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A study of wall slip in the capillary flow of a filled rubber compound
Polymer Testing 37 (2014) 45–50
Contents lists available at ScienceDirect
Polymer Testing journal homepage: www.elsevier.com/locate/polytest
Test method
A study of wall slip in the capillary flow of a filled rubber compound Chuan Yang, Ziran Li* Department of Modern Mechanics, CAS Key Laboratory of Mechanical Behavior and Design of Materials, University of Science and Technology of China, Hefei 230027, China
a r t i c l e i n f o
a b s t r a c t
Article history: Received 13 March 2014 Accepted 21 April 2014
The rheological properties of a filler-reinforced rubber compound were investigated by a dynamic shear rheometer and a rate-controlled capillary rheometer with two different shear modes: oscillatory shear and steady shear. Based on the experiments, we found a deviation between the complex viscosities and the steady shear viscosities, indicating an apparent failure of the Cox-Merz rule, due to the presence of wall slip in the capillary flow. A finite element simulation was utilized to analyze the wall slip in the capillary tests. Numerical results revealed that the capillary rheometer tended to underestimate the shear viscosity in the presence of wall slip, and an explanation to the experimental finding was proposed. Ó 2014 Elsevier Ltd. All rights reserved.
Keywords: Wall slip Filled rubber compound Capillary rheometer Finite element The Cox-Merz rule
1. Introduction Rubber compounds are used in a wide range of applications, the most important being the manufacturing of automotive tires. Since the blends are processed in a molten state before the curing step, rheological properties determine not only their processing behavior but also the performance and appearance of final products. In order to analyze and optimize the production processes, rheological measurements on these polymer systems is of great importance while full of challenge. However, rubber compounds with reinforcing fillers, such as carbon black and silica particles, exhibit highly non-linear rheological properties compared to unfilled elastomers: i.e., the disappearance of Newtonian viscosity plateau [1] and the reversible decrease of the modulus versus strain (so-called “Payne effect”) [2]. The origin of these behaviors is assigned to the rubber-filler interactions and the filler-filler networking. Moreover, high filler content tends to promote slippage at
* Corresponding author. E-mail address: [email protected] (Z. Li). http://dx.doi.org/10.1016/j.polymertesting.2014.04.009 0142-9418/Ó 2014 Elsevier Ltd. All rights reserved.
the wall of laboratory or processing equipment [3]. For shear rates below 1 103 1/s, oscillatory shear flow is commonly used to obtain viscosity data since the steady shear mode of a rotational rheometer is more susceptible to wall slip. The Cox-Merz rule h ðuÞ ¼ hðg_ Þ [4] is applied to convert complex viscosity (h*) as a function of oscillating frequency (u) into shear viscosity (h) as a function of shear rate (g_ ). On the other hand, rubber processing often involves shear rates up to 1 103 1/s, such as the extrusion of tire treads, and a capillary rheometer becomes the only choice for rheological measurements over such a high shear rate range. Occasionally, the viscosity data determined from oscillatory tests and capillary tests cannot be superimposed, suggesting the Cox-Merz rule is not applicable [5]. Ansari et al. [6] pointed out that the slip at the capillary wall was one of the primary reasons for the apparent failure of CoxMerz rule. For some polymer melts, once the capillary data were corrected for slip effects, applicability of the Cox–Merz rule was still validated. Nevertheless, it is difficult to quantify slip effects and, when applied to filled rubber compound, the well-known Mooney method [7] often leads to unphysical results such as a negative wall slip rate and a slip velocity higher than the mean velocity of the flow field
C. Yang, Z. Li / Polymer Testing 37 (2014) 45–50
Table 1 Formulation of the rubber compound. Ingredient
Amount (phr)
Natural rubber (NR) STR10/SIR10 Styrene butadiene rubber (SBR) 1502 Cis-1, 4-polybutadiene rubber (BR) Carbon black (CB) N234 Silica (Si) Z195Gr
75 10 15 40 15
[8,9]. Another alternative, finite element computation, provides a useful tool to analyze the influence of wall slip or other factors on rheological measurements [10–12]. However, to the author’s knowledge, none of these numerical works has ever dealt with filled rubber compounds because of the difficulties inherent to this class of polymers and the limited number of relevant experimental data. In this paper, rheological properties of a typical filled rubber compound were studied with an oscillatory shear rheometer and a rate-controlled capillary rheometer. A numerical model of the capillary rheometer was also built to reproduce the testing procedure and to explain the failure of Cox-Merz rule by introducing a slip boundary condition into the simulation. 2. Experimental 2.1. Materials The rubber compound studied in this paper was provided by Giti Tire Ltd. The detailed formulation is listed in Table 1. The base elastomer was a combination of natural rubber (NR), styrene butadiene rubber (SBR) and cis-1, 4polybutadiene rubber (BR). Carbon black (CB) N234 and silica (Si) Z195Gr were used as reinforcing fillers. The total filler content was as high as 55 parts per hundred parts of rubber (phr). Mixing was performed by means of an internal mixer.
rheometer specially designed for rubber-like viscoelastic materials. Owing to its grooved biconical dies and pressurized testing cavity, wall slip could be neglected during the tests. Frequency sweep tests were performed over a wide range of oscillatory frequency from 1.5 CPM to 1500 CPM (0.157 rad/s to 157 rad/s) at two different temperatures (100 C and 120 C) and a uniform strain of 7%, according to ASTM D6204 [13]. Then, a rate-controlled capillary rheometer Rosand RH2200 (manufactured by Malvern Instruments Ltd., UK) was used to perform steady rheological tests. The apparatus had two bores, one of which was fitted with a capillary die having a length of 24 mm and a diameter of 1.5 mm (L/D ¼ 16/1) while the other, for the sake of Bagley correction, was fitted with an orifice or zero-length die having the same diameter but a length of 0.25 mm. Testing temperature was 100 C and apparent shear rate ranged from 100 1/s to 1000 1/s. Bagley and Rabinowitsch corrections were applied to correct entrance/exit pressure loss and shear-thinning effects, respectively. 2.3. Results and discussion Fig. 1 depicts the master curves of viscoelastic properties obtained from the RPA2000 at a temperature of 100 C. We observe that the molten rubber compound exhibits
a
1000000
Complex viscosity (RPA2000) Shear viscosity (RH2200) 100000
Viscosity (Pa.s)
46
10000
1000
100
2.2. Rheological testing 10
Rheological properties were firstly measured with a Rubber Process Analyzer RPA2000 (manufactured by Alpha Technologies, USA), which is a rotorless oscillatory shear
0.1
1
10
100
1000
Shear rate (1/s) or frequency (rad/s)
b
1000000
Complex viscosity (RPA2000) Shear viscosity (RH2200, with a vertical shift) 100000
100000
100000
10000
Modulus (Pa)
Complex viscosity (Pa.s)
Complex viscosity Storage modulus Loss modulus
Viscosity (Pa.s)
1000000
1000000
10000
1000
100
10 0.1
10000
1000 0.1
1
10
1
10
100
1000
Shear rate (1/s) or frequency (rad/s)
100
Frequency (rad/s)
Fig. 1. Complex viscosity, storage modulus and loss modulus as a function of frequency at 100 C.
Fig. 2. Complex viscosity and shear viscosity as a function of frequency or shear rate at 100 C. (a) Original data showing the existence of inconsistency between the two types of rheological measurements; (b) A vertical shift is applied to the capillary data.
C. Yang, Z. Li / Polymer Testing 37 (2014) 45–50
typical shear-thinning behavior and a Newtonian plateau is hard to capture. In Fig. 2(a), the curve of shear viscosity from RH2200 is compared with the curve of complex viscosity from the RPA2000 with the aid of the Cox-Merz rule at 100 C. Data obtained with the two rheometers are inconsistent (the RPA2000 result is higher than that of the RH2200), while a vertical shift applied to capillary data can eliminate this inconsistency since the slopes of two curves are almost the same on the log-log plot, as shown in Fig. 2(b). It is worthy pointing out that the Cox-Merz rule is an important empirical relationship which has been verified experimentally for a number of materials in the past six decades. On the other hand, the apparent failure of the Cox-Merz rule might be caused by various factors in the experiments, such as wall slip, viscous heating and pressure dependence of viscosity. Referring to Fig. 3, the complex viscosity curves at 100 C and 120 C are indistinguishable from each other, indicating that the rubber compound we studied is not sensitive to small variation of temperatures above 100 C and viscous heating should have a negligible impact on the capillary tests. Also, according to Kazatchkov et al. [14], the dependency of viscosity on pressure tends to cause h*(u) to be lower than hðg_ Þ, which is the opposite of the current situation (h ðuÞ > hðg_ Þ, see Fig. 2(a)). Thus, wall slippage in the capillary rheometer (slippage is negligible in RPA2000, as discussed above) should be the main reason of the inconsistency and its influence on testing results are examined numerically in the present work. 3. Numerical 3.1. Numerical model An axisymmetric finite element model of the twin bore rate-controlled rheometer Rosand RH2200 was built, as depicted in Fig. 4, and all geometrical dimensions were set in accordance with the experimental study. Contraction flows from the reservoir to the capillary as well as extrudate swell are taken into account so that entrance and exit pressure drops can be determined. Mesh of the left bore consists of 6050 quadrilateral elements and the right
47
consists of 5260. Mesh refinement has been checked to guarantee that the finite element solution is meshindependent. Flow in the computational domain is governed by equations of continuity and momentum, written as
V$v ¼ 0
(1)
Vp þ V$s ¼ 0
(2)
where v is the velocity vector, p is the pressure and s is the stress tensor. Inertia items and gravity are neglected due to the high viscosity of the material. A 5-mode Phan-Thien-Tanner (PTT) model [15] was used to describe the viscoelastic behavior of the molten rubber compound. Purely viscous models, e.g. the BirdCarreau model and the Cross model, are excluded from the present study as they tend to severely underestimate the extrusion pressure and die swell ratio. The PTT model is written as
8 5 P > > s¼ sðiÞ > > < i¼1 " # hε l > xi VðiÞ xi DðiÞ > i i ðiÞ ðiÞ > > exp s þ li 1 s þ s tr s ¼ 2hi D : hi 2 2 (3) where the rate of deformation tensor is given by D ¼ 1/ 2(L þ LT), with the velocity gradient tensor L ¼ (Vv)T. li, hi, εi and xi are material parameters. Dynamic data from the RPA2000 at 100 C, referring to Fig. 1, are used to determine these parameters and the Cox-Merz rule is applied to convert complex viscosity into shear viscosity. The PTT model fits the data well over the range of the dynamic tests, as presented in Fig. 5 and Table 2.
a
b Piston (inflow) Axis of symmetry Reservoir
1000000 o
Complex viscosity (Pa.s)
100 C o 120 C 100000
Reservoir wall Capillary die Capillary wall
10000
Extrudate 1000 0.1
1
10
100
Free surface Downstream
Frequency (rad/s)
Fig. 3. Comparison of complex viscosity at two testing temperatures of 100 C and 120 C.
Fig. 4. Finite element model of the twin bore capillary rheometer used in this study. (a) Left bore with a capillary die (L ¼ 16 mm, D ¼ 1.5 mm); (b) Right bore with a zero-length die (L ¼ 0.25 mm, D ¼ 1.5 mm).
48
C. Yang, Z. Li / Polymer Testing 37 (2014) 45–50
A uniform normal velocity profile is imposed at the inflow boundary as the piston speed vp. With the diameter of the reservoir Dr (15 mm for the rheometer used in this work), volumetric flow rate Q can be calculated as
Q ¼
pD2r vp
(4)
4
then, for Newtonian fluids, shear rate at the capillary wall can be obtained and treated as the apparent wall shear rate for non-Newtonian fluids. That is
32Q g_ a ¼ pD3
(5)
where D is the diameter of the capillary die. A no-slip boundary condition is applied at the reservoir wall. At the capillary wall, a Navier’s slip condition is enforced, which can be written as
vs ¼
sw
(6)
k
where vs is the slip velocity, sw is the shear stress at the capillary wall and k is the slip coefficient. If k ¼ N, Eq. (6) reduces to the no-slip condition while with k ¼ 0, Eq. (6) captures the full-slip condition. Other slip models are able to be adopted into the simulation as well, but detailed comparison between different slip conditions is beyond the scope of this paper. In the absence of surface tension, a boundary condition of zero-stress is applied at the free surface and downstream of the extrudate. The position of the free surface is computed as part of the solution and iteration stops when normal velocity on the surface is vanishingly small. Numerical analysis of the above model was carried out using a finite element software package POLYFLOWÒ [16]. 3.2. Results and discussion
Table 2 Material parameters of the 5-mode Phan-Thien-Tanner model. Mode no.
li (s)
hi (Pa.s)
εi
xi
1 2 3 4 5
0.001 0.010 0.110 1.147 12.00
607.3481 2556.049 18282.04 90839.74 727133.4
0.011 0.011 0.026 0.011 0.038
0.227 0.080 0.074 0.070 0.131
shown in Fig. 6 at the apparent shear rate (Eq. (5)) of 1.067 1/s. Higher shear rates or piston speeds were not attempted for the limitation of the well-known high Weissenberg number problem (HWNP), a numerical break down that occurs when the Weissenberg number (Wi) exceeds a critical value. Fortunately, the viscosity curve of the rubber compound is approximately a straight line on a log-log plot for shear rates above 0.1 1/s. Wall slippage and its effect on capillary measurements at high shear rate range can be investigated via extrapolation. As expected, pressure drops in the entrance and exit regions of the capillary die can be found from the pressure-distance plot in Fig. 6. The sum of these pressure losses is determined by the extrusion pressure of the zero-length die Dp0 . Hence, by subtracting Dp0 from the extrusion pressure of the capillary die Dp, the pressure gradient at the capillary die is given by
Dp Dp0 vp ¼ vz L
where z is the downstream direction and L is the capillary length. It should be noted that in the experiments Dp and Dp0 are read directly from the pressure sensors installed at the reservoir wall. Corrected shear stress at the capillary wall can be calculated as
sc ¼
Models with a no-slip condition applied at the capillary wall were analyzed first to check the reliability of the finite element computation. A typical pressure distribution is
(7)
vp D $ vz 4
(8)
where D is the capillary diameter. The end correction method used in this study is different from the traditional Bagley plot, which is indirect and time-consuming. With the corrected wall shear stress sc (Eq. (8)) and the
Viscosity (Pa.s) or modulus (Pa)
1E7
1000000
100000
10000 Shear viscosity (PTT) Storage modulus (PTT) Loss modulus (PTT) Complex viscosity (RPA2000) Storage modulus (RPA2000) Loss modulus (RPA2000)
1000
100 0.01
0.1
1
10
100
1000
Shear rate (1/s) or frequency (rad/s)
Fig. 5. Viscoelastic properties of the rubber compound at 100 C. Lines represent the prediction of the 5-mode Phan-Thien-Tanner model with parameters listed in Table 1 while symbols are experimental data from RPA2000.
Fig. 6. Pressure distribution for the rubber compound at the apparent shear rate of 1.067 1/s with a no-slip condition at the capillary wall.
C. Yang, Z. Li / Polymer Testing 37 (2014) 45–50
apparent wall shear rate g_ a (Eq. (5)) deduced from piston speed, apparent shear viscosity can be obtained as
ha ¼
sc g_ a
(9)
A shear rate sweep was simulated around the apparent shear rate of 1.0 1/s. As revealed by the no-slip results in Fig. 7, the apparent shear viscosity is higher than the shear viscosity predicted by the PTT model or, in other words, the real shear viscosity of the material in numerical experiments. This is because the apparent shear rates are based on Newtonian fluids whose velocity profile is parabolic in the fully-developed region, and shear thinning fluids show a different profile close to plug flow with higher shear rates at the wall. The Rabinowitsch correction is applied to correct the shear thinning effect. A corrected wall shear rate is given by the Rabinowitsch-Mooney equation [17]:
g_ dlng_ a g_ c ¼ a þ3 4 dln sc
(10)
sc g_ c
(11)
Slip condition (Eq. (6)) at the capillary wall was then introduced to the simulation with two different slip coefficients k ¼ 1 109 kg m2 s1 and k ¼ 1 1010 kg m2 s1 representing different levels of slippage. As it is clear from Fig. 7, slip at the capillary wall will make the apparent shear viscosity and corrected shear viscosity lower than the no slip ones. Specifically, the percentage drop in apparent shear viscosity is 3% for k ¼ 1 1010 kg m2 s1 and 27% for k ¼ 1 109 kg m2 s1. In fact, for lower k or higher level of slippage, the underestimation of shear viscosity will be more severe. This confirms that when slippage becomes significant, a capillary rheometer is no longer a reliable device to obtain shear viscosity data unless the slippage can be measured and corrected. However, wall slip correction is
180000 170000
Shear viscosity (PTT) Apparent shear viscosity (no slip) Corrected shear viscosity (no slip) Apparent shear viscosity (k = 1E10) Corrected shear viscosity (k = 1E10) Apparent shear viscosity (k = 1E9) Corrected shear viscosity (k = 1E9)
160000 150000
Shear viscosity (Pa.s)
140000 130000 120000 110000 100000 90000
Increasing slippage 80000
70000 0.7
0.8
0.9
difficult for filled rubber compound since the frequently used Mooney method is not applicable, as discussed before. Meanwhile, in the presence of slippage, the Rabinowitsch correction fails to obtain the true shear rate at the capillary wall, and sometimes the corrected shear viscosity deviates further from PTT prediction than the apparent shear viscosity (see results of k ¼ 1 1010 kg m2 s1 in Fig. 7). It should also be noted that the underestimation of viscosity caused by slip boundary condition can always be corrected via a vertical shift, identical to what we did in Fig. 3(b), indicating that the failure of the Cox-Merz rule in Fig. 3(a) could be (at least partially) caused by wall slip in the capillary tests. For this reason, our future work will be focused on measuring the slippage of filled rubber compound and characterizing the quantified level of slipping with appropriate boundary conditions in finite element computation. 4. Conclusions
and this leads to a corrected shear viscosity fitting the PTT model well, that is
hc ¼
49
1
1.1
1.2
1.3
1.4
1.5 1.6 1.7 1.8
Shear rate (1/s) Fig. 7. Apparent and corrected shear viscosity results obtain from numerical shear rate sweeps with a no-slip condition or a Navier’s slip condition at the capillary wall, compared with the shear viscosity predicted by the PTT model.
In this study, the rheological properties of a filled rubber compound were measured and analyzed by using a dynamic shear rheometer and a capillary rheometer. Comparing the viscosity data obtained from oscillatory shear and steady shear, an apparent failure of the Cox-Merz rule is found, that is, the complex viscosity is higher than the shear viscosity. With the use of a finite element model of the twin bore capillary rheometer, the capillary tests are reproduced and the wall slip at the capillary wall is investigated numerically. Results reveal that the underestimation of shear viscosity becomes significant with increasing level of slippage at the capillary wall, and this could be one of the reasons why the Cox-Merz rule is not applicable. All this will help understand the rheological behavior of filled rubber compounds and be a reminder of the possible error of shear viscosity of similar filler-reinforced elastomers in capillary testing. Acknowledgements This work was funded by the Fundamental Research Funds for the Central Universities (No. WK2090050018) in China. We are grateful to Giti Tire Ltd for supplying the rubber compound used in this study. References [1] J.L. Leblanc, Rubber–filler interactions and rheological properties in filled compounds, Progress in Polymer Science 27 (4) (2002) 627– 687. [2] A. Mongruel, M. Cartault, Nonlinear rheology of styrene-butadiene rubber filled with carbon-black or silica particles, Journal of Rheology 50 (2) (2006) 115–135. [3] D.M. Turner, M.D. Moore, Contribution of wall slip in the flow of rubber, Plastics and Rubber: Processing 5 (1980) 81–84. [4] W.P. Cox, E.H. Merz, Correlation of dynamic and steady flow viscosities, Journal of Polymer Science 28 (118) (1958) 619–622. [5] N. Phewthongin, P. Saeoui, C. Sirisinha, A study of rheological properties in sulfur-vulcanized CPE/NR blends, Polymer Testing 24 (2) (2005) 227–233. [6] M. Ansari, S.G. Hatzikiriakos, A.M. Sukhadia, et al., Rheology of Ziegler–Natta and metallocene high-density polyethylenes: broad molecular weight distribution effects, Rheologica Acta 50 (1) (2011) 17–27. [7] M. Mooney, Explicit formulas for slip and fluidity, Journal of Rheology 2 (2) (1931) 210–222.
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[8] J.L. Leblanc, T. Navarat, Assessing the wall slip rate of complex polymer systems with the Mooney method, in: Proceedings of the 13th International Congress on Rheology, Cambridge, UK, 2000. [9] P. Mourniac, J.F. Agassant, B. Vergnes, Determination of the wall slip velocity in the flow of a SBR compound, Rheologica Acta 31 (6) (1992) 565–574. [10] S. Syrjälä, J. Aho, Capillary rheometry of polymer meltsdsimulation and experiment, Korea-Australia Rheology Journal 24 (3) (2012) 241–247. [11] Y.L. Yeow, Y.K. Leong, A. Khan, Non-Newtonian flow in parallel-disk viscometers in the presence of wall slip, Journal of Non-Newtonian Fluid Mechanics 139 (1) (2006) 85–92. [12] H.M. Laun, Capillary rheometry for polymer melts revisited, Rheologica Acta 43 (5) (2004) 509–528.
[13] ASTM Standard D6204-12, Standard Test Method for RubberdMeasurement of Unvulcanized Rheological Properties Using Rotorless Shear Rheometers, ASTM International, West Conshohocken, PA, USA, 2012. http://www.astm.org/Standards/D6204. htm. [14] I.B. Kazatchkov, S.G. Hatzikiriakos, C.W. Stewart, Extrudate distortion in the capillary/slit extrusion of a molten polypropylene, Journal of Plastic Film and Sheeting 11 (1) (1995) 38–57. [15] N. Phan-Thien, A nonlinear network viscoelastic model, Journal of Rheology 22 (3) (1978) 259–283. [16] ANSYS POLYFLOW, User’s Guide, Release 14.0, ANSYS Inc., Canonsburg, PA, USA, 2011. [17] R.I. Tanner, Engineering Rheology, second ed., University Press, Oxford, 2000.
Contents lists available at ScienceDirect
Polymer Testing journal homepage: www.elsevier.com/locate/polytest
Test method
A study of wall slip in the capillary flow of a filled rubber compound Chuan Yang, Ziran Li* Department of Modern Mechanics, CAS Key Laboratory of Mechanical Behavior and Design of Materials, University of Science and Technology of China, Hefei 230027, China
a r t i c l e i n f o
a b s t r a c t
Article history: Received 13 March 2014 Accepted 21 April 2014
The rheological properties of a filler-reinforced rubber compound were investigated by a dynamic shear rheometer and a rate-controlled capillary rheometer with two different shear modes: oscillatory shear and steady shear. Based on the experiments, we found a deviation between the complex viscosities and the steady shear viscosities, indicating an apparent failure of the Cox-Merz rule, due to the presence of wall slip in the capillary flow. A finite element simulation was utilized to analyze the wall slip in the capillary tests. Numerical results revealed that the capillary rheometer tended to underestimate the shear viscosity in the presence of wall slip, and an explanation to the experimental finding was proposed. Ó 2014 Elsevier Ltd. All rights reserved.
Keywords: Wall slip Filled rubber compound Capillary rheometer Finite element The Cox-Merz rule
1. Introduction Rubber compounds are used in a wide range of applications, the most important being the manufacturing of automotive tires. Since the blends are processed in a molten state before the curing step, rheological properties determine not only their processing behavior but also the performance and appearance of final products. In order to analyze and optimize the production processes, rheological measurements on these polymer systems is of great importance while full of challenge. However, rubber compounds with reinforcing fillers, such as carbon black and silica particles, exhibit highly non-linear rheological properties compared to unfilled elastomers: i.e., the disappearance of Newtonian viscosity plateau [1] and the reversible decrease of the modulus versus strain (so-called “Payne effect”) [2]. The origin of these behaviors is assigned to the rubber-filler interactions and the filler-filler networking. Moreover, high filler content tends to promote slippage at
* Corresponding author. E-mail address: [email protected] (Z. Li). http://dx.doi.org/10.1016/j.polymertesting.2014.04.009 0142-9418/Ó 2014 Elsevier Ltd. All rights reserved.
the wall of laboratory or processing equipment [3]. For shear rates below 1 103 1/s, oscillatory shear flow is commonly used to obtain viscosity data since the steady shear mode of a rotational rheometer is more susceptible to wall slip. The Cox-Merz rule h ðuÞ ¼ hðg_ Þ [4] is applied to convert complex viscosity (h*) as a function of oscillating frequency (u) into shear viscosity (h) as a function of shear rate (g_ ). On the other hand, rubber processing often involves shear rates up to 1 103 1/s, such as the extrusion of tire treads, and a capillary rheometer becomes the only choice for rheological measurements over such a high shear rate range. Occasionally, the viscosity data determined from oscillatory tests and capillary tests cannot be superimposed, suggesting the Cox-Merz rule is not applicable [5]. Ansari et al. [6] pointed out that the slip at the capillary wall was one of the primary reasons for the apparent failure of CoxMerz rule. For some polymer melts, once the capillary data were corrected for slip effects, applicability of the Cox–Merz rule was still validated. Nevertheless, it is difficult to quantify slip effects and, when applied to filled rubber compound, the well-known Mooney method [7] often leads to unphysical results such as a negative wall slip rate and a slip velocity higher than the mean velocity of the flow field
C. Yang, Z. Li / Polymer Testing 37 (2014) 45–50
Table 1 Formulation of the rubber compound. Ingredient
Amount (phr)
Natural rubber (NR) STR10/SIR10 Styrene butadiene rubber (SBR) 1502 Cis-1, 4-polybutadiene rubber (BR) Carbon black (CB) N234 Silica (Si) Z195Gr
75 10 15 40 15
[8,9]. Another alternative, finite element computation, provides a useful tool to analyze the influence of wall slip or other factors on rheological measurements [10–12]. However, to the author’s knowledge, none of these numerical works has ever dealt with filled rubber compounds because of the difficulties inherent to this class of polymers and the limited number of relevant experimental data. In this paper, rheological properties of a typical filled rubber compound were studied with an oscillatory shear rheometer and a rate-controlled capillary rheometer. A numerical model of the capillary rheometer was also built to reproduce the testing procedure and to explain the failure of Cox-Merz rule by introducing a slip boundary condition into the simulation. 2. Experimental 2.1. Materials The rubber compound studied in this paper was provided by Giti Tire Ltd. The detailed formulation is listed in Table 1. The base elastomer was a combination of natural rubber (NR), styrene butadiene rubber (SBR) and cis-1, 4polybutadiene rubber (BR). Carbon black (CB) N234 and silica (Si) Z195Gr were used as reinforcing fillers. The total filler content was as high as 55 parts per hundred parts of rubber (phr). Mixing was performed by means of an internal mixer.
rheometer specially designed for rubber-like viscoelastic materials. Owing to its grooved biconical dies and pressurized testing cavity, wall slip could be neglected during the tests. Frequency sweep tests were performed over a wide range of oscillatory frequency from 1.5 CPM to 1500 CPM (0.157 rad/s to 157 rad/s) at two different temperatures (100 C and 120 C) and a uniform strain of 7%, according to ASTM D6204 [13]. Then, a rate-controlled capillary rheometer Rosand RH2200 (manufactured by Malvern Instruments Ltd., UK) was used to perform steady rheological tests. The apparatus had two bores, one of which was fitted with a capillary die having a length of 24 mm and a diameter of 1.5 mm (L/D ¼ 16/1) while the other, for the sake of Bagley correction, was fitted with an orifice or zero-length die having the same diameter but a length of 0.25 mm. Testing temperature was 100 C and apparent shear rate ranged from 100 1/s to 1000 1/s. Bagley and Rabinowitsch corrections were applied to correct entrance/exit pressure loss and shear-thinning effects, respectively. 2.3. Results and discussion Fig. 1 depicts the master curves of viscoelastic properties obtained from the RPA2000 at a temperature of 100 C. We observe that the molten rubber compound exhibits
a
1000000
Complex viscosity (RPA2000) Shear viscosity (RH2200) 100000
Viscosity (Pa.s)
46
10000
1000
100
2.2. Rheological testing 10
Rheological properties were firstly measured with a Rubber Process Analyzer RPA2000 (manufactured by Alpha Technologies, USA), which is a rotorless oscillatory shear
0.1
1
10
100
1000
Shear rate (1/s) or frequency (rad/s)
b
1000000
Complex viscosity (RPA2000) Shear viscosity (RH2200, with a vertical shift) 100000
100000
100000
10000
Modulus (Pa)
Complex viscosity (Pa.s)
Complex viscosity Storage modulus Loss modulus
Viscosity (Pa.s)
1000000
1000000
10000
1000
100
10 0.1
10000
1000 0.1
1
10
1
10
100
1000
Shear rate (1/s) or frequency (rad/s)
100
Frequency (rad/s)
Fig. 1. Complex viscosity, storage modulus and loss modulus as a function of frequency at 100 C.
Fig. 2. Complex viscosity and shear viscosity as a function of frequency or shear rate at 100 C. (a) Original data showing the existence of inconsistency between the two types of rheological measurements; (b) A vertical shift is applied to the capillary data.
C. Yang, Z. Li / Polymer Testing 37 (2014) 45–50
typical shear-thinning behavior and a Newtonian plateau is hard to capture. In Fig. 2(a), the curve of shear viscosity from RH2200 is compared with the curve of complex viscosity from the RPA2000 with the aid of the Cox-Merz rule at 100 C. Data obtained with the two rheometers are inconsistent (the RPA2000 result is higher than that of the RH2200), while a vertical shift applied to capillary data can eliminate this inconsistency since the slopes of two curves are almost the same on the log-log plot, as shown in Fig. 2(b). It is worthy pointing out that the Cox-Merz rule is an important empirical relationship which has been verified experimentally for a number of materials in the past six decades. On the other hand, the apparent failure of the Cox-Merz rule might be caused by various factors in the experiments, such as wall slip, viscous heating and pressure dependence of viscosity. Referring to Fig. 3, the complex viscosity curves at 100 C and 120 C are indistinguishable from each other, indicating that the rubber compound we studied is not sensitive to small variation of temperatures above 100 C and viscous heating should have a negligible impact on the capillary tests. Also, according to Kazatchkov et al. [14], the dependency of viscosity on pressure tends to cause h*(u) to be lower than hðg_ Þ, which is the opposite of the current situation (h ðuÞ > hðg_ Þ, see Fig. 2(a)). Thus, wall slippage in the capillary rheometer (slippage is negligible in RPA2000, as discussed above) should be the main reason of the inconsistency and its influence on testing results are examined numerically in the present work. 3. Numerical 3.1. Numerical model An axisymmetric finite element model of the twin bore rate-controlled rheometer Rosand RH2200 was built, as depicted in Fig. 4, and all geometrical dimensions were set in accordance with the experimental study. Contraction flows from the reservoir to the capillary as well as extrudate swell are taken into account so that entrance and exit pressure drops can be determined. Mesh of the left bore consists of 6050 quadrilateral elements and the right
47
consists of 5260. Mesh refinement has been checked to guarantee that the finite element solution is meshindependent. Flow in the computational domain is governed by equations of continuity and momentum, written as
V$v ¼ 0
(1)
Vp þ V$s ¼ 0
(2)
where v is the velocity vector, p is the pressure and s is the stress tensor. Inertia items and gravity are neglected due to the high viscosity of the material. A 5-mode Phan-Thien-Tanner (PTT) model [15] was used to describe the viscoelastic behavior of the molten rubber compound. Purely viscous models, e.g. the BirdCarreau model and the Cross model, are excluded from the present study as they tend to severely underestimate the extrusion pressure and die swell ratio. The PTT model is written as
8 5 P > > s¼ sðiÞ > > < i¼1 " # hε l > xi VðiÞ xi DðiÞ > i i ðiÞ ðiÞ > > exp s þ li 1 s þ s tr s ¼ 2hi D : hi 2 2 (3) where the rate of deformation tensor is given by D ¼ 1/ 2(L þ LT), with the velocity gradient tensor L ¼ (Vv)T. li, hi, εi and xi are material parameters. Dynamic data from the RPA2000 at 100 C, referring to Fig. 1, are used to determine these parameters and the Cox-Merz rule is applied to convert complex viscosity into shear viscosity. The PTT model fits the data well over the range of the dynamic tests, as presented in Fig. 5 and Table 2.
a
b Piston (inflow) Axis of symmetry Reservoir
1000000 o
Complex viscosity (Pa.s)
100 C o 120 C 100000
Reservoir wall Capillary die Capillary wall
10000
Extrudate 1000 0.1
1
10
100
Free surface Downstream
Frequency (rad/s)
Fig. 3. Comparison of complex viscosity at two testing temperatures of 100 C and 120 C.
Fig. 4. Finite element model of the twin bore capillary rheometer used in this study. (a) Left bore with a capillary die (L ¼ 16 mm, D ¼ 1.5 mm); (b) Right bore with a zero-length die (L ¼ 0.25 mm, D ¼ 1.5 mm).
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C. Yang, Z. Li / Polymer Testing 37 (2014) 45–50
A uniform normal velocity profile is imposed at the inflow boundary as the piston speed vp. With the diameter of the reservoir Dr (15 mm for the rheometer used in this work), volumetric flow rate Q can be calculated as
Q ¼
pD2r vp
(4)
4
then, for Newtonian fluids, shear rate at the capillary wall can be obtained and treated as the apparent wall shear rate for non-Newtonian fluids. That is
32Q g_ a ¼ pD3
(5)
where D is the diameter of the capillary die. A no-slip boundary condition is applied at the reservoir wall. At the capillary wall, a Navier’s slip condition is enforced, which can be written as
vs ¼
sw
(6)
k
where vs is the slip velocity, sw is the shear stress at the capillary wall and k is the slip coefficient. If k ¼ N, Eq. (6) reduces to the no-slip condition while with k ¼ 0, Eq. (6) captures the full-slip condition. Other slip models are able to be adopted into the simulation as well, but detailed comparison between different slip conditions is beyond the scope of this paper. In the absence of surface tension, a boundary condition of zero-stress is applied at the free surface and downstream of the extrudate. The position of the free surface is computed as part of the solution and iteration stops when normal velocity on the surface is vanishingly small. Numerical analysis of the above model was carried out using a finite element software package POLYFLOWÒ [16]. 3.2. Results and discussion
Table 2 Material parameters of the 5-mode Phan-Thien-Tanner model. Mode no.
li (s)
hi (Pa.s)
εi
xi
1 2 3 4 5
0.001 0.010 0.110 1.147 12.00
607.3481 2556.049 18282.04 90839.74 727133.4
0.011 0.011 0.026 0.011 0.038
0.227 0.080 0.074 0.070 0.131
shown in Fig. 6 at the apparent shear rate (Eq. (5)) of 1.067 1/s. Higher shear rates or piston speeds were not attempted for the limitation of the well-known high Weissenberg number problem (HWNP), a numerical break down that occurs when the Weissenberg number (Wi) exceeds a critical value. Fortunately, the viscosity curve of the rubber compound is approximately a straight line on a log-log plot for shear rates above 0.1 1/s. Wall slippage and its effect on capillary measurements at high shear rate range can be investigated via extrapolation. As expected, pressure drops in the entrance and exit regions of the capillary die can be found from the pressure-distance plot in Fig. 6. The sum of these pressure losses is determined by the extrusion pressure of the zero-length die Dp0 . Hence, by subtracting Dp0 from the extrusion pressure of the capillary die Dp, the pressure gradient at the capillary die is given by
Dp Dp0 vp ¼ vz L
where z is the downstream direction and L is the capillary length. It should be noted that in the experiments Dp and Dp0 are read directly from the pressure sensors installed at the reservoir wall. Corrected shear stress at the capillary wall can be calculated as
sc ¼
Models with a no-slip condition applied at the capillary wall were analyzed first to check the reliability of the finite element computation. A typical pressure distribution is
(7)
vp D $ vz 4
(8)
where D is the capillary diameter. The end correction method used in this study is different from the traditional Bagley plot, which is indirect and time-consuming. With the corrected wall shear stress sc (Eq. (8)) and the
Viscosity (Pa.s) or modulus (Pa)
1E7
1000000
100000
10000 Shear viscosity (PTT) Storage modulus (PTT) Loss modulus (PTT) Complex viscosity (RPA2000) Storage modulus (RPA2000) Loss modulus (RPA2000)
1000
100 0.01
0.1
1
10
100
1000
Shear rate (1/s) or frequency (rad/s)
Fig. 5. Viscoelastic properties of the rubber compound at 100 C. Lines represent the prediction of the 5-mode Phan-Thien-Tanner model with parameters listed in Table 1 while symbols are experimental data from RPA2000.
Fig. 6. Pressure distribution for the rubber compound at the apparent shear rate of 1.067 1/s with a no-slip condition at the capillary wall.
C. Yang, Z. Li / Polymer Testing 37 (2014) 45–50
apparent wall shear rate g_ a (Eq. (5)) deduced from piston speed, apparent shear viscosity can be obtained as
ha ¼
sc g_ a
(9)
A shear rate sweep was simulated around the apparent shear rate of 1.0 1/s. As revealed by the no-slip results in Fig. 7, the apparent shear viscosity is higher than the shear viscosity predicted by the PTT model or, in other words, the real shear viscosity of the material in numerical experiments. This is because the apparent shear rates are based on Newtonian fluids whose velocity profile is parabolic in the fully-developed region, and shear thinning fluids show a different profile close to plug flow with higher shear rates at the wall. The Rabinowitsch correction is applied to correct the shear thinning effect. A corrected wall shear rate is given by the Rabinowitsch-Mooney equation [17]:
g_ dlng_ a g_ c ¼ a þ3 4 dln sc
(10)
sc g_ c
(11)
Slip condition (Eq. (6)) at the capillary wall was then introduced to the simulation with two different slip coefficients k ¼ 1 109 kg m2 s1 and k ¼ 1 1010 kg m2 s1 representing different levels of slippage. As it is clear from Fig. 7, slip at the capillary wall will make the apparent shear viscosity and corrected shear viscosity lower than the no slip ones. Specifically, the percentage drop in apparent shear viscosity is 3% for k ¼ 1 1010 kg m2 s1 and 27% for k ¼ 1 109 kg m2 s1. In fact, for lower k or higher level of slippage, the underestimation of shear viscosity will be more severe. This confirms that when slippage becomes significant, a capillary rheometer is no longer a reliable device to obtain shear viscosity data unless the slippage can be measured and corrected. However, wall slip correction is
180000 170000
Shear viscosity (PTT) Apparent shear viscosity (no slip) Corrected shear viscosity (no slip) Apparent shear viscosity (k = 1E10) Corrected shear viscosity (k = 1E10) Apparent shear viscosity (k = 1E9) Corrected shear viscosity (k = 1E9)
160000 150000
Shear viscosity (Pa.s)
140000 130000 120000 110000 100000 90000
Increasing slippage 80000
70000 0.7
0.8
0.9
difficult for filled rubber compound since the frequently used Mooney method is not applicable, as discussed before. Meanwhile, in the presence of slippage, the Rabinowitsch correction fails to obtain the true shear rate at the capillary wall, and sometimes the corrected shear viscosity deviates further from PTT prediction than the apparent shear viscosity (see results of k ¼ 1 1010 kg m2 s1 in Fig. 7). It should also be noted that the underestimation of viscosity caused by slip boundary condition can always be corrected via a vertical shift, identical to what we did in Fig. 3(b), indicating that the failure of the Cox-Merz rule in Fig. 3(a) could be (at least partially) caused by wall slip in the capillary tests. For this reason, our future work will be focused on measuring the slippage of filled rubber compound and characterizing the quantified level of slipping with appropriate boundary conditions in finite element computation. 4. Conclusions
and this leads to a corrected shear viscosity fitting the PTT model well, that is
hc ¼
49
1
1.1
1.2
1.3
1.4
1.5 1.6 1.7 1.8
Shear rate (1/s) Fig. 7. Apparent and corrected shear viscosity results obtain from numerical shear rate sweeps with a no-slip condition or a Navier’s slip condition at the capillary wall, compared with the shear viscosity predicted by the PTT model.
In this study, the rheological properties of a filled rubber compound were measured and analyzed by using a dynamic shear rheometer and a capillary rheometer. Comparing the viscosity data obtained from oscillatory shear and steady shear, an apparent failure of the Cox-Merz rule is found, that is, the complex viscosity is higher than the shear viscosity. With the use of a finite element model of the twin bore capillary rheometer, the capillary tests are reproduced and the wall slip at the capillary wall is investigated numerically. Results reveal that the underestimation of shear viscosity becomes significant with increasing level of slippage at the capillary wall, and this could be one of the reasons why the Cox-Merz rule is not applicable. All this will help understand the rheological behavior of filled rubber compounds and be a reminder of the possible error of shear viscosity of similar filler-reinforced elastomers in capillary testing. Acknowledgements This work was funded by the Fundamental Research Funds for the Central Universities (No. WK2090050018) in China. We are grateful to Giti Tire Ltd for supplying the rubber compound used in this study. References [1] J.L. Leblanc, Rubber–filler interactions and rheological properties in filled compounds, Progress in Polymer Science 27 (4) (2002) 627– 687. [2] A. Mongruel, M. Cartault, Nonlinear rheology of styrene-butadiene rubber filled with carbon-black or silica particles, Journal of Rheology 50 (2) (2006) 115–135. [3] D.M. Turner, M.D. Moore, Contribution of wall slip in the flow of rubber, Plastics and Rubber: Processing 5 (1980) 81–84. [4] W.P. Cox, E.H. Merz, Correlation of dynamic and steady flow viscosities, Journal of Polymer Science 28 (118) (1958) 619–622. [5] N. Phewthongin, P. Saeoui, C. Sirisinha, A study of rheological properties in sulfur-vulcanized CPE/NR blends, Polymer Testing 24 (2) (2005) 227–233. [6] M. Ansari, S.G. Hatzikiriakos, A.M. Sukhadia, et al., Rheology of Ziegler–Natta and metallocene high-density polyethylenes: broad molecular weight distribution effects, Rheologica Acta 50 (1) (2011) 17–27. [7] M. Mooney, Explicit formulas for slip and fluidity, Journal of Rheology 2 (2) (1931) 210–222.
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[8] J.L. Leblanc, T. Navarat, Assessing the wall slip rate of complex polymer systems with the Mooney method, in: Proceedings of the 13th International Congress on Rheology, Cambridge, UK, 2000. [9] P. Mourniac, J.F. Agassant, B. Vergnes, Determination of the wall slip velocity in the flow of a SBR compound, Rheologica Acta 31 (6) (1992) 565–574. [10] S. Syrjälä, J. Aho, Capillary rheometry of polymer meltsdsimulation and experiment, Korea-Australia Rheology Journal 24 (3) (2012) 241–247. [11] Y.L. Yeow, Y.K. Leong, A. Khan, Non-Newtonian flow in parallel-disk viscometers in the presence of wall slip, Journal of Non-Newtonian Fluid Mechanics 139 (1) (2006) 85–92. [12] H.M. Laun, Capillary rheometry for polymer melts revisited, Rheologica Acta 43 (5) (2004) 509–528.
[13] ASTM Standard D6204-12, Standard Test Method for RubberdMeasurement of Unvulcanized Rheological Properties Using Rotorless Shear Rheometers, ASTM International, West Conshohocken, PA, USA, 2012. http://www.astm.org/Standards/D6204. htm. [14] I.B. Kazatchkov, S.G. Hatzikiriakos, C.W. Stewart, Extrudate distortion in the capillary/slit extrusion of a molten polypropylene, Journal of Plastic Film and Sheeting 11 (1) (1995) 38–57. [15] N. Phan-Thien, A nonlinear network viscoelastic model, Journal of Rheology 22 (3) (1978) 259–283. [16] ANSYS POLYFLOW, User’s Guide, Release 14.0, ANSYS Inc., Canonsburg, PA, USA, 2011. [17] R.I. Tanner, Engineering Rheology, second ed., University Press, Oxford, 2000.