ON THE ROLE OF EXTENSIONAL RHEOLOGY, ELASTICITY AND DEBORAH NUMBER ON NECK-IN PHENOMENON DURING FLAT FILM PRODUCTION
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1 ON THE ROLE OF EXTENSIONAL RHEOLOGY, ELASTICITY AND DEBORAH NUMBER ON NECK-IN PHENOMENON DURING FLAT FILM PRODUCTION Martin Zatloukal 1, Tomas Barborik 1 and Costas Tzoganakis 2 1 Polymer Centre, Faculty of Technology, Tomas Bata University in Zlin, Vavreckova 275, 76 1 Zlin, Czech Republic 2 Department of Chemical Engineering, University of Waterloo, 2 University Avenue West, Waterloo, Ontario, Canada Abstract In this work, viscoelastic, isothermal extrusion film casting modeling utilizing 1D membrane model and modified Leonov model was performed in order to understand the role of planar and uniaxial extensional viscosities, extensional strain hardening, Deborah number and die exit stress state (captured here via the second to first normal stress difference ratio N 2 /N 1 ). It has been found that the neck-in can be expressed via simple set of dimensionless analytical equations utilizing all above mentioned variables, and thus providing detail view into complicated relationship between polymer melt rheology, die design, process conditions and unwanted neck-in phenomenon. Introduction Extrusion film casting is a continuous, high-speed manufacturing process during which a thin, highly oriented polymer film is produced [1], [2]. In this widely used and dominant process, a molten polymer is extruded through a slit die with a narrow gap to form a thick sheet of polymer that is subsequently intensively stretched in the machine direction by means of rotating take-up drum, which linear velocity is higher than extrusion one providing the macromolecular orientation and decrease in the sheet thickness. Finally, the dimensions of this just created thin film are fixed by cooling down on a chill roll. The intensity of film drawing is usually measured by means of draw ratio that is defined as velocity at the takeup drum divided by polymer exit velocity at the extrusion die. Under certain processing conditions, several phenomena that hamper the production in film quality and quantity requested may be encountered. One of them is called neck-in causing the undesirable reduction in the film width and interrelated phenomenon of edge-beads (also called dog-bone defect) making the edge portions of the film substantially thicker than its central part (see Figure 1). Both phenomena result in needs for postproduction film trimming since only central part of the film features uniformity in thickness. From the practical viewpoint, it is of great importance, therefore, to understand the underlying mechanism of neck-in formation in order to minimize these unwanted effects. Figure 1. Neck-in and edge-beads phenomena during extrusion film casting process. There are many key findings with respect to neck-in phenomenon, which are available in the open literature. Dobroth and Erwin [3] pointed out that the deformation flow in the drawing length comprises from two related regions and the extent of edge-beads and interrelated neck-in phenomenon is determined by interplay between them by an edge stress effect. While the center of the film undergoes planar extensional deformation flow, the edge sections are subjected to uniaxial extensional one (see Figure 2). Figure 2. Visualization of planar and uniaxial extensional flows during extrusion film casting process. SPE ANTEC Anaheim 217 / 1131
2 Some authors endeavored to relate and quantify the gauge of the observed necking in terms of rheological parameters, such as shear, uniaxial and planar viscosity. Many authors reported that the strain hardening in uniaxial extensional viscosity may depress the extent of necking phenomena [4] [6]. This idea was continued by Ito [7], who related the neck-in extent to rheological parameters as ratio of planar viscosities in axial and transverse direction, and derived an analytical equation for the edge line of a molten film of Newtonian and Maxwell fluid. Based on the lines of Dobroth and Erwin s [3] article, who as the first recognized deformation flow type in the drawing area, Shiromoto [8], recently, drafted the idea that the extent of the necking should not have been described by uniaxial extensional viscosity only in addition to take-up length but as the ratio of planar and uniaxial extensional viscosity reflecting the deformation flow type in the central and edge portion of the film in drawing section. In more detail, the following simple relationship between neck-in, NI, and planar to uniaxial extensional viscosity ratio has recently been found [9], [1]: 1 / 2 1 E, P NI W D WF X (1) 2 E, U where W D represents the die exit width, W F the final film width, respectively, X is distance between the flat die and the chill/nip rolls and <η E,P /η E,U > is averaged planar to uniaxial extensional viscosity ratio. Just recently, it has been revealed that die exit stress state (characterized by N 2 /N 1 ratio) increases in the neck-in phenomenon for polymer melts with high relaxation time [11], [12]. The aim of this paper is generalization of Eq. 1 considering the effect die exit stress state, Deborah number and extensional strain hardening effect via process modeling. X Modeling In this work, the one-dimensional membrane model developed by Silagy et al. [13] was used to model isothermal extrusion film casting process. The model essentially features two hypotheses to facilitate the description of the stress and velocity field development in the drawing film. In the membrane hypothesis, where the one of the dimensions of the solved domain is small compared to the other dimensions, is possible to assume that the total stress in this direction is equaled to zero. This dimension is, in this case, the film thickness. Further used supplementary kinematic hypothesis, formerly adopted in the work of Narayanaswamy [14] for the case of glass manufacturing by the float process, is helpful in reducing dimensionality of the task. Although, the model can be used for modeling lateral film width and thickness reduction, it is considered as one-dimensional model in sense of variables that are dependent on x only. The whole system of the first-order ordinary differential equations was then numerically solved by the fourth order Runge- Kutta method incorporating the shooting method. The input parameters were updated in each iteration step to meet the appropriate solution in terms of DR. It was preferred to develop the solver itself in the C++ programming language, to avoid a black box effect, which could have appeared in the case of using an inbuilt solver in any other commercial mathematical-modeling software. The equation evaluation was performed on the PC with the following hardware parameters: CPU Intel Core 2 Quad Q965 (3. GHz); RAM 8 GB DDR2; GPU Sapphire Radeon HD 387; SSD Crucial 256 GB. Typical computational time for one calculation was about 2 minutes. In this work, modified Leonov model, which relates the stress and elastic strain stored in the polymer melt, was used as the constitutive equation as: W 1 W 2 c c (2) I1 I 2 where is the stress, and W, the elastic potential, which depends on the invariants I 1 and I 2 of the recoverable Finger tensor c, W 2 n 3 1 n 1 G I 1 I n (3) where G denotes linear Hookean elastic modulus, and n are numerical parameters. In this work, the Mooney potential (i.e. n = in Eq. 3) and the following dissipation function b proposed in [15], have been employed. 1 sinh I1 3 b I1 exp I1 3 4 I 31 (4) 1 In our previous work [11], [12], the utilized film casting model was validated by using experimental data for LDPE, taken from [16], [17], and model predictions based on the single-mode modified Leonov model (this work) were compared with multi-mode XPP model. It has been found, that behavior of both models is comparable. Such a reasonably good agreement with the experimental data is believed to justify utilization of the single-mode Leonov model for detailed investigation of the neck-in phenomenon. Theoretical analysis of neck-in phenomenon In order to understand the neck-in phenomenon in more detail, uniaxial/planar extensional viscosity (see Figure 3), Deborah number, De, (from.1 up to.1) and die exit stress state N 2 /N 1 (from.1 up to 2.), see Figure 4, were systematically varied and for each individual case, the neck-in was recorded. It has been found that within the investigated range of given parameters, the normalized neck-in can be described by the following set of simple dimensionless analytical equations: SPE ANTEC Anaheim 217 / 1132
3 NI 1 X 1 u De X max (5) 3 E,P 6 1 exp 2De exp 5 De N (6) E,U max 2 E,U max (7) N1 3 where X is the stretching distance (take-up length) represents the maximum value in the steady uniaxial extensional viscosity, E, P and represents planar and uniaxial extensional viscosity (at the average extensional strain rate), respectively, is the Newtonian viscosity, λ is the relaxation time, N 1 is the first normal stress difference at the die exit, N 2 is the second normal stress difference at the die exit, u is the velocity in the machine direction at the die exit, α 1 =.553, α 4 =.512, α 2 = , α 5 =66.712, α 7 =-.43, α 9 =.19, α 3 =2.323, α 6 =1.87, α 8 =-.5, α 1 =1.22. Here, the alpha parameters can be considered as universal for the considered range of extensional strain hardening, Deborah number and N 2 /N 1 ratio. Based on the Figure 4 it can be stated that the analytical model given by Eqs. 5-7 (lines) is capable to represents neck-in predictions of the utilized 1D viscoelastic membrane model predictions (symbols) very well. It is important to mention that utilization of Eqs. 5-7 requires experimental determination of planar to uniaxial extensional viscosity ratio, which is one of the most challenging rheological task because generation and control of the extensional flow is difficult. Just recently, it has been showed that planar and uniaxial extensional viscosity can be measured in wide temperature and deformation rate range by using standard twin bore capillary rheometer, novel rectangle and circular orifice (zero-length) dies and Cogswell model [18]. Conclusion In this work, viscoelastic, isothermal extrusion film casting modeling utilizing 1D membrane model and single-mode modified Leonov model was performed in order to understand the role of viscoelastic stress state at the die exit, extensional viscosity and elasticity on the neck-in phenomenon. It has been found that the neck-in can be expressed via simple set of dimensionless analytical equations utilizing planar to uniaxial E,P extensional viscosity ratio,, uniaxial extensional strain hardening parameter, 3 max max, Deborah number, De, and die exit stress state, N 2 /N 1. It is believed that this model can be used for material, die design and process conditions optimization in order to minimize unwanted neck-in phenomenon in the film production. Acknowledgments The authors wish to acknowledge Grant Agency of the Czech Republic (Grant registration No S, Title: Investigation the effect of polymer melt shear and elongational rheology on production stability of meltblown nanofibers and films) for the financial support. References 1. T. Kanai and G. A. Campbell, Film Processing Advances (Carl Hanser Verlag GmbH & Co. KG., München, 214), T. Kanai and G. A. Campbell, Film Processing (Hanser Publishers, 1999), T. Dobroth and L. Erwin, Polym. Eng. Sci., 26(7), , (1986). 4. B. Debbaut, J. M. Marchal, and M. J. Crochet, Z. Angew. Math. Phys., 46(SPEC. ISSUE), , (1995). 5. N. Toft and M. Rigdahl, Int. Polym. Process., 17(3), , (22). 6. S. Kouda, Polym. Eng. Sci., 48(6), , (28). 7. H. Ito, M. Doi, T. Isaki, and M. Takeo, J. Soc. Rheol. Japan, 31(3), , (23). 8. S. Shiromoto, Y. Masutani, M. Tsutsubuchi, Y. Togawa, and T. Kajiwara, Polym. Eng. Sci., 5(1), 22 31, (21). 9. S. Shiromoto, Y. Masutani, M. Tsutsubuchi, Y. Togawa, and T. Kajiwara, Rheol. Acta, 49(7), , (21). 1. S. Shiromoto, Int. Polym. Process., 29(2), , (214). 11. T. Barborik and M. Zatloukal, AIP Conference Proceedings, 1662, , (215). 12. M. Zatloukal and T. Barborik, Annual Technical Conference - ANTEC, Conference Proceedings, , (216). 13. D. Silagy, Y. Demay, and J. F. Agassant, Polym. Eng. Sci., 36(21), , (1996). 14. O. S. Narayanaswamy, J. Am. Ceram. Soc., 6(1 2), 1 5, (1977). 15. M. Zatloukal, J. Nonnewton. Fluid Mech., 113(2 3), , (23). 16. H. V. Pol, S. S. Thete, P. Doshi, and A. K. Lele, J. Rheol., 57(2), , (213). 17. H. Pol, S. Banik, L. B. Azad, S. S. Thete, P. Doshi, and A. Lele, Rheol. Acta, 53(1), 85 11, (214). 18. M. Zatloukal, Polymer, 14, (216). SPE ANTEC Anaheim 217 / 1133
4 Figure 3. Uniaxial and planar extensional viscosities of different virtual polymer melts utilized in this work having high (top), medium (middle) and low (bottom) level of extensional strain hardening. SPE ANTEC Anaheim 217 / 1134
5 Figure 4. The effect of Deborah number (left) and die exit stress state N 2 /N 1 (right) on the normalized neck-in vs. planar to uniaxial extensional viscosity ratio for virtual polymer melts having high (top), medium (middle) and low (bottom) level of extensional strain hardening. Here, symbols and lines represent utilized viscoelastic 1D membrane model and simple model (Eqs. 5-7) predictions, respectively. SPE ANTEC Anaheim 217 / 1135
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