# Dependence of bulk viscosity of polypropylene on strain, strain rate, and melt temperature.

INTRODUCTIONThe role of bulk viscosity or volume viscosity has been neglected for several decades in the flow analysis of injection molding. This is because the flow of molten plastic in the filling stage is virtually divergence free. Despite the compressional and nondivergence free flow in the packing-holding stage, still the bulk viscosity has been ignored. The rationale is that the flow is mostly influenced by the melt compressibility governed by the equation of state. In the packing-holding stage the melt could be compressed up to 15% of its volume as evidenced by the PVT diagram according to Zoller and Walsh [1], and Wang et al. [2]. Thus, the divergence and the bulk viscosity coefficient should be considered in the packing-holding stage. The bulk viscosity of a fluid was measured by means of wave attenuation measurements and uniaxial compression. Narasimham [3] modeled the bulk viscosity of several chemical organic compounds as a function of the frequency and the amplitude of a travelling wave in the fluid through the acoustic excitation. Hoover [4] developed a bulk viscosity model of fluid based specially on cyclic compression experiments. However, the proposed models from both studies are difficult to be implemented in the simulation of injection molding. This is because the model parameters, such as the frequency, amplitude and pressure waves, are not used in injection molding process modeling. Furthermore, the magnitude of the bulk viscosity based on the mentioned studies has never been explicitly stated. Lesbats et al. [5] proposed a more practical approach to measure the bulk viscosity of epoxide monomer from uniaxial compression in analogy to the PVT measurement. In this experiment, the bulk viscosity was modeled as Kelvin-Voigt model (spring and dashpot in parallel) [5]. The bulk viscosity of epoxide monomer was found to be inversely proportional to the compression rate [5]. In contrast, the bulk viscosity increased with the melt temperature [5]. Thus, the bulk viscosity of epoxide monomer was thermal and volume deformation dependent [5], Aleman [6] used the same approach to investigate the dependence of the bulk viscosity on the elongational and shear viscosities of several thermoplastic polymers. Aleman [6] discovered the same compression rate-bulk viscosity dependence as in previous work. Alternative to Kelvin-Voigt model, Aleman [6] showed that the bulk viscosity could be determined from the combination of the shear and the elongational viscosities [6]. Aleman [7] measured the bulk viscosity of high and low-density polyethylene (HDPE and LDPE). The bulk viscosities of HDPE and LDPE were [10.sup.4] times greater than the shear viscosities. In contrary to epoxide monomer, the bulk viscosities of HDPE and LDPE decreased when the melt temperature was increased [7], Aleman [7] used Arrhenius equation to model the bulk viscosity-temperature dependence. In the studies of Lesbats et al. [5] and Aleman [6, 7], the magnitude of the bulk viscosities of the proposed polymers were significantly greater than the shear viscosity. Not only the bulk viscosity of the polymer melts but also the bulk viscosity of ethanol, benzene and carbon disulfide is much higher than the shear viscosity [8], Apart from Kelvin-Voigt-based bulk viscosity model, Aksel [9] introduced an alternative approach modeling the bulk viscosity as a non-Newtonian fluid containing compressible bubble. However, the association between the bulk viscosity and temperature is not stated in this study. Okumura and Yonezawa [10] and Fernandez et al. [11] focused on determining the bulk viscosities of several gas mixtures excited by pressure waves at microscopic level. Holmes et al. [12] developed another approach measuring the bulk viscosity of water from attenuation of ultrasonic wave. The bulk viscosity of water exponentially decreased with the rise of the water temperature in analogy to the work of Aleman [7]. Also, the bulk viscosity of the water was found to be three times greater than the shear viscosity [12]. In addition. Sata et al. [13] discovered that the use of ultrasonic wave during the packing-holding stage could improve the final part quality in term of the part weight and shrinkage reduction. However, there was no further research on relating this to the bulk viscosity.

From the previous studies, the magnitude of the bulk viscosity of any fluid, especially the melt, is significantly greater than the shear viscosity. Hence, the bulk viscosity should not be neglected in the case of the compressional flow with large dilatation or compression deformation. The methods to measure and determine the bulk viscosity of fluid are distinctive. Each method is for the specific type of fluid. The bulk viscosity models from earlier studies are difficult to apply to the simulation of plastic injection molding since these models' parameters are used on microscopic scale. None of the previous models related the bulk viscosity with compressibility, compression rate and temperature of the fluid explicitly in a single model. In this research, for the first time the bulk viscosity of polypropylene and its dependence on the strain, strain rate, and melt temperature is analyzed. We propose a more practical bulk viscosity model for the simulation of injection molding based on three variables, the normal strain, normal strain rate, and melt temperature. The proposed model is logarithmic in analogy to the second-order shear viscosity model in Moldflow [14]. The uniaxial compression experiment proposed by Lesbats et al. [5] is used to measure the bulk viscosity since it is close to the real process of the packing-holding stage in injection molding. Instead of Kelvin-Voigt model, we propose the alternative approach to determine the bulk viscosity from experimental parameters based on Navier-Stokes equation but finally proved to be the same model as Kelvin-Voigt.

THEORETICAL BACKGROUND

The bulk viscosity appears in the momentum and energy equations in the form of normal stress and dilatational heat dissipation terms. In cylindrical coordinates, the momentum and energy equations can be expressed in tensor notation as [14]:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2)

where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4)

For uniaxial compression, the melt is compressed in a close cylindrical chamber. Thus, the melt is allowed to deform in the axial direction only. Also, a nonslip wall constraint is assumed. Thus, only the divergence in the axial direction is considered. The deviatoric rate of strain tensor, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is then reduced to:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (5)

Substituting Eq. 5 and one dimensional divergence into Eq. 3, the divergence of the stress tensor, [nabla][[sigma].bar] becomes:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (6)

The term with the summation of the shear and bulk viscosities is called the longitudinal bulk viscosity. It is the viscous property of the melt due to the deformation in one direction [5-7]. For the sake of simplicity, the term bulk viscosity in this experiment is used to refer the longitudinal bulk viscosity throughout this article. The stress tensor because of both the viscosities is:

[[sigma].bar] = -[P.sub.z]+(4/3[[eta].sub.s] + [[eta].sub.k])[partial derivative][v.sub.z]/[partial derivative]z + [[eta].sub.s] [partial derivative][v.sub.z]/[partial derivative]r (7)

The bulk viscosity is associated with the resultant stress from the stress tensor in Eq. 7. Based on Aleman [7], the shear viscosity is [10.sup.4] times less than the bulk viscosity. Also, the shear rate could be very low relative to the normal strain rate in the uniaxial compression. Hence, the terms with the shear viscosity can be neglected. This negligence is later proved by the shear viscosity measurement in the next section. Without the shear viscosity, the bulk viscosity solely depends on the normal stress component in the axial direction:

[[sigma].sub.z] = -[P.sub.z] + [[eta].sub.k] [partial derivative][v.sub.z]/[partial derivative]z (8)

Eq. 8 can be rearranged and the bulk viscosity is obtained as follows:

[[eta].sub.k] = [[[sigma].sub.z] + [P.sub.z]]/[[??].sub.z] (9)

[[??].sub.z] = [partial derivative][v.sub.z]/[partial derivative]z [approximately equal to] [DELTA]L/Lt (10)

Equation 9 is used to determine the bulk viscosity from the uniaxial compression. In the packing-holding stage, the pressure [P.sub.z] is the thermodynamic pressure which can be determined by the equation of state (Melt compressibility) [15], The normal strain rate [[??].sub.z] can be approximated as the compression deformation of the melt [DELTA]L over its original length L and the compression time. The normal stress [[sigma].sub.z] is the stress measured from the experiment, in this case, the total pressure measured during the compression. This normal stress has negative sign since it is considered as the compressional stress. Equation 9 is in fact, the same as Kelvin-Voigt-based bulk viscosity model proposed from the earlier studies [5-7], This derivation shows that the use of Kelvin-Voigt model can also be proved by the momentum equation.

EXPERIMENTAL

Polypropylene (PP) 1100 NK homopolymer from IRPC public co., ltd. was used in this experiment. The specification of the sample is presented in Table 1. In order to examine the consistency of the flow properties between the sample and supplier's specification, the material melt flow index (MFI) and melt volume flow rate (MVR) were measured according to ASTM D1238 (Table 2) by Davenport MFI device. The melt flow index from the measurement and the supplier data was 11.58 and 11, respectively.

To validate the assumption that the shear viscosity was significantly lower than the bulk viscosity, the shear viscosity of the sample was measured by the capillary rheometer Rosand RH7. The bulk viscosity of the sample was measured with the same capillary rheometer, but different configuration as shown in Fig. 1. To avoid the air trap during the measurement, the sample was pre-heated three minutes at the specified melt temperature according to Table 3 and purged out twice with 0.1 MPa of pressure. The capillary rheometer was then plugged by the steel plug to completely close the compression chamber. The sample was compressed by the plunger with different constant compression speeds and melt temperatures (Table 3). The total melt pressure and deformation during the compression were then recorded by the pressure transducer as shown in Fig. 1.

EXPERIMENTAL RESULTS

The flow curve of the sample is shown in Fig. 2. The maximum and minimum shear viscosities from the experimental data were 590 and 1.4 Pa s at the shear rate of 10 and 18,000 [s.sup.-1] respectively. This result confirms the experimental result of Aleman [7] that the magnitude of the shear viscosity of the melt flow in the cylindrical chamber is significantly lower than that of the bulk viscosity. Hence, the shear viscosity can be neglected from the stress tensor in the case of uniaxial compression.

The experimental data shows that the pressure of the molten sample increases with the compression deformation in similar fashion to Aleman [7] as shown in Fig. 3. The pressure seems to have no correlation with the compression speed at any melt temperature, except at the compression speed of 0.5 mm/min where the pressure is the lowest relative to the other compression speeds (Fig. 3). At low compression speed, the melt may experience the relaxation of stress or resultant pressure through the slow compression, thereby resulting in the lowest pressure. Moreover, this pressure is independent of the melt temperature as shown in Fig. 4. In short, the melt pressure is solely associated with the compression deformation.

The thermodynamic pressure of the melt, [P.sub.z] is considered as the pressure at the equilibrium state and isothermal condition. The equilibrium state means there is no further exchange of heat and the pressure is kept at the specified compression deformation. Thus, the thermodynamic pressure can be determined at the state of no flow. Due to the limitation of capillary rheometer system, it is virtually impossible to measure the pressure at different compression deformation without the compression speed concern. The only method to obtain the thermodynamic pressure, in this case, is to extrapolate the profile of measured pressure at zero strain rate as shown in Fig. 5. The thermodynamic pressure increased with the rise of compression deformation and the melt temperature in analogy to the PVT diagram of a polymer in liquid phase (Fig. 6).

As the thermodynamic pressure represents the time-independent stress, the elasticity of melt can be predicted as follows:

[P.sub.z] = E([epsilon], T)[epsilon] [approximately equal to] E([epsilon], T) [DELTA]L/L (11)

The melt elasticity E([epsilon], T) is a function of strain and temperature as shown in Fig. 7. The increase of the elasticity with the temperature is due to the expansion and trapped melt within the close chamber attempting to repel the molecular contraction during the compression.

Effect of Strain Rate on the Bulk Viscosity

Once the thermodynamic pressure is obtained, the bulk viscosity can be determined from Eq. 9. The bulk viscosity exponentially decreased with the increase of the strain rate as shown in Fig. 8. This is similar to the bulk viscosity of epoxide monomer, HDPE and LDPE in the studies of Lesbats et al [5] and Aleman [7], respectively. However, it is difficult to compare the magnitude of the bulk viscosity of the sample with both studies since the polymers were measured at the different range of compression speeds and melt temperatures. Because the bulk viscosity is the resistance to the volume deformation [11], the ability to occupy the free space within the molecular chains by the movement of the bond angles and the entanglement of the chains is directly associated with the magnitude of the bulk viscosity. The behavior of bulk viscosity, therefore, can be directly related to the transformation of the original and deformed structure of the molecular chains, for example, molecular orientation, the variation and number of functional groups and the compactness of molecular chains. Based on Fig. 9, the bulk viscosity of polypropylene is smaller than the bulk viscosities of HDPE and LDPE from the study of Aleman [7] at the very low strain rate and it is almost the same at higher strain rate. One possible reason for this difference is molecular compactness which can be interpreted as density. Aleman [7] measured densities of solid HDPE and LDPE as 0.947 and 0.920 g/[cm.sup.3], respectively. Meanwhile, the mean density of the polypropylene sample in the solid state is 0.900 g/[cm.sup.1]. As the densities of solid HDPE and LDPE are slightly greater than the density of the sample polypropylene, the molecular structure of both HDPE and LDPE in the liquid state are potentially denser and more resistant to the volume deformation than the liquid polypropylene. The compactness of molecular chains more highly affects the bulk viscosity at the low strain rate than at the high strain rate because the molecular chains have sufficient time to self-recoil and kink with each other by Van der Waals interaction forces during compression deformation. This self-recoiling and kinking of the dense molecular chains lead to higher resistance to volume deformation (Fig. 8). On the other hand, at the high strain rate, the randomly coiled and kinking molecular chains have shorter time for re-entanglement and tends to be more oriented in the compression direction. This molecular orientation facilitates the volume deformation and reduces the effect of the molecular entanglement, likewise reducing the bulk viscosity.

Effect of Strain on the Bulk Viscosity

The bulk viscosity increased when the compression deformation or strain increased as shown in Fig. 8. Once again, this phenomenon can be explained by the compactness of molecular chains which are originally coiled or kinked. Higher strain results in greater reduction of the free space. The lower free space means the distance between atoms may become smaller beyond the Van der Waals contact distance and leads to increase of the repulsive force. Hence, the molecules of the polymer have difficulty in travelling from one location to another due to this repulsion. Figure 10 shows the effect of pressure induced by the movement of the plunger on the bulk viscosity. Since this pressure is, in fact, a function of strain or compression deformation, the bulk viscosity also rises with the increase of the total melt pressure.

Effect of Temperature on the Bulk Viscosity

The effect of the temperature on the bulk viscosity of polypropylene is shown in Fig. 11a. Overall, the bulk viscosity declined with the increase of the temperature. This temperature dependence is identical to the experimental results of Aleman [7], The rationale behind this temperature effects is that the addition of heat leads to the expansion of the kinking molecular chains. Therefore, the entanglement of molecular structure becomes more relax leaving more free space for the neighbor bond molecules. Some fluctuation of the bulk viscosity with the temperature was observed at low compression deformation (Fig. 11a). This fluctuation was also found in the case of epoxide monomer [5]. This phenomenon is attributed to the fact that the heat within this range of temperature may not be able to initiate significant molecular expansion. Instead, this range of temperature may lead to small vibration of entangled molecular chains, thereby repelling each other and enhancing resistance to compression deformation. Moreover, the bulk viscosity at any temperature reaches an asymptote for high strain rate. The molecular expansion has no longer effect on the polymer chains deformation since they are more dominated by orientation from high compression speed.

Empirical Bulk Viscosity Model

The bulk viscosity as a function of strain rate at specific melt temperature and different deformation is firstly transformed into logarithmic base 10 value as shown in Fig. 11b. The closeness of the logarithm of bulk viscosity to the logarithm of strain rate approximately varied in quadratic fashion, likewise the bulk viscosity-temperature dependence in Fig. 11a. Hence, the bulk viscosity empirical model can be formulated from the tri-quadratic logarithm base 10 function in analogy to Moldflow second-order model [14] as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (11)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (12)

[DELTA][epsilon]=[epsilon]-[[epsilon].sub.0] (13)

[DELTA]T = T - [T.sub.0] (14)

where [b.sub.i] are data fitted coefficients. [T.sub.0] is the reference temperature, in this case, it is set at 185[degrees]C. [[eta].sub.0] is the bulk viscosity at zero strain, zero strain rate and the reference temperature. This parameter can be only obtained by extrapolation near zero strain Co, zero strain rate [[??].sub.o] and at the reference temperature. All coefficients are listed in Table 4.

Figure 12 shows the comparison between the proposed empirical model and the experimental data. At the temperature of 195[degrees]C, the empirical model overestimated the bulk viscosity (Fig. 12a). Meanwhile, at the higher temperature the bulk viscosity was underestimated (Fig. 12b). However, from both cases the empirical model predicted the bulk viscosity within the same order of magnitude as the experimental data. The accuracy of the empirical model can be improved by increasing the order.

CONCLUSIONS

The bulk viscosity of polypropylene was measured by the uniaxial compression method as proposed by Lesbats et al. [5] and Aleman [7], The empirical model of bulk viscosity dependence on strain, strain rate and melt temperature was developed for simulation of the packing-holding stage of injection molding. The bulk viscosity is associated with the occupancy of the bonded molecules within the free interstitial space. From the experiment, the bulk viscosity decreased exponentially with the strain rate and reached an asymptote at the high strain rate at all compression deformation. In addition, the bulk viscosity decreased with the increase of the melt temperature. Meanwhile, the bulk viscosity increased with the increase of the compression deformation. The largest magnitude of the bulk viscosity was observed at the highest compression deformation and the lowest strain rate. This magnitude was significantly greater than the shear viscosity. Hence, apart from shear viscosity, the bulk viscosity should be considered in the flow analysis of the packing-holding stage of the plastic injection process when large compression deformation of the melt occurs. From the energy equation, the bulk viscosity would produce additional viscous heat dissipation apart from shear viscosity because of volume dilatation or compression effects. Hence, the bulk viscosity would increase the melt temperature and total stress during the packing-holding stage, especially in the highly deformed region.

ACKNOWLEDGMENTS

The authors would like to thank to Dr. Asira Fuongfuchat, Ms. Phatchareeya Raksa, and Ms. Thidarat Makmoon, Rheology laboratory, National Metal and Materials Technology Center, National Science and Technology Development Agency Thailand for the experimental equipments and other supports for the research.

NOMENCLATURE [rho] Melt density [c.sub.p] Specific heat at constant pressure E Modulus of elasticity [logical]g Body force per unit mass I Unit tensor [bar.k] Thermal conductivity L Original length of the melt sample [DELTA]L Deformation P Thermodynamic pressure T Melt temperature t Compression time [logical]v Velocity vector [v.sub.z] Velocity along axial direction [[delta].bar] Stress tensor [[sigma].sub.z] Total stress in axial direction [beta] Isothermal expansitivity [[eta].sub.s] Shear strain [[eta].sub.k] Bulk viscosity [[[??].bar].sub.d] Deviatoric rate of strain tensor [??] Rate of strain tensor [epsilon] Strain in axial direction [[??].sub.z] Strain rate in axial direction

REFERENCES

[1.] P. Zoller and D.J. Walsh, Standard Pressure-Volume-Temperature Data for Polymers, Technomic Publishing Company Inc., Lancaster, PA (1995).

[2.] J. Wang, P. Xie, Y. Ding, and W. Yang, J. Appl. Polym. Sci., 118, 200 (2010).

[3.] A.V. Narasimham, A model for the bulk viscosity of a liquid, IEEE Trans. Sonic and Ultrasonic, 16, 182 (1969).

[4.] W.G. Hoover, A.J.C. Ladd, R.B. Hickman, and B.L. Holian, Phys. Rev. -4, 21, 1756 (1980).

[5.] J.P. Lesbats, R. Legros, and J.V. Aleman, J. Polym. Sci., 20, 1971 (1982).

[6.] J.V. Aleman, Rheol. Acta, 27, 238 (1988).

[7.] J.V. Aleman, Polym. Eng. Sci., 30, 326 (1990).

[8.] P.A. Longwell, Mechanics of Fluid Flow, McGraw-Hill, New York (1966)

[9.] N. Aksel, Continuum Mech. Thermodyn., 7, 333 (1995).

[10.] H. Okumura and F. Yonezawa, Phys. A, 321, 207 (2003).

[11.] G.A. Fernandez, J. Vrabec, and H. Hasse, Fluid Phase Equilibria, 221, 157 (2004).

[12.] M.J. Holmes, N.G. Parker, and M.J.W. Povey, J. Phys. Conf. Ser., 269, 1 (2011).

[13.] A. Sata, H. Ito, and K. Koyama, Polym. Eng. Sci., 49, 768 (2009).

[14.] P. Kennedy, Flow Analysis of Injection Molds, Carl Hanser Verlag, New York (1995).

[15.] C. L. Tucker, Computer Modeling for Polymer Processing, Carl Hanser Verlag, New York (1989).

P. Chivapornthip, E.L.J. Bohez

Industrial System Engineering, School of Engineering and Technology, Asian Institute of Technology, Klong Luang, Pathumthani 12120, Thailand

DOI 10.1002/pen.24459

Correspondence to: P. Chivapornthip; e-mail: prapol_c@hotmail.com

Caption: FIG. 1. Uniaxial compression experimental equipment.

Caption: FIG. 2. Shear viscosity of polypropylene (Pa.s) at 190[degrees]C and 215[degrees]C.

Caption: FIG. 3. Pressure of polypropylene (MPa) as a function of compression deformation percentage (k%) and different compression speed (mm/min) at (a) 190[degrees]C and (b) 200[degrees]C.

Caption: FIG. 4. Pressure of polypropylene (MPa) as a function of compression deformation percentage (k%) and different melt temperatures ([degrees]C) at the compression speed of (a) 5 and (b) 40 mm/min.

Caption: FIG. 5. Pressure of polypropylene (MPa) as a function of normal strain rate (I/s) and different compression deformation (k%) at (a) 190[degrees]C and (b) 205[degrees]C.

Caption: FIG. 6. Thermodynamic pressure of polypropylene (MPa) as a function of compression deformation (k%) and different melt temperatures ([degrees]C).

Caption: FIG. 7. Elasticity of polypropylene (MPa) as a function of compression deformation (k%) and different melt temperatures ([degrees]C).

Caption: FIG. 8. Longitudinal bulk viscosity of polypropylene (MPa-s) as a function of normal strain rate (1/s) and different compression deformation (k%) at the melt temperature of (a) 190[degrees]C, (b) 205[degrees]C, (c) enlarged Fig 8(a) at strain rate interval of 0.008-0.020 (Ms) and (d) enlarged Fig. 8(b) at strain rate interval of 0.008-0.024 (1/s).

Caption: FIG. 9. Longitudinal bulk viscosity of HDPE, LDPE from Aleman [7] and polypropylene 1100NK (MPa.s) as a function of strain rate (Ms) at the compression deformation of 5% and melt temperature of 190[degrees]C.

Caption: FIG. 10. Bulk viscosity of polypropylene (MPa.s) as a function of compressional stress (MPa) at the melt temperature of 205 ([degrees]C) and different compression speeds (mm/min).

Caption: FIG. 11. (a) Bulk viscosity as a function of the melt temperature and different deformation percentage at the compression speed of 20 mm/min. (b) logarithmic based 10 of the bulk viscosity and strain rate at the melt temperature of 205[degrees]C and different deformation percentage.

Caption: FIG. 12. Comparison of the empirical model and the experimental data al different normal strains and the melt temperature of (a) 195 [degrees]C and (b) 210 [degrees]C.

TABLE 1. Specification of polypropylene (PP) 1100 NK. Properties Value Supplier MFI (230[degree]C) 11 Melting temperature 130-170 Processing temperature 190-240 Chemical formula [([C.sub.3[H.sub.6]).sub.n] Melting temperature 130-170 Density 0.85-0.95 Water absorption <0.05 Properties Unit Supplier MFI (230[degree]C) g/10 min Melting temperature [degrees]C Processing temperature [degrees]C Chemical formula Melting temperature [degrees]C Density g/[cm.sup.3] Water absorption wt% TABLE 2. MFI and MVR of polypropylene (PP) 1100 NK at various temperature followed ASTM D1238. Measured MFI Measured MVR Temperature ([degrees]C) (g/10 min) ([cm.sup.3]/10 min) 190 6.63 6.99 195 7.53 8.92 200 7.82 9.75 205 8.62 11.25 210 9.04 12.5 230 11.58 17.69 TABLE 3. Experimental conditions. Conditions Value Melt temperature ([degrees]C) 190, 195, 200, 205, 210 Compressional speed (mm/min) 0.5-70 Compression deformation (%) 1-10 TABLE 4. List of data fitted coefficients for second-order bulk viscosity empirical model. Coefficients Value [b.sub.1] -0.1887 (MPa [s.sup.3]) [b.sub.2] 0.7022 (MPa s) [b.sub.3] -0.0017 (MPas/[degrees][C.sup.2]) [b.sub.4] 0.1193 (MPa [s.sup.2]) [b.sub.5] 0.0124 (MPa [s.sup.2]/[degrees]C) [b.sub.6] -0.0138 (MPa s/[degrees]C) [b.sub.7] -1.6066 (MPa [s.sup.2]) [b.sub.8] 3.3302 (MPa s) [b.sub.9] 0.0524 (MPas/[degrees]C) [[eta].sub.0] 524.2110 (MPa s)

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Author: | Chivapornthip, P.; Bohez, E.L.J. |
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Publication: | Polymer Engineering and Science |

Article Type: | Report |

Date: | Aug 1, 2017 |

Words: | 4420 |

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