Investigation of Polymer Long Chain Branching on Film Blowing Process Stability by using Variational Principle ROMAN KOLARIK a,b and MARTIN ZATLOUKAL a,b a Centre of Polymer Systems, University Institute Tomas Bata University in Zlin Nad Ovcirnou 3685, 760 0 Zlin CZECH REPUBLIC b Polymer Centre, Faculty of Technology Tomas Bata University in Zlin TGM 275, 762 72 Zlin CZECH REPUBLIC rkolarik@ft.utb.cz a http://web.uni.utb.cz/, b http://web.ft.utb.cz/ Abstract: This work is focused on a theoretical stability analysis of the film blowing process. The variational principle based film blowing model employing non-isothermal processing conditions, non-newtonian polymer behavior and physically limiting criteria (maximum tensile and/or hoop stress) is applied to investigate the long chain branching effect on film blowing process stability. Key-Words: Mathematical modeling, Variational principle, Stability analysis, Non-isothermal film blowing, Non-Newtonian fluids, Long chain branching, Films. Introduction The film blowing process is characterized by a production of biaxially oriented films of higher quality physical and optical properties which are applicable to daily used products, such as carrier bags, wrapping foils, garbage bags, and also it can be used to scientific balloons and electrolytic membranes production. All these products are made from typical film blowing materials such as polyethylene, polypropylene, polystyrene and polyamide [, 2]. In this process, the polymer melt is extruded through an annular die to a rising continuous tube which is cooled by an air ring and is simultaneously axially stretched and circumferentially inflated by the take-up force, F, and the internal bubble pressure, p, respectively, until the freezeline height is reached. Then, the stable bubble is created, as can be seen in Fig.. The film blowing process is mostly characterized by the following three ratios. First, the blow-up ratio, BUR, is defined as a ratio of the final bubble diameter at the freezeline height to the die diameter. This ratio determines the magnitude of melt stretching in the transverse direction and normally takes values in the range from to 5 [3, 4]. Second, the take-up ratio, TUR, is expressed as a ratio between the film velocity above the freezeline height (nip velocity) and melt velocity at the die exit. This ratio determines the magnitude of melt stretching in the axial machine direction which is usually kept between 5 and 40 [3-4]. Third, the thickness reduction, TR, is the ratio of the die gap to the final film thickness and is typically in the range of 20 to 200 [4]. Fig.. Stable bubble formation. During the film blowing process, some of the below presented bubble instabilities (Fig. 2) can happen and limit a production of the blown film. The most popular bubble stabilization methods consist in proper: set-up of screw and nip rolls speed, adjustment of air ring, height of freeze line, melt temperature, die design and material properties []. ISBN: 978--6804-065-7 74
a) Draw resonance b) Helical instability c) Freezeline height d) Bubble tear instability Fig. 2. Different types of film blowing bubble instabilities. For better understanding of the conditions at which these unwanted instabilities occur, as well as stability and multiplicity analyses, the Pearson-Petrie [5-7] and Cain-Denn [8] formulations are usually employed. However, it has been shown [8] that the Pearson-Petrie model has limited capability for describing a full range of bubble shapes observed experimentally and, equally important, may lead to a variety of numerical instabilities. With the aim to overcome some of these limitations, stable numerical scheme has been recently developed and applied for the variational principle based Zatloukal-Vlcek film blowing model [9-4] (where the stable bubble satisfies minimum energy requirements) considering non-isothermal processing conditions and non-newtonian polymer behavior [5]. The goal of this work is to perform a theoretical stability analysis of the film blowing process by using the Zatloukal-Vlcek film blowing model considering non-isothermal processing conditions, non-newtonian polymer behavior and physically limiting criteria (maximum tensile and/or hoop stress). In more detail, theoretical investigation of the long chain branching effect on the film blowing stability window size will be performed and the obtained results will be discussed with experimental data taken from the open literature. parameter because the membrane is very thin. Then, two bubble shapes can occur. One of them is the bubble shape before deformation (Fig. 3) where this shape is described as a line element of the membrane dx. Consequently, after deformation (Fig. 3), shape of the element is changed and expressed by the following equation 2 y dx 2 y dx. () 2 Thus, in order to describe the bubble shape, the following four parameters have to be known: freezeline height, L, bubble curvature pj (which is given by membrane compliance, J, and the internal load, p), the blow up ratio, BUR, and the die radius, R 0. Important to know is, that the equations describing the freezeline height and also temperature profile have been derived in [5] from the cross-sectionally averaged energy equation [6] where the axial conduction, dissipation, radiation effects and crystallization were neglected. 2 Mathematical Modeling 2. The Zatloukal-Vlcek formulation The Zatloukal-Vlcek formulation [9] describes film blowing process as a stable bubble which is satisfying minimum energy requirements by using variational principles. The bubble is considered as a deformed elastic membrane due to the load, p, and the take-up force, F, where thickness is a neglected Fig. 3. Membrane before deformation (left) and after deformation (right). The whole derivation of the Zatloukal-Vlcek formulation is described in detail in [9]. Further, the non-isothermal conditions and non-newtonian polymer behavior are detailed described in [5, 7, ISBN: 978--6804-065-7 75
8]. Finally, it should be summarized, that in these works as well as in this study, the following model assumptions are included: the process is non-isothermal, the polymer melt is considered as the non-newtonian fluid, heat transfer coefficient along the bubble is not varying, velocity profile along the bubble is non-linear, creation of high stalk bubble is not allowed and occurrence of the crystallization is not taken to account. 2.2 Stability Contours Determination In this work, the stability processing window is defined as the closed area in the figure at which the relative final film thickness, H /H 0, (film thickness at the freezeline height divided by the film thickness at the die) is plotted as the function of the blow-up ratio, BUR (see Fig. 4 as the example). In this work, the film blowing process is viewed as an unstable process when the process does not satisfy minimum energy requirements [9] or if the film stresses in machine or circumference direction reach the rupture stress. Then, a typical stability processing window for a particular internal bubble pressure range p p ; p can be formed. min max Rt x h x 33 x p x (3) h x R x m where R m and R t are the curvature radius in the machine and transverse direction, respectively. 2.2 Numerical Scheme Firstly, it is necessary to create a regular grid of H /H 0 versus BUR with equidistant steps in both variables. Thus, the basic film blowing variables are calculated for all grid nodes according to the numerical scheme which is presented in Fig 5. The whole computing procedure is described in detail in [5]. Then, a continuous field of the given film blowing variables is determined by using linear interpolation between grid nodes. Finally, the film blowing stability window is then generated on the H /H 0 versus BUR mesh for a given internal bubble pressure range by all grid nodes for which the energy based stability limit and/or the rupture stress in the machine/circumferential film direction is reached. Just note that if the energy based stability limit is reached, the film blowing process is viewed as unstable because it does not satisfy the minimum energy requirements [9]. H /H 0 0. 0.0 0.00 STABLE AREA UNSTABLE AREA 3 Results and Discussion The film blowing model developed in [9, 5, 7] and briefly introduced in the previous section has been applied to investigate the effect of extensional viscosity, E, (characterized by the extensional strain hardening parameter,, included in the constitutive equation [7]) on the film blowing process stability. For this theoretical stability analysis, the particular processing/material parameters, summarized in Tables -3, taken from Tas s Ph.D. thesis [9] for LDPE material (L8 experiment No. 23), were applied. 0.000 0 2 3 4 5 6 7 8 9 BUR Fig. 4. A typical stability processing window defined by the Zatloukal-Vlcek model for fixed freezeline height, i.e. heat transfer coefficient is a calculated parameter. It also should be mentioned that the machine as well as circumferential stresses are calculated according to the Pearson and Petrie formulation [5] as 2 2 F p r f r x x (2) 2 r x h x cos x Table. The Zatloukal/Vlcek film blowing model parameters. L p min m R 0 H 0 max A (m) (Pa) (kg. s - ) (m) (m) (MPa) (-) 0. 0 0.00 0.078 0.0022 - Table 2. Generalized Newtonian constitutive equation parameters (A =, = 20). 0 λ a n α β (Pa. s) (s) (-) (-) (s) (-) (-) 2,365 0.72 0.76 0.37. 0-5 9.2. 0-7 0.0544 Table 3. Temperature parameters. T air T solid T die T r E a R C p ( C) ( C) ( C) ( C) (J. mol - ) (J. K -. mol - ) (J. kg -. K - ) 25 92 45 90 59,000 8.34 2,300 ISBN: 978--6804-065-7 76
been done. Then, predicted effect of the extensional viscosity on the film blowing stability window wideness is presented in Fig.7. As can be seen, by comparing the uniaxial extensional viscosity curves (Fig.6) and corresponding calculated film blowing stability contours (Fig.7), the relationship between the level of long chain branching and the film blowing stability window size has non-monotonic character, i.e. there is an existence of some optimum value of the extensional viscosity to reach maximum stability processing window. This non-monotonic trend is more obvious in the Fig.8 where stability window size is quantified as an area of each curve. It should be mentioned here, that in order to evaluate the optimum value of long chain branching, the film blowing stability window size has been calculated from 0 to 0.08 in step 0.005 of the extensional strain hardening parameter,. Thus, as can be seen in Fig.8, to achieve an optimum processing window, the extensional strain hardening parameter,, has to be varied from 0.05 to 0.030. Interestingly, the calculated results depicted in Fig.9 suggest that the relationship between the extensional strain hardening parameter and minimum achievable film thickness during the film blowing process has no monotonic character. Then, the optimal value of the minimum film thickness is reached for the extensional strain hardening parameter equal to 0.03. On the other hand, in the case of maximum attainable film thickness, the extensional strain hardening parameter has no effect. 0 5 Fig. 5. Iteration scheme of the Zatloukal-Vlcek model applied for the film blowing stability analysis. The Figs. 6-0 summarize results of the numerical film blowing stability. It should be mentioned that, stability window wideness is characterized here as its area, which is calculated for each value of the investigated parameter,. In Fig. 6., it is clear, that change in the long chain branching level, characterized by the extensional strain hardening parameter,, of the generalized Newtonian model, leads to different shapes of the uniaxial extensional viscosity curves. Thus, for these five virtual materials (described in Table 2, differing in long chain branching level only) the film blowing process stability analysis has Shear and uniaxial extensional viscosities (Pa. s) 0 4 0 3 Uniaxial extensional viscosity, = 0 Uniaxial extensional viscosity, = 0.02 Uniaxial extensional viscosity, = 0.04 Uniaxial extensional viscosity, = 0.06 Uniaxial extensional viscosity, = 0.08 Shear viscosity 0 2 0-3 0-2 0-0 0 0 0 2 0 3 Shear and extensional strain rates (/s) Fig. 6. Shear and uniaxial extensional viscosity prediction of the generalized Newtonian model for five virtual materials varying in different values of the extensional strain hardening parameter, ζ. ISBN: 978--6804-065-7 77
0 H /H 0 (-) 0. 0.0 0.00 STABLE = 0 = 0.0 = 0.02 = 0.04 = 0.06 = 0.08 Relative final film thickness, H /H 0 0. 0.0 0.00 Maximum achievable final thickness Minimum achievable final thickness UNSTABLE 0.000 0 2 3 4 5 6 7 8 9 BUR (-) Fig. 7. Predicted film blowing stability window shape for different levels of the extensional strain hardening parameter, ζ. Film blowing stability window size (-) 4 3.8 3.880 3.6 3.4 3.2 3.007 3 2.8 2.6 2.4 2.2 2.28 2.872.8.6.4.27.2 0.8 0.6 0.4 0.2 0 0 0.0 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 Extensional strain hardening parameter, (-) Fig. 8. Predicted film blowing stability window size for different levels of the extensional strain hardening parameter, ζ. In this work, also investigation of rupture stress effect on the film blowing process stability has been performed. It is clear (see Fig. 0), that increasing rupture stress leads to increasing processing window which is in good agreement with the experimental work of Micic and Bhattacharya [20]. It also should be mentioned, that at very low levels of strain hardening, characterized by the extensional strain hardening parameter, (lower than 0.05), the film blowing processing window is surprisingly bigger for lower rupture stress values than for higher values. 0.000 0 0.0 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 Extensional strain hardening parameter, (-) Fig. 9. Predicted maximum/minimum attainable film thickness for different levels of the extensional strain hardening parameter, ζ. Film blowing stability window size (-) 5 4.5 4 3.5 3 2.5 2.5 0.5 =.0 MPa = 2.0 MPa = 3.0 MPa 0 0 0.0 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 Extensional strain hardening parameter, (-) Fig. 0. Effect of the extensional strain hardening parameter ζ on the predicted film blowing stability window size for three investigated rupture stress values. 4 Conclusion In this work, the Zatloukal-Vlcek film blowing model based on variational principle, extended for the non-isothermal processing conditions and non-newtonian polymer behavior, together with the presented stable numerical scheme has been successfully applied to investigate the effect of extensional viscosity on the film blowing process stability. It has been found that certain range of long ISBN: 978--6804-065-7 78
chain branching level exists to achieve an optimal film blowing processing window. Further, investigation of rupture stress effect on process stability has been successfully compared with the open literature, i.e. increasing rupture stress value leads to increasing processing window. Thus, this model can be considered as the useful tool for the film blowing process optimization. Acknowledgement The authors wish to acknowledge the Grant Agency of the Czech Republic (grant No. P08/0/325) and Operational Program Research and Development for Innovations co-funded by the European Regional Development Fund (ERDF) and national budget of Czech Republic, within the framework of project Centre of Polymer Systems (reg. number: CZ..05/2..00/03.0) for the financial support. References: [] Cantor, K., Blown Film Extrusion, Carl Hanser Verlag, 2006. [2] Butler, T. I., Film extrusion Manual: Process, Materials, Properties, Tappi press, 2005. [3] Baird, D. G., Collias, D. I., Polymer Processing: Principles and Design, John Wiley & Sons, Inc., 998. [4] Beaulne, M., Mitsoulis, E., Effect of Viscoelasticity in the Film-Blowing Process, J. Appl. Polym. Sci., Vol.05, 2007, pp. 2098-22. [5] Pearson, J. R. A., Petrie, C. J. S., The Flow of a Tubular Film. Part. Formal Mathematical Representation, J. Fluid. Mech., Vol.40, No., 970, pp. -9 [6] Pearson, J. R. A., Petrie, C. J. S., The Flow of a Tubular Film. Part 2. Interpretation of the Model and Discussion of Solutions, J. Fluid Mech., Vol.42, No.3, 970, pp. 609-625. [7] Pearson, J. R. A., Petrie, C. J. S., A Fluid- Mechanical Analysis of the Film-Blowing Process, Plast. Polymer, Vol.38, 970, pp. 85-94. [8] Cain, J. J., Denn, M. M., Multiplicities and Instabilities in Film Blowing, Polym. Eng. Sci., Vol.28, No.23, 988, pp. 527-54. [9] Zatloukal, M., Vlcek, J., Modeling of the Film Blowing Process by Using Variational Principles, J. Non-Newtonian Fluid Mech., Vol.23, 2004, pp. 20-23. [0] Zatloukal, M., Vlcek, J., Application of Variational Principles in Modeling of the Film Blowing Process for High Stalk Bubbles, J. Non-Newtonian Fluid Mech., Vol.33, 2006, pp. 63-72. [] Zatloukal, M., Vlcek, J., Sáha, P., Modeling of the Film Blowing Process by using Variational Principles, Annual Technical Conference - ANTEC, Conference Proceedings, Vol., 2004, pp. 235-239. [2] Zatloukal, M., Vlcek, Modeling of the Film Blowing Process for High Stalk Bubbles, Annual Technical Conference - ANTEC, Conference Proceedings, Vol., 2005, pp. 39-42. [3] Zatloukal, M., Mavridis, H., Vlcek, J., Saha, P., Modeling of Non-Isothermal Film Blowing Process by using Variational Principles, Annual Technical Conference - ANTEC, Conference Proceedings, Vol.2, 2006, pp. 825-829. [4] Zatloukal, M., Mavridis, H., Vlcek, J., Saha, P., Modeling of Non-Isothermal High Stalk Film Blowing Process by using Variational Principles for Non-Newtonian Fluids, Annual Technical Conference - ANTEC, Conference Proceedings, Vol.3, 2007, pp. 57-575. [5] Kolarik, R., Zatloukal, M., Modeling of Nonisothermal Film Blowing Process for Non-Newtonian Fluids by Using Variational Principles, J. Appl. Polym. Sci., Vol.22, 20, pp. 2807-2820. [6] Doufas, A. K., McHugh, A. J., Simulation of Film Blowing Including Flow-Induced Crystallization, J. Rheol., Vol.45, No.5, 200, pp. 085-04. [7] Zatloukal, M., A Simple Phenomenological Non-Newtonian Fluid Model, J. Non- Newtonian Fluid Mech., Vol.65, 200, pp. 592-595. [8] Kolarik, R., Zatloukal, M., Variational Principle Based Stability Analysis of Non-isothermal Film Blowing Process for Non-Newtonian Fluids, AIP Conference Proceedings, Vol.375, 20, pp. 56-74. [9] Tas, P. P., Film Blowing from Polymer to Product, Ph.D. Thesis, Technische Universitat Eindhoven, 994. [20] Micic, P., Bhattacharya, S. N., Transient Elongational Viscosity of LLDPE/LDPE Blends and Its Relevance to Bubble Stability in the Film Blowing Process, Polym. Eng. Sci., Vol.38, No.0, 998, pp. 685-693. ISBN: 978--6804-065-7 79