viscosity
10 2
10 1 10 -4
10 -3
10 -2
10 -1
10 0
101
102
10 3
Shear and extensional rates (s-1)
Shear and extensional rates (s-1)
Fig. 2. Shear and uniaxial extensional viscosity for materials used in the theoretical study. a) Comparison between steady shear and uniaxial extensional viscosity data (symbols) and modified White-Metzner model predictions (lines) for LDPE Escorene LD 165 BW1 at 200 C. Experimental data are taken from [11]. b) Steady shear and uniaxial extensional viscosity curves predicted by modified White-Metzner model for LDPE Escorene LD 165 BW1 and four additional virtual materials (M1, M2, M3 and M4).
n,a are constants. It should be mentioned that pffiffiffi l0, K2 have to satisfy the following constraint: l0 =K2 < 3=2 to avoid an infinite steady uniaxial extensional viscosity. 2.1. FEM analysis of Binding, Cogswell and Gibson models In this part, the capability of Binding, Cogswell and Gibson models to determine extensional rheology for different polyolefin melts will be investigated theoretically by modelling of the abrupt contraction flow though an ideal orifice die by FEM. For this aim, LDPE Escorene LD 165BW 1 [11] together with four additional virtual materials having continuously decreasing extensional strain hardening level were considered see Fig. 2a,b). This has been done by holding parameters in Eq. (2) as constant, whereas l0 and K2 parameters in Eq. (3) were varied. The modified White-Metzner model parameters for all these samples are provided in Table 1. The sketch of the ‘‘ideal orifice die’’ (i.e. it is assumed that the downstream region of the orifice is not filled by the melt – the extrudate flows as a free jet through the expansion) together with corresponding boundary conditions and FE mesh is provided in Fig. 3. The die has 90 entrance angle, 2 mm die diameter
and capillary length equal to 0.24 mm (ideally this length should be zero but from machinery point of view the considered orifice die has certain very small length). According to [12], the maximum achievable extensional strain in this domain is given by Eq. (4) i.e. 4.03 in this specific case.
R
Barrel 3 ¼ 2ln R capillary
(4)
The FE mesh used for abrupt contraction flow modeling consists of 2949 elements with 6094 nodes, which sufficiently describes the flow domain and gives precise enough results. Edge length was chosen as 0.8 mm and, due to low flow rates, iteration level in our research was 1012. The number of iterations was 100, or 200 for shear rates lower than 0.03 s1. It should be mentioned that in all calculations, the procedure is the same as proposed in [9] i.e., it is assumed that the polymer melt does not swell and that the free surface boundary conditions for free jet flow can be replaced with a wall with absolute slip (only the axial wall velocity component is allowed to vary, and the wall radial velocity is equal to zero). This assumption has several advantages. First, it allows simulation of free jet flow in
Table 1 Parameters of modified White-Metzner model for both, real (LDPE Escorene LD 165 BW1) and virtual (M1, M2, M3 and M4) materials. Material
modified White-Metzner model
h0 (Pa.s)
K1 (s)
a ()
n ()
l0 (s)
K2 (s)
LDPE Escorene LD 165BW 1 Virtual material M1 Virtual material M2 Virtual material M3 Virtual material M4
77 103 77 103 77 103 77 103 77 103
7.6286 7.6286 7.6286 7.6286 7.6286
0.4747 0.4747 0.4747 0.4747 0.4747
0.3284 0.3284 0.3284 0.3284 0.3284
248.32 236.00 220.00 193.00 0.00
293.63 293.63 293.63 293.63 293.63
846
M. Zatloukal, J. Musil / Polymer Testing 28 (2009) 843–853
O15
50
Input Axis Output Wall F u lly 0. 24
wall
12. 5
O2
lubricated
a
b
c Fig. 3. Details for theoretical ideal orifice die analysis where downstream region is not filled by polymer melt. a) Geometrical sketch of ideal orifice die, b) Boundary conditions, c) FEM mesh.
which the velocity rearrangement at the end of the die is taken into account. This is the main advantage in comparison with simulations that do not consider any free jet flow. Secondly, the no extrudate swell assumption simplifies the flow situation, so that calculations can be made at much higher flow rates. In order to demonstrate the role of material extensional properties on the generation of the entrance pressure drop for different polymer melts flow through an abrupt contraction, LDPE Escorene LD 165 BW1 and four virtual materials (see Fig. 2b) were used. The calculated results, in terms of entrance viscosity (entrance pressure drop divided by apparent shear rate), are depicted in Fig. 4 and the parameters of entrance pressure drop model are summarized in Table 2. It is clearly visible that all materials showing extensional strain hardening in the uniaxial extensional viscosity (see Fig. 2b) also yield overshoot in the entrance viscosities (see Fig. 4). With the aim of testing the applicability of Binding, Cogswell and Gibson model to predict uniaxial extensional viscosity, firstly, the entrance pressure drop model proposed in [9], Eq. (5), was used to fit
theoretical entrance viscosity for each particular materials (see Fig. 4). ) ( x
logðhENT Þ ¼ log
hENT;0
a
1 þ ðlg_ APP Þ
tanhðag_ APP þ 1Þ tanhð1Þ
(5)
where hENT;0 represents the plateau-value of entrance viscosity, and l, a, a, x are adjustable parameters. Secondly, Cogswell, Binding and Gibson models were utilized to calculate uniaxial extensional viscosity according to equations provided in [9]. All data needed for such calculation were taken from the numerical simulations. Moreover, effective entry length correction proposed in [9] was used for all considered models. Comparisons between the calculated extensional and theoretical viscosities are given in Fig. 5a–e. It is shown that all the entrance techniques predict the same extensional viscosity plateau at low shear rates due to the applied effective entry length correction. The comparison between theoretical and calculated extensional viscosities with the applied correction for extensional strain hardening polymer melts (see Fig. 5a–d) leads to the following conclusions:
M. Zatloukal, J. Musil / Polymer Testing 28 (2009) 843–853
10 6
Apparent entrance viscosity (Pa.s)
LDPE Escorene LD 165 BW1 at 200°C and virtual materials M1,M2,M3 and M4 at 200°C
10 5
10 4
10 3 10 -4
LDPE Escorene LD 165 BW1 Virtual material M1 Virtual material M2 Virtual material M3 Virtual material M4 Entrance pressure drop model fit
10 -3
10 -2
10 -1
10 0
10 1
10 2
10 3
Apparent shear rate (s-1) Fig. 4. Apparent entrance viscosity curves determined by viscoelastic FEM calculations for ‘‘ideal orifice die’’ depicted in Fig. 3a).
All methods seem to capture the same and proper slope of the extensional viscosity at high extensional rates. The Binding model over-predicts and under-predicts hE at low and high 3_ , respectively. The Gibson model significantly under-predicts hE at both, low and high 3_ . The Cogswell model under-predicts hE for highly extensional strain hardening melts, i. e. if the maximum uniaxial extensional viscosity divided by 3 times Newtonian viscosity, hEmax/(3h0), is higher than 2. On the other hand, for low extensional strain hardening materials where hEmax/(3h0) 2, the Cogswell model predicts hE very precisely within a wide range of extensional strain rates. Fig. 5e compares theoretical and calculated extensional viscosities with the applied correction for extensional strain thinning polymer melt. It is nicely visible that both Binding and Gibson models provide reasonably good agreement between calculated and predicted hE. On the other hand, the Cogswell model tends to over-predict the uniaxial extensional viscosity, especially at high extensional strain rates. 2.2. Theoretical effect of die design on the measured extensional rheology In this part, the effect of the orifice die design on the measured entrance viscosity and consequent extensional
847
viscosity calculation has been investigated. Due to a quite narrow downstream orifice die region (see Fig. 6 for more detail for conventional orifice die) and extrudate swell, the polymer melt may fill this region. Therefore, if this happens, the question is how much this flow situation influences the measured entrance viscosity/uniaxial extensional viscosity. In order to answer this question, virtual material M2 (having medium level of extensional strain hardening) has been chosen for FEM analysis for the case considering that downstream orifice die region is filled by the polymer melt. The sketch of such a low situation for a conventional orifice die together with corresponding boundary conditions and FE mesh is provided in Fig. 7 (grid consists of 7400 nodes and 3553 elements). The problem has been solved for different flow rates and the theoretical results obtained were plotted in Fig. 8 and compared with corresponding theoretical data considering that the downstream orifice die region is not filled by the polymer melt. In this Figure, it is clearly visible that entrance viscosities for both cases differ significantly within a wide range of apparent shear rates. In addition, entrance viscosity parameters for this material are provided in Table 3. In more detail, the fact that the polymer melt fills the downstream region of the orifice die leads to significant entrance viscosity increase due to an additional shear flow component occurring in this region. Fig. 9 shows the impact of such artificial entrance viscosity increase on the uniaxial extensional viscosity determined by the corrected Cogswell model. Clearly, artificial entrance viscosity increase leads to artificial uniaxial extensional viscosity increase, leading to a higher level of disagreement between Cogswell model prediction and true uniaxial extensional viscosity. In the other words, the main conclusion from this theoretical study is that a very narrow downstream region of the used orifice die may lead to erroneous estimation of the entrance viscosity, leading to artificially high uniaxial extensional viscosity. 3. Experimental analysis 3.1. Novel orifice die design In order to prevent unwanted artificial entrance viscosity increase due to a very narrow downstream channel, we have proposed a novel (patent pending) orifice die design (see Fig. 10). The main advantage of this novel orifice die design is a very open downstream region which consists of highly diverging channel (Fig. 10a and b) or flat exit region (Fig. 10c–d) and four holes which enables the use of a special key to screw-up the orifice die to the rheometer barrel. This downstream orifice die geometry
Table 2 Entrance pressure drop model, Eq. (5), parameters for both, real (LDPE Escorene LD 165 BW1) and virtual (M1, M2, M3 and M4) materials. Material
LDPE Escorene LD 165BW 1 Virtual material M1 Virtual material M2 Virtual material M3 Virtual material M4
Entrance pressure drop model
hENT,0 (Pa.s)
l (s)
a ()
a (s)
x ()
59 550.14209 57 758.09353 55 776.31681 57 806.94014 61 906.54042
0.581123 0.782182 0.976292 1.497522 0.940443
0.6064556417 0.5951470027 0.5902599899 0.5749431565 0.6261065864
15.834387 17.037560 18.775613 16.035987 0.000000
0.299492 0.292668 0.276932 0.244389 0.000000
848
10
7
Uniaxial extensional viscosity (Pa.s)
a 10
6
10
5
b
LDPE Escorene LD 165 BW1 at 200°C
c
Virtual material M1 at 200°C
Virtual material M2 at 200°C
104
102 -4 10
10
-3
10
-2
-1
0
10 10 Extensional rate (s -1)
10
10
modified White-Metzner model prediction Corrected Cogswell model prediction Corrected Binding model prediction Corrected Gibson model prediction
1
10
10
-4
10
-3
10
-2
-1
0
10 10 Extensional rate (s -1)
10
1
10
2
10
-4
10
-3
10
-2
-1
0
10 10 Extensional rate (s -1)
10
1
10
2
10
3
7
d
Uniaxial extensional viscosity (Pa.s)
2
modified White-Metzner model prediction Corrected Cogswell model prediction Corrected Binding model prediction Corrected Gibson model prediction
10
6
10
5
10
4
10
3
10
2
10
e
Virtual material M3 at 200°C
Virtual material M4 at 200°C
modified White-Metzner model prediction Corrected Cogswell model prediction Corrected Binding model prediction Corrected Gibson model prediction
-4
10
-3
10
-2
-1
0
10 10 Extensional rate (s -1)
10
modified White-Metzner model prediction Corrected Cogswell model prediction Corrected Binding model prediction Corrected Gibson model prediction
1
10
2
10
-4
10
-3
10
-2
-1
0
10 10 Extensional rate (s -1)
10
1
10
2
10
3
Fig. 5. Comparison between modified White-Metzner model and Cogswell, Binding and Gibson extensional viscosity predictions for different polymer melts by using ideal orifice die depicted in Fig. 3a. Here, corrected Cogswell/Binding/Gibson model means that the effective entry length correction has been applied according to [9]. a) LDPE Escorene LD 165 BW1, b) Virtual material M1, c) Virtual material M2, d) Virtual material M3, e) Virtual material M4.
M. Zatloukal, J. Musil / Polymer Testing 28 (2009) 843–853
10
Uniaxial extensional viscosity (SER-HV-A01) modified White-Metzner model prediction Corrected Cogswell model prediction Corrected Binding model prediction Corrected Gibson model prediction
3
M. Zatloukal, J. Musil / Polymer Testing 28 (2009) 843–853
849
Fig. 6. Conventional orifice die. a) Section view, b) Bottom view.
Apparent entrance viscosity (Pa.s)
106
105
104
Conventional orifice die, filled downstream region Novel orifice die, unfilled downstream region
eliminates any possibility for artificial pressure increase due to polymer melt touching the downstream wall. Moreover, a detail view of the extrudate leaving the die being possible during the measurements allowing direct measurement of the free surface development during the extrudate swell, which is practically impossible if one uses the conventional orifice die design depicted in Fig. 6. Another expected advantage of this novel die design is a more precise detection of the melt rupture for rupture stress determination in comparison with conventional orifice design, where the melt rupture can be hidden if the downstream region is filled by the polymer melt. 3.2. Experimental evaluation of the novel orifice die design
50
O15
0.25
16
33
O2
Input Axis Output Wall
110° 40° O12
b
10-3
100
101
102
103
(s-1)
Table 3 Parameters of entrance pressure drop model, Eq. (5) for virtual material M2. Down Entrance pressure drop model stream a hENT,0 l die region (Pa.s) (s) () type Unfilled Filled
a
x
(s)
()
55 776.31681 0.976292 0.5902599899 18.775613 0.276932 70 727.18176 0.654474 0.6011542038 18.599694 0.279597
107
106
105
104
Uncorrected Cogswell model (unfilled downstream region) Corrected Cogswell model (unfilled downstream region) Corrected Cogswell model (filled downstream region) modified White-Metzner model prediction
103
102 10-4
Fig. 7. Details for theoretical conventional orifice die analysis where downstream region is filled by polymer melt. a) Geometrical sketch of conventional orifice die, b) Boundary conditions, c) FEM mesh.
10-1
Fig. 8. Theoretically predicted effect of die design (filled/unfilled orifice downstream region) on entrance viscosity (entrance pressure drop divided by apparent shear rate) for virtual polymer melt M2 at 200 C. Line represents entrance pressure drop model fit.
a
c
10-2
Apparent shear rate
Uniaxial extensional viscosity (Pa.s)
In order to evaluate the novel orifice die with respect to conventional orifice die, the entrance viscosity (entrance pressure drop divided by the apparent shear rate) has been measured for LDPE Lupolen 1840H at 250 C by using a Rosand RH7-2 twin bore capillary rheometer and both orifice dies. The results together with the closer view of the orifice downstream region during measurements are
103 10-4
10-3
10-2
10-1
100
Extensional rate (s -1)
101
102
103
Fig. 9. Comparison between modified White-Metzner model and Cogswell extensional viscosity predictions for virtual polymer melt M2 by using two different orifice dies. Here, corrected Cogswell model means that the effective entry length correction has been applied for the Cogswell model according to [9].
850
M. Zatloukal, J. Musil / Polymer Testing 28 (2009) 843–853
Fig. 10. Novel patent pending orifice dies. a) Abrupt entry (section view), b) Abrupt entry (bottom view), c) Convergent entry (section view), d) Convergent entry (bottom view).
3.3. Experimental analysis of Binding, Cogswell and Gibson models In this section, the novel orifice die will be utilized to determine uniaxial extensional viscosity by using Binding, Cogswell and Gibson models for highly branched (LDPE Lupolen 1840H), slightly branched (mLLDPE Exact 0201) and linear polymers (HDPE Tipelin FS 450 – 26). In the first step, entrance viscosity has been determined for all three samples by the use of the novel orifice die and
4000 LDPE Lupolen 1840H at 250°C
Apparent entrance viscosity (Pa.s)
provided in Fig. 11 and Fig. 12, respectively. As expected from the theoretical study, entrance viscosity measured by the conventional orifice die is much higher in comparison with the novel die design, due to the fact that polymer melt fills the downstream region of the conventional orifice die. Also for that reason, the level of the extrudate swell from the conventional orifice die is artificially high. This leads us to the conclusion that the measurements of the extensional rheology by using a conventional orifice die can be highly erroneous and the novel orifice die design should be preferred.
3000
2000
Conventional orifice die design (D = 1 mm) Novel orifice die design (D = 1 mm)
103 101
20
50
10 2
200
500
10 3
2000
-1
Apparent shear rate (s ) Fig. 11. Comparison between experimentally determined entrance viscosities (entrance pressure drop divided by apparent shear rate) by using conventional orifice die (green symbols) and novel orifice die with abrupt entry (red symbols) for LDPE Lupolen 1840H at 250 C.
M. Zatloukal, J. Musil / Polymer Testing 28 (2009) 843–853
Fig. 12. Closer view of the orifice downstream region during measurements depicted in Fig. 11. a) Conventional orifice die design, b) Novel patent pending orifice die design with abrupt entry.
a
b
851
subsequently fitted by the entrance pressure drop model, Eq. (5) as depicted in Fig. 13a–c and with fitting parameters provided in Table 4. It is nicely visible that in all three cases, Eq. (5) represents the measured entrance viscosity data very well. In the second step, effective entry length correction has been applied for entrance viscosity data with respect to Binding, Cogswell and Gibson models. Finally, the extensional viscosities have been calculated for the all three samples according to all three techniques, and compared with SER steady state extensional viscosity data reported in the literature [4,11] as seen in Fig. 14a–i (the shear viscosity data needed for the uniaxial extensional viscosity calculation were measured on a Rosand RH7-2 twin – bore capillary rheometer; corresponding CarreauYasuda model fitting parameters are provided in Table 5). It also has to be mentioned that the dotted line occurring in Fig. 14a–i represents extensional viscosity calculated from the entrance viscosity fitting lines provided in Fig. 13a–c, i.e. the fitting lines were taken as the measurements in this case. Closer analysis of these Figures reveals the following conclusions. For highly branched LDPE Lupolen 1840H material where hEmax/(3h0) ¼ 3.9, Binding model predicts extensional viscosity in good agreement with SER extensional viscosity data at low extensional extensional rates, whereas at medium and higher deformation rates the calculated data are under-predicted (see Fig. 14a). Gibson model behaviour is even worse in this case, because calculated extensional viscosity is under-predicted within whole extensional rate range (see Fig. 14b). Finally, the Cogswell model gives good agreement between measured and calculated extensional viscosity at low and high extensional deformation rates but at medium deformation rates the model under-predicts the measured data (see Fig. 14c). Obviously, these observations are in good agreement with the conclusions obtained from theoretical analysis. Thus, we could conclude that for branched LDPE Lupolen 1840H polymer the Cogswell model predictions are more realistic in comparison with Binding and Gibson models. The incapability of the Cogswell model to describe extensional viscosity data at medium deformation rates can be
c
Fig. 13. Entrance viscosity (entrance pressure drop divided by apparent shear rate) measured by novel orifice die with abrupt entry. a) LDPE Lupolen 1840H at 150 C, b) mLLDPE Exact 0201 at 180 C, c) HDPE Tipelin FS 450–26 at 180 C.
852
M. Zatloukal, J. Musil / Polymer Testing 28 (2009) 843–853
Table 4 Parameters of entrance pressure drop model, Eq. (5), for LDPE Lupolen 1840H, mLLDPE Exact 0201 and HDPE Tipelin FS 450-26. Material
Entrance pressure drop model
LDPE Lupolen 1840H mLLDPE Exact 0201 HDPE Tipelin FS 450 - 26
hENT,0 (Pa.s)
l (s)
a ()
a (s)
x ()
29 647.25498 19 010.49081 1000 000.00000
0.051010 0.420384 8.004018
0.6599534062 0.3817786899 0.5778474780
19.951935 6.012095 0.000000
0.256709 0.306068 0.000000
7
10
a
b
LDPE Lupolen 1840H at 150°C
c
LDPE Lupolen 1840H at 150°C
LDPE Lupolen 1840H at 150°C
Shear and uniaxial extensional viscosities (Pa.s)
106
5
10
104
103
2
10
101 10-4
10-3
10-2 10-1 100 101 Shear and extensional rates (s-1)
Shear viscosity (rotational rheometer) Shear viscosity (capillary rheometer) Extensional viscosity (uncorrected Cogswell-measurement) Extensional viscosity (uncorrected Cogswell-extrapolation) Extensional viscosity (corrected Cogswell-measurement) Extensional viscosity (corrected Cogswell-extrapolation) Uniaxial extensional viscosity (SER-HV-A01) Carreau-Yasuda model fit
Shear viscosity (rotational rheometer) Shear viscosity (capillary rheometer) Extensional viscosity (uncorrected Gibson-measurement) Extensional viscosity (uncorrected Gibson-extrapolation) Extensional viscosity (corrected Gibson-measurement) Extensional viscosity (corrected Gibson-extrapolation) Extensional viscosity (SER-HV-A01) Carreau-Yasuda model fit
Shear viscosity (rotational rheometer) Shear viscosity (capillary rheometer) Extensional viscosity (uncorrected Binding-measurement) Extensional viscosity (uncorrected Binding-extrapolation) Extensional viscosity (corrected Binding-measurement) Extensional viscosity (corrected Binding-extrapolation) Extensional viscosity (SER-HV-A01) Carreau-Yasuda model fit 102
10-4
10-3
10-2 10-1 100 101 Shear and extensional rates (s -1)
102
10-4
10-3
10-2 10-1 100 101 Shear and extensional rates (s -1)
102
103
6
10
Shear and uniaxial extensional viscosities (Pa.s)
d
e
mLLDPE Exact 0201 at 180°C
f
mLLDPE Exact 0201 at 180°C
mLLDPE Exact 0201 at 180°C
105
104
Shear viscosity (rotational rheometer) Shear viscosity (capillary rheometer) Extensional viscosity (uncorrected Binding-measurement) Extensional viscosity (uncorrected Binding-extrapolation) Extensional viscosity (corrected Binding-measurement) Extensional viscosity (corrected Binding-extrapolation) Extensional viscosity (SER-HV-A01) Carreau-Yasuda model fit
103
102 10-4
10-3
10-2 10-1 100 101 Shear and extensional rates (s-1)
Shear viscosity (rotational rheometer) Shear viscosity (capillary rheometer) Extensional viscosity (uncorrected Gibson-measurement) Extensional viscosity (uncorrected Gibson-extrapolation) Extensional viscosity (corrected Gibson-measurement) Extensional viscosity (corrected Gibson-extrapolation) Extensional viscosity (SER-HV-A01) Carreau-Yasuda model fit
102
10-4
10-3
10-2 10-1 100 101 Shear and extensional rates (s -1)
Shear viscosity (rotational rheometer) Shear viscosity (capillary rheometer) Extensional viscosity (uncorrected Cogswell-measurement) Extensional viscosity (uncorrected Cogswell-extrapolation) Extensional viscosity (corrected Cogswell-measurement) Extensional viscosity (corrected Cogswell-extrapolation) Extensional viscosity (SER-HV-A01) Carreau-Yasuda model fit
102
10-4
10-3
10-2 10-1 100 101 Shear and extensional rates (s -1)
102
103
8
10
g
h
HDPE Tipelin FS 450-26 at 180°C
i
HDPE Tipelin FS 450-26 at 180°C
HDPE Tipelin FS 450-26 at 180°C
7
Shear and uniaxial extensional viscosities (Pa.s)
10
6
10
5
10
104
3
10
2
10
101 -4 10
Shear viscosity (rotational rheometer) Shear viscosity (capillary rheometer) Extensional viscosity (uncorrected Binding-measurement) Extensional viscosity (corrected Binding-measurement) Extensional viscosity (SER-HV-A01) Carreau-Yasuda model fit
10
-3
-2
10
-1
0
10 10 10 Shear and extensional rates (s-1)
1
10
Shear viscosity (rotational rheometer) Shear viscosity (capillary rheometer) Extensional viscosity (uncorrected Cogswell-measurement) Extensional viscosity (corrected Cogswell-measurement) Extensional viscosity (SER-HV-A01) Carreau-Yasuda model fit
Shear viscosity (rotational rheometer) Shear viscosity (capillary rheometer) Extensional viscosity (uncorrected Gibson-measurement) Extensional viscosity (corrected Gibson-measurement) Extensional viscosity (SER-HV-A01) Carreau-Yasuda model fit
2
10
-4
-3
10
10
-2
-1
0
1
10 10 10 Shear and extensional rates (s -1)
10
2
-4
10
-3
10
10
-2
-1
0
1
10 10 10 Shear and extensional rates (s -1)
2
10
3
10
Fig. 14. Comparison between Binding/Gibson/Cogswell extensional viscosity data and SER measurements for different polyolefines taken from [11,13] (shear viscosity data obtained from rotational and capillary rheometer are also provided in this Figure). Here, corrected Binding/Gibson/Cogswell model means that the effective entry length correction has been applied according to [9]. a-c) LDPE Lupolen 1840H at 150 C, d-f) mLLDPE Exact 0201 at 180 C, g-i) HDPE Tipelin FS 450–26 at 180 C.
M. Zatloukal, J. Musil / Polymer Testing 28 (2009) 843–853 Table 5 Fitting parameters of Carreau – Yasuda model, Eq. (2), for LDPE Lupolen 1840H, mLLDPE Exact 0201 and HDPE Tipelin FS 450-26. Material
Carreau – Yasuda model
h0 (Pa.s)
K1 (s)
a (-)
n (-)
LDPE Lupolen 1840H mLLDPE Exact 0201 HDPE Tipelin FS 450 – 26
59 950 17 998 5 033 405
0.5834 5.6062 3.8745
0.3282 1.3639 0.1401
0.1000 0.6029 0.0653
explained by the fact that the extensional strain hardening is too high in this case i.e. that hEmax/(3h0) > 2, which is consistent with the findings based on theoretical analysis. For slightly branched mLLDPE Exact 0201 material where hEmax/(3h0) ¼ 1.6, Binding model over-predicts extensional viscosity at low extensional rates, whereas at medium and high deformation rates the extensional viscosity is under-predicted (see Fig. 14d). Gibson model predicts extensional viscosity in good agreement with the measured SER data at low extensional rates, but at medium and high deformation rates, the extensional viscosity is under-predicted (see Fig. 14e). Cogswell model predictions are surprisingly in excellent agreement with the measured SER extensional viscosity data as seen in Fig. 14f. For slightly branched mLLDPE Exact 0201 polymer, all above mentioned conclusions are in good agreement with the findings based on theoretical analysis for all three tested model. Here, it should be pointed out that the extensional strain hardening parameter for mLLDPE Exact 0201 is hEmax/(3h0) < 2 i.e. it is in the range where Cogswell model predictions are very precise as shown in the theoretical analysis. For linear HDPE Tipelin FS 450 – 26 where hEmax/ (3h0) ¼ 1, both, Binding and Gibson models gives excellent agreement with the measured SER extensional viscosity data within the experimentally determined extensional rates range (see Fig. 14g–h). On the other hand, the Cogswell model failed in this case because the model overpredicts SER extensional viscosity data (see Fig. 14i) within the whole deformation rate range. Again, all above mentioned conclusions for the linear HDPE Tipelin FS 450 – 26 polymer are in very good agreement with corresponding conclusions obtained during theoretical analysis. 4. Conclusion Novel patent pending orifice die design has been proposed and tested for the entrance pressure drop measurements. It has been theoretically and experi-
853
mentally demonstrated that the novel orifice die design allows more precise entrance pressure drop measurement (and thus more precise extensional viscosity determination) in comparison with standard and conventional orifice die. Based on theoretical and experimental analysis, it has been revealed that by using novel orifice die design corrected Cogswell model is more precise in uniaxial extensional viscosity prediction than Binding and Gibson models for highly branched and slightly branched polymer melts. In more detail, it has been demonstrated that corrected Cogswell model provides excellent capability correctly calculate extensional viscosity, especially, if the extensional strain hardening level given by hEmax/(3h0) ratio is lower than 2. On the other hand, for linear polymer melts, the situation has been found to be the opposite, i.e. that Binding and Gibson models were found to provide excellent capability to predict extensional viscosity data from the entrance pressure drop measurements, whereas the Cogswell model extensional viscosity predictions were found to be poor (predicted extensional viscosity data were always lower in comparison with SER measurements).
Acknowledgements The support of the projects by the Ministry of Education CR (KONTAKT ME08090, MSM 7088352101) is gratefully acknowledged. References [1] F.A. Morrison, Understanding Rheology, Oxford University Press, Inc., New York, USA, 2001. [2] H. Mu¨nstedt, J. Rheol. 23 (1979) 421–436. [3] M.L. Sentmanat, Rheol. Acta 43 (2004) 657–669. [4] M.L. Sentmanat, B.N. Wang, G.H. McKinley, J. Rheol. 49 (3) (2005) 585–606. [5] F.N. Cogswell, Polym. Eng. Sci. 12 (1972) 64–73. [6] D.M. Binding, J. Non-Newtonian Fluid Mech. 27 (1988) 173–189. [7] A.G. Gibson, Composites 20 (1989) 57–64. [8] S. Kim, J.M. Dealy, J. Rheol. 45 (6) (2001) 1413–1419. [9] M. Zatloukal, J. Vlcek, C. Tzoganakis, P. Saha, J. Non-Newtonian Fluid Mech. 107 (2002) 13–37. [10] H.A. Barnes, G.P. Roberts, J. Non-Newtonial Fluid Mech. 44 (1992) 113. [11] R. Pivokonsky, M. Zatloukal, P. Filip, J. Non-Newtonial Fluid Mech. 135 (1) (2006) 58. [12] A.D. Gotsis, A. Odriozola, Rheol. Acta 37 (1998) 430. [13] R. Pivokonsky, M. Zatloukal, P. Filip, J. Non-Newtonial Fluid Mech. 150 (1) (2008) 56.
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Test Method
Analysis of entrance pressure drop techniques for extensional viscosity determination Martin Zatloukal*, Jan Musil Polymer Centre, Faculty of Technology, Tomas Bata University in Zlı´n, TGM 275, Zlı´n 76272, Czech Republic
a r t i c l e i n f o
a b s t r a c t
Article history: Received 6 June 2009 Accepted 23 July 2009
In this work, a novel (patent pending) orifice die design for precise extensional viscosity data determination from entrance pressure drop measurements has been developed and tested both theoretically (through Finite Element Analysis) and experimentally. It has been demonstrated that the proposed novel orifice die allows much more precise extensional viscosity measurements for polymer melts in comparison with conventional orifice dies. Moreover, it has been found that, for extensional strain hardening and extensional strain thinning polymer melts, the corrected Cogswell model and Binding/Gibson model should be preferred, respectively. Otherwise, the extensional viscosity determination can be rather erroneous. Ó 2009 Elsevier Ltd. All rights reserved.
Keywords: Extensional viscosity Entrance pressure drop Orifice die Cogswell model Binding model Gibson model Polymer melts
1. Introduction Extensional viscosity is an important rheological property which expresses the resistance of the polymer melt to extensional flow (stretching). The knowledge of extensional viscosity is crucially important for better understanding of polymer molecular structure, and it significantly helps to optimize the polymer processing conditions and equipment design to achieve stable flow conditions, especially in technologies with predominantly extensional flow (the flow through converging channels in extrusion/coextrusion dies, injection moulding, calendering and post die processes such as fibre spinning, tubular film blowing, cast film). It should be mentioned that the extensional viscosity of the polymer melt is much more difficult to measure in comparison with the shear viscosity [1]. In the open literature, several types of extensional rheometers and experimental techniques have been developed [1–9] to measure this very important property. However, each of them is applicable for only * Corresponding author. E-mail address: [email protected] (M. Zatloukal). 0142-9418/$ – see front matter Ó 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.polymertesting.2009.07.007
a limited range of extensional rates or stresses and a low experimental error can be observed only in the case when the elongation flow is stable for a sufficient time for large enough volume of the sample to pass. Interestingly, the only available experimental technique to determine extensional viscosity at high extensional strain rates is based on entrance pressure drop measurements [1,5–9] by using capillary rheometers (see Fig. 1). The use of an orifice die for entrance pressure drop measurements is a very attractive methodology because it minimizes errors due to pressure dependence of viscosity, wall slip and viscous heating, which can occur during alternative extrapolation based on the Bagley plot approach. On the other hand, Kim and Dealy [8] recently pointed out that entrance pressure drop measurements can be highly erroneous by using orifice dies for the cases when the orifice die exit section is wetted by the polymer melt. In order to understand more deeply advantages and disadvantages of the entrance pressure drop techniques for extensional viscosity determination, the effect of orifice die design on the entrance pressure drop determination as well as applicability of the different models relating entrance pressure drop and extensional viscosity will be investigated theoretically and
844
M. Zatloukal, J. Musil / Polymer Testing 28 (2009) 843–853
Fig. 1. Transparent section view of barrel part of twin-bore capillary rheometer.
experimentally for polyolefins having different level of extensional strain hardening. 2. Theoretical analysis In the theoretical part of this work, isothermal viscoelastic steady state two-dimensional finite element simulations were performed by solving the well known mass and momentum conservation equations using the commercially available Compuplast software VEL 6.3 together with the modified White-Metzner (mWM) model according to Barnes and Roberts [10]. The numerical scheme used for this model is a mixed scheme with the stresses as unknowns (u-v-p-s scheme). The main advantage of this phenomenological model is, firstly, the existence of analytical expressions for both shear and uniaxial extensional viscosities, which simplifies the model parameters identification process from the measured steady shear and steady uniaxial extensional viscosity data by a direct last square minimization method. Secondly, it has very good capability to describe steady shear and uniaxial extensional viscosities for various polymeric materials such as unfilled polymers, polymers filled with
glass beads and polymer blends [10]. Finally, the numerical solution remains stable even for strong mixed shear and extensional flows, mainly due to the fact that recoverable shear, N1/(2sxy), is predicted by the model to be constant at very high shear rates [10]. The isothermal mWM model is given by Eqs. (1)–(3) V
s þ lðIId Þs ¼ hðIId Þd ho
hðIId Þ ¼
(1)
a
1 þ ðK1 IId Þ
lðIId Þ ¼
lo 1 þ K2 IId
1n a
(2)
(3)
where d is the rate of deformation tensor, IId is the second invariant of the rate of deformation tensor, s is the stress V tensor, s is the upper-convected time derivative of the stress tensor. l(IId) is the deformation rate-dependent relaxation time and h(IId) is the deformation rate-dependent viscosity, h0 is zero shear-rate viscosity and l0, K1, K2,
M. Zatloukal, J. Musil / Polymer Testing 28 (2009) 843–853 10 7
10 7
a
LDPE Escorene LD 165 BW1 at 200°C
Shear and uniaxial extensional viscosities (Pa.s)
Shear and uniaxial extensional viscosities (Pa.s)
845
10 6
10 5
10 4
10 3 Uniaxial extensional viscosity (SER-HV-A01) Shear viscosity (rotational rheometer) Shear viscosity (capillary rheometer) modified White-Metzner model prediction
10 2
10 1 10 -5
10 -4
10 -3
10 -2
10 -1
10 0
10 1
10 2
10 3
10 4
10 5
b
LDPE Escorene LD 165 BW1 at 200°C and virtual materials M1, M2, M3 and M4 at 200°C
10 6
10 5
10 4
10 3 Extensional viscosity for LDPE Escorene LD 165BW1 Extensional viscosity for virtual material M1 Extensional viscosity for virtual material M2 Extensional viscosity for virtual material M3 Extensional viscosity for virtual material M4 Shear viscosity
10 2
10 1 10 -4
10 -3
10 -2
10 -1
10 0
101
102
10 3
Shear and extensional rates (s-1)
Shear and extensional rates (s-1)
Fig. 2. Shear and uniaxial extensional viscosity for materials used in the theoretical study. a) Comparison between steady shear and uniaxial extensional viscosity data (symbols) and modified White-Metzner model predictions (lines) for LDPE Escorene LD 165 BW1 at 200 C. Experimental data are taken from [11]. b) Steady shear and uniaxial extensional viscosity curves predicted by modified White-Metzner model for LDPE Escorene LD 165 BW1 and four additional virtual materials (M1, M2, M3 and M4).
n,a are constants. It should be mentioned that pffiffiffi l0, K2 have to satisfy the following constraint: l0 =K2 < 3=2 to avoid an infinite steady uniaxial extensional viscosity. 2.1. FEM analysis of Binding, Cogswell and Gibson models In this part, the capability of Binding, Cogswell and Gibson models to determine extensional rheology for different polyolefin melts will be investigated theoretically by modelling of the abrupt contraction flow though an ideal orifice die by FEM. For this aim, LDPE Escorene LD 165BW 1 [11] together with four additional virtual materials having continuously decreasing extensional strain hardening level were considered see Fig. 2a,b). This has been done by holding parameters in Eq. (2) as constant, whereas l0 and K2 parameters in Eq. (3) were varied. The modified White-Metzner model parameters for all these samples are provided in Table 1. The sketch of the ‘‘ideal orifice die’’ (i.e. it is assumed that the downstream region of the orifice is not filled by the melt – the extrudate flows as a free jet through the expansion) together with corresponding boundary conditions and FE mesh is provided in Fig. 3. The die has 90 entrance angle, 2 mm die diameter
and capillary length equal to 0.24 mm (ideally this length should be zero but from machinery point of view the considered orifice die has certain very small length). According to [12], the maximum achievable extensional strain in this domain is given by Eq. (4) i.e. 4.03 in this specific case.
R
Barrel 3 ¼ 2ln R capillary
(4)
The FE mesh used for abrupt contraction flow modeling consists of 2949 elements with 6094 nodes, which sufficiently describes the flow domain and gives precise enough results. Edge length was chosen as 0.8 mm and, due to low flow rates, iteration level in our research was 1012. The number of iterations was 100, or 200 for shear rates lower than 0.03 s1. It should be mentioned that in all calculations, the procedure is the same as proposed in [9] i.e., it is assumed that the polymer melt does not swell and that the free surface boundary conditions for free jet flow can be replaced with a wall with absolute slip (only the axial wall velocity component is allowed to vary, and the wall radial velocity is equal to zero). This assumption has several advantages. First, it allows simulation of free jet flow in
Table 1 Parameters of modified White-Metzner model for both, real (LDPE Escorene LD 165 BW1) and virtual (M1, M2, M3 and M4) materials. Material
modified White-Metzner model
h0 (Pa.s)
K1 (s)
a ()
n ()
l0 (s)
K2 (s)
LDPE Escorene LD 165BW 1 Virtual material M1 Virtual material M2 Virtual material M3 Virtual material M4
77 103 77 103 77 103 77 103 77 103
7.6286 7.6286 7.6286 7.6286 7.6286
0.4747 0.4747 0.4747 0.4747 0.4747
0.3284 0.3284 0.3284 0.3284 0.3284
248.32 236.00 220.00 193.00 0.00
293.63 293.63 293.63 293.63 293.63
846
M. Zatloukal, J. Musil / Polymer Testing 28 (2009) 843–853
O15
50
Input Axis Output Wall F u lly 0. 24
wall
12. 5
O2
lubricated
a
b
c Fig. 3. Details for theoretical ideal orifice die analysis where downstream region is not filled by polymer melt. a) Geometrical sketch of ideal orifice die, b) Boundary conditions, c) FEM mesh.
which the velocity rearrangement at the end of the die is taken into account. This is the main advantage in comparison with simulations that do not consider any free jet flow. Secondly, the no extrudate swell assumption simplifies the flow situation, so that calculations can be made at much higher flow rates. In order to demonstrate the role of material extensional properties on the generation of the entrance pressure drop for different polymer melts flow through an abrupt contraction, LDPE Escorene LD 165 BW1 and four virtual materials (see Fig. 2b) were used. The calculated results, in terms of entrance viscosity (entrance pressure drop divided by apparent shear rate), are depicted in Fig. 4 and the parameters of entrance pressure drop model are summarized in Table 2. It is clearly visible that all materials showing extensional strain hardening in the uniaxial extensional viscosity (see Fig. 2b) also yield overshoot in the entrance viscosities (see Fig. 4). With the aim of testing the applicability of Binding, Cogswell and Gibson model to predict uniaxial extensional viscosity, firstly, the entrance pressure drop model proposed in [9], Eq. (5), was used to fit
theoretical entrance viscosity for each particular materials (see Fig. 4). ) ( x
logðhENT Þ ¼ log
hENT;0
a
1 þ ðlg_ APP Þ
tanhðag_ APP þ 1Þ tanhð1Þ
(5)
where hENT;0 represents the plateau-value of entrance viscosity, and l, a, a, x are adjustable parameters. Secondly, Cogswell, Binding and Gibson models were utilized to calculate uniaxial extensional viscosity according to equations provided in [9]. All data needed for such calculation were taken from the numerical simulations. Moreover, effective entry length correction proposed in [9] was used for all considered models. Comparisons between the calculated extensional and theoretical viscosities are given in Fig. 5a–e. It is shown that all the entrance techniques predict the same extensional viscosity plateau at low shear rates due to the applied effective entry length correction. The comparison between theoretical and calculated extensional viscosities with the applied correction for extensional strain hardening polymer melts (see Fig. 5a–d) leads to the following conclusions:
M. Zatloukal, J. Musil / Polymer Testing 28 (2009) 843–853
10 6
Apparent entrance viscosity (Pa.s)
LDPE Escorene LD 165 BW1 at 200°C and virtual materials M1,M2,M3 and M4 at 200°C
10 5
10 4
10 3 10 -4
LDPE Escorene LD 165 BW1 Virtual material M1 Virtual material M2 Virtual material M3 Virtual material M4 Entrance pressure drop model fit
10 -3
10 -2
10 -1
10 0
10 1
10 2
10 3
Apparent shear rate (s-1) Fig. 4. Apparent entrance viscosity curves determined by viscoelastic FEM calculations for ‘‘ideal orifice die’’ depicted in Fig. 3a).
All methods seem to capture the same and proper slope of the extensional viscosity at high extensional rates. The Binding model over-predicts and under-predicts hE at low and high 3_ , respectively. The Gibson model significantly under-predicts hE at both, low and high 3_ . The Cogswell model under-predicts hE for highly extensional strain hardening melts, i. e. if the maximum uniaxial extensional viscosity divided by 3 times Newtonian viscosity, hEmax/(3h0), is higher than 2. On the other hand, for low extensional strain hardening materials where hEmax/(3h0) 2, the Cogswell model predicts hE very precisely within a wide range of extensional strain rates. Fig. 5e compares theoretical and calculated extensional viscosities with the applied correction for extensional strain thinning polymer melt. It is nicely visible that both Binding and Gibson models provide reasonably good agreement between calculated and predicted hE. On the other hand, the Cogswell model tends to over-predict the uniaxial extensional viscosity, especially at high extensional strain rates. 2.2. Theoretical effect of die design on the measured extensional rheology In this part, the effect of the orifice die design on the measured entrance viscosity and consequent extensional
847
viscosity calculation has been investigated. Due to a quite narrow downstream orifice die region (see Fig. 6 for more detail for conventional orifice die) and extrudate swell, the polymer melt may fill this region. Therefore, if this happens, the question is how much this flow situation influences the measured entrance viscosity/uniaxial extensional viscosity. In order to answer this question, virtual material M2 (having medium level of extensional strain hardening) has been chosen for FEM analysis for the case considering that downstream orifice die region is filled by the polymer melt. The sketch of such a low situation for a conventional orifice die together with corresponding boundary conditions and FE mesh is provided in Fig. 7 (grid consists of 7400 nodes and 3553 elements). The problem has been solved for different flow rates and the theoretical results obtained were plotted in Fig. 8 and compared with corresponding theoretical data considering that the downstream orifice die region is not filled by the polymer melt. In this Figure, it is clearly visible that entrance viscosities for both cases differ significantly within a wide range of apparent shear rates. In addition, entrance viscosity parameters for this material are provided in Table 3. In more detail, the fact that the polymer melt fills the downstream region of the orifice die leads to significant entrance viscosity increase due to an additional shear flow component occurring in this region. Fig. 9 shows the impact of such artificial entrance viscosity increase on the uniaxial extensional viscosity determined by the corrected Cogswell model. Clearly, artificial entrance viscosity increase leads to artificial uniaxial extensional viscosity increase, leading to a higher level of disagreement between Cogswell model prediction and true uniaxial extensional viscosity. In the other words, the main conclusion from this theoretical study is that a very narrow downstream region of the used orifice die may lead to erroneous estimation of the entrance viscosity, leading to artificially high uniaxial extensional viscosity. 3. Experimental analysis 3.1. Novel orifice die design In order to prevent unwanted artificial entrance viscosity increase due to a very narrow downstream channel, we have proposed a novel (patent pending) orifice die design (see Fig. 10). The main advantage of this novel orifice die design is a very open downstream region which consists of highly diverging channel (Fig. 10a and b) or flat exit region (Fig. 10c–d) and four holes which enables the use of a special key to screw-up the orifice die to the rheometer barrel. This downstream orifice die geometry
Table 2 Entrance pressure drop model, Eq. (5), parameters for both, real (LDPE Escorene LD 165 BW1) and virtual (M1, M2, M3 and M4) materials. Material
LDPE Escorene LD 165BW 1 Virtual material M1 Virtual material M2 Virtual material M3 Virtual material M4
Entrance pressure drop model
hENT,0 (Pa.s)
l (s)
a ()
a (s)
x ()
59 550.14209 57 758.09353 55 776.31681 57 806.94014 61 906.54042
0.581123 0.782182 0.976292 1.497522 0.940443
0.6064556417 0.5951470027 0.5902599899 0.5749431565 0.6261065864
15.834387 17.037560 18.775613 16.035987 0.000000
0.299492 0.292668 0.276932 0.244389 0.000000
848
10
7
Uniaxial extensional viscosity (Pa.s)
a 10
6
10
5
b
LDPE Escorene LD 165 BW1 at 200°C
c
Virtual material M1 at 200°C
Virtual material M2 at 200°C
104
102 -4 10
10
-3
10
-2
-1
0
10 10 Extensional rate (s -1)
10
10
modified White-Metzner model prediction Corrected Cogswell model prediction Corrected Binding model prediction Corrected Gibson model prediction
1
10
10
-4
10
-3
10
-2
-1
0
10 10 Extensional rate (s -1)
10
1
10
2
10
-4
10
-3
10
-2
-1
0
10 10 Extensional rate (s -1)
10
1
10
2
10
3
7
d
Uniaxial extensional viscosity (Pa.s)
2
modified White-Metzner model prediction Corrected Cogswell model prediction Corrected Binding model prediction Corrected Gibson model prediction
10
6
10
5
10
4
10
3
10
2
10
e
Virtual material M3 at 200°C
Virtual material M4 at 200°C
modified White-Metzner model prediction Corrected Cogswell model prediction Corrected Binding model prediction Corrected Gibson model prediction
-4
10
-3
10
-2
-1
0
10 10 Extensional rate (s -1)
10
modified White-Metzner model prediction Corrected Cogswell model prediction Corrected Binding model prediction Corrected Gibson model prediction
1
10
2
10
-4
10
-3
10
-2
-1
0
10 10 Extensional rate (s -1)
10
1
10
2
10
3
Fig. 5. Comparison between modified White-Metzner model and Cogswell, Binding and Gibson extensional viscosity predictions for different polymer melts by using ideal orifice die depicted in Fig. 3a. Here, corrected Cogswell/Binding/Gibson model means that the effective entry length correction has been applied according to [9]. a) LDPE Escorene LD 165 BW1, b) Virtual material M1, c) Virtual material M2, d) Virtual material M3, e) Virtual material M4.
M. Zatloukal, J. Musil / Polymer Testing 28 (2009) 843–853
10
Uniaxial extensional viscosity (SER-HV-A01) modified White-Metzner model prediction Corrected Cogswell model prediction Corrected Binding model prediction Corrected Gibson model prediction
3
M. Zatloukal, J. Musil / Polymer Testing 28 (2009) 843–853
849
Fig. 6. Conventional orifice die. a) Section view, b) Bottom view.
Apparent entrance viscosity (Pa.s)
106
105
104
Conventional orifice die, filled downstream region Novel orifice die, unfilled downstream region
eliminates any possibility for artificial pressure increase due to polymer melt touching the downstream wall. Moreover, a detail view of the extrudate leaving the die being possible during the measurements allowing direct measurement of the free surface development during the extrudate swell, which is practically impossible if one uses the conventional orifice die design depicted in Fig. 6. Another expected advantage of this novel die design is a more precise detection of the melt rupture for rupture stress determination in comparison with conventional orifice design, where the melt rupture can be hidden if the downstream region is filled by the polymer melt. 3.2. Experimental evaluation of the novel orifice die design
50
O15
0.25
16
33
O2
Input Axis Output Wall
110° 40° O12
b
10-3
100
101
102
103
(s-1)
Table 3 Parameters of entrance pressure drop model, Eq. (5) for virtual material M2. Down Entrance pressure drop model stream a hENT,0 l die region (Pa.s) (s) () type Unfilled Filled
a
x
(s)
()
55 776.31681 0.976292 0.5902599899 18.775613 0.276932 70 727.18176 0.654474 0.6011542038 18.599694 0.279597
107
106
105
104
Uncorrected Cogswell model (unfilled downstream region) Corrected Cogswell model (unfilled downstream region) Corrected Cogswell model (filled downstream region) modified White-Metzner model prediction
103
102 10-4
Fig. 7. Details for theoretical conventional orifice die analysis where downstream region is filled by polymer melt. a) Geometrical sketch of conventional orifice die, b) Boundary conditions, c) FEM mesh.
10-1
Fig. 8. Theoretically predicted effect of die design (filled/unfilled orifice downstream region) on entrance viscosity (entrance pressure drop divided by apparent shear rate) for virtual polymer melt M2 at 200 C. Line represents entrance pressure drop model fit.
a
c
10-2
Apparent shear rate
Uniaxial extensional viscosity (Pa.s)
In order to evaluate the novel orifice die with respect to conventional orifice die, the entrance viscosity (entrance pressure drop divided by the apparent shear rate) has been measured for LDPE Lupolen 1840H at 250 C by using a Rosand RH7-2 twin bore capillary rheometer and both orifice dies. The results together with the closer view of the orifice downstream region during measurements are
103 10-4
10-3
10-2
10-1
100
Extensional rate (s -1)
101
102
103
Fig. 9. Comparison between modified White-Metzner model and Cogswell extensional viscosity predictions for virtual polymer melt M2 by using two different orifice dies. Here, corrected Cogswell model means that the effective entry length correction has been applied for the Cogswell model according to [9].
850
M. Zatloukal, J. Musil / Polymer Testing 28 (2009) 843–853
Fig. 10. Novel patent pending orifice dies. a) Abrupt entry (section view), b) Abrupt entry (bottom view), c) Convergent entry (section view), d) Convergent entry (bottom view).
3.3. Experimental analysis of Binding, Cogswell and Gibson models In this section, the novel orifice die will be utilized to determine uniaxial extensional viscosity by using Binding, Cogswell and Gibson models for highly branched (LDPE Lupolen 1840H), slightly branched (mLLDPE Exact 0201) and linear polymers (HDPE Tipelin FS 450 – 26). In the first step, entrance viscosity has been determined for all three samples by the use of the novel orifice die and
4000 LDPE Lupolen 1840H at 250°C
Apparent entrance viscosity (Pa.s)
provided in Fig. 11 and Fig. 12, respectively. As expected from the theoretical study, entrance viscosity measured by the conventional orifice die is much higher in comparison with the novel die design, due to the fact that polymer melt fills the downstream region of the conventional orifice die. Also for that reason, the level of the extrudate swell from the conventional orifice die is artificially high. This leads us to the conclusion that the measurements of the extensional rheology by using a conventional orifice die can be highly erroneous and the novel orifice die design should be preferred.
3000
2000
Conventional orifice die design (D = 1 mm) Novel orifice die design (D = 1 mm)
103 101
20
50
10 2
200
500
10 3
2000
-1
Apparent shear rate (s ) Fig. 11. Comparison between experimentally determined entrance viscosities (entrance pressure drop divided by apparent shear rate) by using conventional orifice die (green symbols) and novel orifice die with abrupt entry (red symbols) for LDPE Lupolen 1840H at 250 C.
M. Zatloukal, J. Musil / Polymer Testing 28 (2009) 843–853
Fig. 12. Closer view of the orifice downstream region during measurements depicted in Fig. 11. a) Conventional orifice die design, b) Novel patent pending orifice die design with abrupt entry.
a
b
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subsequently fitted by the entrance pressure drop model, Eq. (5) as depicted in Fig. 13a–c and with fitting parameters provided in Table 4. It is nicely visible that in all three cases, Eq. (5) represents the measured entrance viscosity data very well. In the second step, effective entry length correction has been applied for entrance viscosity data with respect to Binding, Cogswell and Gibson models. Finally, the extensional viscosities have been calculated for the all three samples according to all three techniques, and compared with SER steady state extensional viscosity data reported in the literature [4,11] as seen in Fig. 14a–i (the shear viscosity data needed for the uniaxial extensional viscosity calculation were measured on a Rosand RH7-2 twin – bore capillary rheometer; corresponding CarreauYasuda model fitting parameters are provided in Table 5). It also has to be mentioned that the dotted line occurring in Fig. 14a–i represents extensional viscosity calculated from the entrance viscosity fitting lines provided in Fig. 13a–c, i.e. the fitting lines were taken as the measurements in this case. Closer analysis of these Figures reveals the following conclusions. For highly branched LDPE Lupolen 1840H material where hEmax/(3h0) ¼ 3.9, Binding model predicts extensional viscosity in good agreement with SER extensional viscosity data at low extensional extensional rates, whereas at medium and higher deformation rates the calculated data are under-predicted (see Fig. 14a). Gibson model behaviour is even worse in this case, because calculated extensional viscosity is under-predicted within whole extensional rate range (see Fig. 14b). Finally, the Cogswell model gives good agreement between measured and calculated extensional viscosity at low and high extensional deformation rates but at medium deformation rates the model under-predicts the measured data (see Fig. 14c). Obviously, these observations are in good agreement with the conclusions obtained from theoretical analysis. Thus, we could conclude that for branched LDPE Lupolen 1840H polymer the Cogswell model predictions are more realistic in comparison with Binding and Gibson models. The incapability of the Cogswell model to describe extensional viscosity data at medium deformation rates can be
c
Fig. 13. Entrance viscosity (entrance pressure drop divided by apparent shear rate) measured by novel orifice die with abrupt entry. a) LDPE Lupolen 1840H at 150 C, b) mLLDPE Exact 0201 at 180 C, c) HDPE Tipelin FS 450–26 at 180 C.
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M. Zatloukal, J. Musil / Polymer Testing 28 (2009) 843–853
Table 4 Parameters of entrance pressure drop model, Eq. (5), for LDPE Lupolen 1840H, mLLDPE Exact 0201 and HDPE Tipelin FS 450-26. Material
Entrance pressure drop model
LDPE Lupolen 1840H mLLDPE Exact 0201 HDPE Tipelin FS 450 - 26
hENT,0 (Pa.s)
l (s)
a ()
a (s)
x ()
29 647.25498 19 010.49081 1000 000.00000
0.051010 0.420384 8.004018
0.6599534062 0.3817786899 0.5778474780
19.951935 6.012095 0.000000
0.256709 0.306068 0.000000
7
10
a
b
LDPE Lupolen 1840H at 150°C
c
LDPE Lupolen 1840H at 150°C
LDPE Lupolen 1840H at 150°C
Shear and uniaxial extensional viscosities (Pa.s)
106
5
10
104
103
2
10
101 10-4
10-3
10-2 10-1 100 101 Shear and extensional rates (s-1)
Shear viscosity (rotational rheometer) Shear viscosity (capillary rheometer) Extensional viscosity (uncorrected Cogswell-measurement) Extensional viscosity (uncorrected Cogswell-extrapolation) Extensional viscosity (corrected Cogswell-measurement) Extensional viscosity (corrected Cogswell-extrapolation) Uniaxial extensional viscosity (SER-HV-A01) Carreau-Yasuda model fit
Shear viscosity (rotational rheometer) Shear viscosity (capillary rheometer) Extensional viscosity (uncorrected Gibson-measurement) Extensional viscosity (uncorrected Gibson-extrapolation) Extensional viscosity (corrected Gibson-measurement) Extensional viscosity (corrected Gibson-extrapolation) Extensional viscosity (SER-HV-A01) Carreau-Yasuda model fit
Shear viscosity (rotational rheometer) Shear viscosity (capillary rheometer) Extensional viscosity (uncorrected Binding-measurement) Extensional viscosity (uncorrected Binding-extrapolation) Extensional viscosity (corrected Binding-measurement) Extensional viscosity (corrected Binding-extrapolation) Extensional viscosity (SER-HV-A01) Carreau-Yasuda model fit 102
10-4
10-3
10-2 10-1 100 101 Shear and extensional rates (s -1)
102
10-4
10-3
10-2 10-1 100 101 Shear and extensional rates (s -1)
102
103
6
10
Shear and uniaxial extensional viscosities (Pa.s)
d
e
mLLDPE Exact 0201 at 180°C
f
mLLDPE Exact 0201 at 180°C
mLLDPE Exact 0201 at 180°C
105
104
Shear viscosity (rotational rheometer) Shear viscosity (capillary rheometer) Extensional viscosity (uncorrected Binding-measurement) Extensional viscosity (uncorrected Binding-extrapolation) Extensional viscosity (corrected Binding-measurement) Extensional viscosity (corrected Binding-extrapolation) Extensional viscosity (SER-HV-A01) Carreau-Yasuda model fit
103
102 10-4
10-3
10-2 10-1 100 101 Shear and extensional rates (s-1)
Shear viscosity (rotational rheometer) Shear viscosity (capillary rheometer) Extensional viscosity (uncorrected Gibson-measurement) Extensional viscosity (uncorrected Gibson-extrapolation) Extensional viscosity (corrected Gibson-measurement) Extensional viscosity (corrected Gibson-extrapolation) Extensional viscosity (SER-HV-A01) Carreau-Yasuda model fit
102
10-4
10-3
10-2 10-1 100 101 Shear and extensional rates (s -1)
Shear viscosity (rotational rheometer) Shear viscosity (capillary rheometer) Extensional viscosity (uncorrected Cogswell-measurement) Extensional viscosity (uncorrected Cogswell-extrapolation) Extensional viscosity (corrected Cogswell-measurement) Extensional viscosity (corrected Cogswell-extrapolation) Extensional viscosity (SER-HV-A01) Carreau-Yasuda model fit
102
10-4
10-3
10-2 10-1 100 101 Shear and extensional rates (s -1)
102
103
8
10
g
h
HDPE Tipelin FS 450-26 at 180°C
i
HDPE Tipelin FS 450-26 at 180°C
HDPE Tipelin FS 450-26 at 180°C
7
Shear and uniaxial extensional viscosities (Pa.s)
10
6
10
5
10
104
3
10
2
10
101 -4 10
Shear viscosity (rotational rheometer) Shear viscosity (capillary rheometer) Extensional viscosity (uncorrected Binding-measurement) Extensional viscosity (corrected Binding-measurement) Extensional viscosity (SER-HV-A01) Carreau-Yasuda model fit
10
-3
-2
10
-1
0
10 10 10 Shear and extensional rates (s-1)
1
10
Shear viscosity (rotational rheometer) Shear viscosity (capillary rheometer) Extensional viscosity (uncorrected Cogswell-measurement) Extensional viscosity (corrected Cogswell-measurement) Extensional viscosity (SER-HV-A01) Carreau-Yasuda model fit
Shear viscosity (rotational rheometer) Shear viscosity (capillary rheometer) Extensional viscosity (uncorrected Gibson-measurement) Extensional viscosity (corrected Gibson-measurement) Extensional viscosity (SER-HV-A01) Carreau-Yasuda model fit
2
10
-4
-3
10
10
-2
-1
0
1
10 10 10 Shear and extensional rates (s -1)
10
2
-4
10
-3
10
10
-2
-1
0
1
10 10 10 Shear and extensional rates (s -1)
2
10
3
10
Fig. 14. Comparison between Binding/Gibson/Cogswell extensional viscosity data and SER measurements for different polyolefines taken from [11,13] (shear viscosity data obtained from rotational and capillary rheometer are also provided in this Figure). Here, corrected Binding/Gibson/Cogswell model means that the effective entry length correction has been applied according to [9]. a-c) LDPE Lupolen 1840H at 150 C, d-f) mLLDPE Exact 0201 at 180 C, g-i) HDPE Tipelin FS 450–26 at 180 C.
M. Zatloukal, J. Musil / Polymer Testing 28 (2009) 843–853 Table 5 Fitting parameters of Carreau – Yasuda model, Eq. (2), for LDPE Lupolen 1840H, mLLDPE Exact 0201 and HDPE Tipelin FS 450-26. Material
Carreau – Yasuda model
h0 (Pa.s)
K1 (s)
a (-)
n (-)
LDPE Lupolen 1840H mLLDPE Exact 0201 HDPE Tipelin FS 450 – 26
59 950 17 998 5 033 405
0.5834 5.6062 3.8745
0.3282 1.3639 0.1401
0.1000 0.6029 0.0653
explained by the fact that the extensional strain hardening is too high in this case i.e. that hEmax/(3h0) > 2, which is consistent with the findings based on theoretical analysis. For slightly branched mLLDPE Exact 0201 material where hEmax/(3h0) ¼ 1.6, Binding model over-predicts extensional viscosity at low extensional rates, whereas at medium and high deformation rates the extensional viscosity is under-predicted (see Fig. 14d). Gibson model predicts extensional viscosity in good agreement with the measured SER data at low extensional rates, but at medium and high deformation rates, the extensional viscosity is under-predicted (see Fig. 14e). Cogswell model predictions are surprisingly in excellent agreement with the measured SER extensional viscosity data as seen in Fig. 14f. For slightly branched mLLDPE Exact 0201 polymer, all above mentioned conclusions are in good agreement with the findings based on theoretical analysis for all three tested model. Here, it should be pointed out that the extensional strain hardening parameter for mLLDPE Exact 0201 is hEmax/(3h0) < 2 i.e. it is in the range where Cogswell model predictions are very precise as shown in the theoretical analysis. For linear HDPE Tipelin FS 450 – 26 where hEmax/ (3h0) ¼ 1, both, Binding and Gibson models gives excellent agreement with the measured SER extensional viscosity data within the experimentally determined extensional rates range (see Fig. 14g–h). On the other hand, the Cogswell model failed in this case because the model overpredicts SER extensional viscosity data (see Fig. 14i) within the whole deformation rate range. Again, all above mentioned conclusions for the linear HDPE Tipelin FS 450 – 26 polymer are in very good agreement with corresponding conclusions obtained during theoretical analysis. 4. Conclusion Novel patent pending orifice die design has been proposed and tested for the entrance pressure drop measurements. It has been theoretically and experi-
853
mentally demonstrated that the novel orifice die design allows more precise entrance pressure drop measurement (and thus more precise extensional viscosity determination) in comparison with standard and conventional orifice die. Based on theoretical and experimental analysis, it has been revealed that by using novel orifice die design corrected Cogswell model is more precise in uniaxial extensional viscosity prediction than Binding and Gibson models for highly branched and slightly branched polymer melts. In more detail, it has been demonstrated that corrected Cogswell model provides excellent capability correctly calculate extensional viscosity, especially, if the extensional strain hardening level given by hEmax/(3h0) ratio is lower than 2. On the other hand, for linear polymer melts, the situation has been found to be the opposite, i.e. that Binding and Gibson models were found to provide excellent capability to predict extensional viscosity data from the entrance pressure drop measurements, whereas the Cogswell model extensional viscosity predictions were found to be poor (predicted extensional viscosity data were always lower in comparison with SER measurements).
Acknowledgements The support of the projects by the Ministry of Education CR (KONTAKT ME08090, MSM 7088352101) is gratefully acknowledged. References [1] F.A. Morrison, Understanding Rheology, Oxford University Press, Inc., New York, USA, 2001. [2] H. Mu¨nstedt, J. Rheol. 23 (1979) 421–436. [3] M.L. Sentmanat, Rheol. Acta 43 (2004) 657–669. [4] M.L. Sentmanat, B.N. Wang, G.H. McKinley, J. Rheol. 49 (3) (2005) 585–606. [5] F.N. Cogswell, Polym. Eng. Sci. 12 (1972) 64–73. [6] D.M. Binding, J. Non-Newtonian Fluid Mech. 27 (1988) 173–189. [7] A.G. Gibson, Composites 20 (1989) 57–64. [8] S. Kim, J.M. Dealy, J. Rheol. 45 (6) (2001) 1413–1419. [9] M. Zatloukal, J. Vlcek, C. Tzoganakis, P. Saha, J. Non-Newtonian Fluid Mech. 107 (2002) 13–37. [10] H.A. Barnes, G.P. Roberts, J. Non-Newtonial Fluid Mech. 44 (1992) 113. [11] R. Pivokonsky, M. Zatloukal, P. Filip, J. Non-Newtonial Fluid Mech. 135 (1) (2006) 58. [12] A.D. Gotsis, A. Odriozola, Rheol. Acta 37 (1998) 430. [13] R. Pivokonsky, M. Zatloukal, P. Filip, J. Non-Newtonial Fluid Mech. 150 (1) (2008) 56.