Influence of molecular parameters on material processability in extrusion processes.
Extrusion is one of the most versatile polymer processing methods. Therefore, not surprisingly, considerable efforts have been made to increase its efficiency, especially in terms of productivity and energy use. It is well known that the production rates, the energy consumption and product quality are strongly influenced not only by equipment design, but also by material parameters such as molecular weight and polydispersity. Polymer "processability," as it is usually referred to in industry, will also affect the processing conditions when using a particular machine. When changing materials, trial-and-error experimental procedures are usually used to determine an optimum set of operating conditions. Very often, such procedures are tedious and highly expensive.
An alternative approach to establish some correlation between material properties and operating conditions is through the use of numerical simulations. Such simulations can vary in their degree of complexity, depending on the number of simplifying assumptions invoked. Take the example of a single screw melt extruder. The body of literature on flow analysis in single screw extruders is huge and is well refereed in textbooks (1-3). Most of the published literature on flow in single screw extruders is based on either the Newtonian fluid or, at the next level of complexity, on the Generalized Newtonian Fluid. Constitutive equations describing viscoelastic flow phenomena are generally numerically insoluble in multidimensional flows. Yet, another drawback of the existing literature is the lack of an explicit formulation of the flow patterns in terms of material parameters such as molecular weight or molecular weight distribution. This last problem is solely based on the lack of a constitutive model which can express the influence of material parameters on the polymer melt flow curve, yet is numerically tractable in complex 3D flow geometries.
Recently, we have developed a simple Generalized Newtonian Fluid model to predict shear-thinning behavior of linear, polydisperse polymeric systems (4). The model, based on a superposition principle, allows direct correlation between molecular weight distributions and shear-thinning flow behavior of polymeric melts, yet is simple enough to be used in complex 3D flow simulations.
In this work we make use of this model to analyze the influence of material parameters, such as molecular weight and degree of polydispersity on the operating point, power consumption and residence time distribution in a single screw extruder.
Details for the derivation of the viscosity model used in the flow simulations can be found in a previous paper (4). Here we discuss only the most salient results.
We start with a viscosity model for monodisperse systems that describes a shear-thinning behavior of the melt within a range of shear rates between a lower limit, [Mathematical Expression Omitted] and an upper limit, [Mathematical Expression Omitted]. The flow behavior of the polymer is Newtonian at shear rates outside this range. At the lower end of shear rates, i.e. [Mathematical Expression Omitted], the Newtonian zero-shear viscosity, [[Eta].sub.0], is related to the polymer molecular weight, M, through a power-law model, whereas at the upper end of shear rates, i.e. [Mathematical Expression Omitted], the viscosity, termed as [[Eta].sub.[infinity]], depends exclusively on the polymer critical molecular weight for entanglements, [M.sub.e]. The constitutive model for monodisperse systems can be written as:
[Mathematical Expression Omitted] (1)
where the parameter n is related to disentanglement phenomena and was found to satisfy the relationship n = 1/[Alpha] and [[Phi].sub.e] is an interaction coefficient, reflective of the friction force between the polymer molecules. Using the assumption that for each polymer homologous series, the critical shear stress for the onset of non-Newtonian behavior is a constant, independent of the molecular weight (5), a relationship between the interaction coefficient and molecular weight can be derived:
[Mathematical Expression Omitted] (2)
In Eq 2 [[Phi].sub.0] is defined as an interaction coefficient per entanglement and will depend on the physical chemical nature of the polymer.
In the case of a polydisperse system, we view its rheological behavior as a superposition of contributions from all monodisperse fractions present in the system. Taken the example of a polydisperse system whose molecular weight distribution can be described in terms of a log-normal function, the Wesslau equation can be used to specify the MWD:
[Omega](M) = 1 / [-square root of 2[Pi]] [multiplied by] M [multiplied by] [Sigma] [multiplied by] exp [- 1/2 [(ln(M) - ln([m.sub.0]) / [Sigma]).sup.2]] (3)
where [m.sub.0] characterizes the location of the distribution and the parameter [Sigma] defines the breadth of the distribution.
Equation 1 can be generalized for a system characterized by a continuous function for the molecular weight distribution by considering a mapping between molecular weight and shear rate. The molecular weight of a monodisperse fraction can be calculated as:
[Mathematical Expression Omitted] (4)
For such a system the viscosity can be written as:
[Mathematical Expression Omitted] (5)
Parameters A, B and C are introduced for notation simplicity and are given in Eqs 6-8, [Mathematical Expression Omitted] is the weight average molecular weight and [Mathematical Expression Omitted] is the polydispersity index.
[Mathematical Expression Omitted] (6)
[Mathematical Expression Omitted] (7)
[Mathematical Expression Omitted] (8)
Figure 1 illustrates the effect of polydispersity on the viscosity of the system. The plots are presented in a dimensionless form showing the dependence of [Mathematical Expression Omitted] on [Mathematical Expression Omitted].
The complex geometries encountered in most of the polymer processing equipment make impossible an analytical solution for the flow problem. In this work a 3D numerical analysis was carried out for the flow field in a single screw extruder with the following geometrical features: diameter of the barrel = 9.0 cm: diameter of the screw at the top of the flight = 8.975 cm; diameter of the screw root = 5 cm; flight width = 0.543 cm; pitch = 12 cm.
The equations of continuity and motion for the steady-state, isothermal flow of an incompressible fluid:
[Mathematical Expression Omitted] (9)
[Mathematical Expression Omitted] (10)
were solved using no-slip boundary conditions on the screw and barrel surfaces and a nominal value for the normal stress difference in the axial direction (the traction was applied normal to the mesh face). We also used a coordinate system moving in the axial direction toward the die with a constant velocity [U.sub.0] = [L.sub.p]N where [L.sub.p] is the pitch of the screw and N is the screw rotational speed. In all simulations we used a filling factor of 1 and a variable rotational speed (30, 60, or 120 rpm).
A fluid dynamics analysis package - FIDAP (6) using the finite element method was employed to simulate the flow patterns. The mesh, shown in Fig. 2, was built of 4530 brick elements and has 5880 nodal points.
The materials used in the study were samples of polystyrene (PS) with average molecular weights [Mathematical Expression Omitted] = 150,000-400,000 and various degrees of polydispersity, I = 1.05-8. The viscosity model described in the previous section was used to describe the rheological behavior of the system. The simulations were carried out on a CRAY-YMP supercomputer at Ohio Supercomputer Center.
We solved for the velocity and pressure profiles and calculated the screw characteristic curves (plots of output, Q versus the pressure rise, [Delta]P). The power consumption for melt conveying, [Mathematical Expression Omitted], was calculated according to:
[Mathematical Expression Omitted] (11)
where S is the surface area of the screw, [Mathematical Expression Omitted] is a normal unit vector to the surface, [Mathematical Expression Omitted] is the stress tensor at the screw surface and [Mathematical Expression Omitted] is the screw velocity.
Residence time distributions were measured by following the motion of a statistically significant number of massless tracers in the system. Since the flow field is completely deterministic, the locations of the particles at any time can be found by integrating the velocity vectors with respect to time along the particle pathlines. A total of 5000 tracers, randomly distributed across the inlet plane, were followed during their motion through the single screw extruder for a total length of 1 pitch. The cumulative residence time distribution function, F(t), was calculated considering the fraction of exiting particles with a residence time less or equal to
Finally we looked at the operating points for a 2 pitch extruder-die combination. We chose an annular die and varied its flow conductance by changing the internal diameter. The die geometrical specifications are: internal diameter [D.sub.i] = 2-3.2 cm; external diameter [D.sub.0] = 4 cm; die length [L.sub.D] = 5 cm. Die characteristics, plotting the output versus the pressure drop over the die, were obtained by the same finite element method and intersected with the screw characteristics in order to determine the operating points for different systems.
RESULTS AND DISCUSSION
We evaluated the screw characteristics curves for five monodisperse samples of PS of molecular weights 1.5 x [10.sup.5], 2.0 x [10.sup.5], 2.5 x [10.sup.5], 3.0 x [10.sup.5], and 4.0 x [10.sup.5] respectively. The screw characteristics obtained at a rotational speed of 60 rpm were plotted as solid lines in Fig. 3. Also shown in Fig. 3 are die characteristics obtained for the same samples using an annular die with a [D.sub.i]/[D.sub.0] ratio of 0.75 (dashed lines). It is interesting to note that pressure at the working points (intersecting points between screw and die characteristics) shows a linear dependence on molecular weight. Figure 4 shows this dependence for dies of various [D.sub.i]/[D.sub.0] ratios. As the "flow conductance" of the die decreases (thinner annulus) an expected increase in the slope is observed.
We also investigated the influence of material polydispersity on the working point for a given combination extruder-die. We selected 8 different samples of PS with the same average molecular weight (3.0 x [10.sup.5]) but different degrees of polydispersity (the index I in a range from 1 to 8). Figure 5 plots the screw characteristics (solid lines) for the case of an annular die with [D.sub.i]/[D.sub.0] = 0.75 and a rotational speed of 60 rpm. Samples with broader molecular weight distributions (higher values for the polydispersity index) require less pressure difference and consequently less power consumption for processing. Indeed, the power consumption calculated at the working points for the different materials investigated decreases with material polydispersity and increases, as expected, with the average molecular weight [ILLUSTRATION FOR FIGURE 6 OMITTED]. It is interesting to note that the power consumption varies linearly with the sample molecular weight, but exponentially with the degree of polydispersity.
On the other hand, for the same rotational speed, samples with narrower molecular weight distributions will generate higher output values. Figure 7 displays plots of output (at working point, using a die with [D.sub.i]/[D.sub.0] = 0.75) versus rotational speed for two extreme cases, i.e. I = 1.05 and I = 8. The output is 20% higher for the narrow molecular weight distribution sample. Similar results were reported experimentally (7, 8).
Material parameters will affect not only processability, but product quality. Although this subject is beyond the limits of our study, we decided to look at one additional aspect of material processing which can be closer related to product quality. If one considers residence time distribution in the equipment, important information on potential material degradation or degree of chemical conversion (if the equipment is used as a continuous chemical reactor) can be obtained. In this work we calculated cumulative residence time distributions for 3 different materials, changing molecular weight and the degree of polydispersity. The results obtained are shown in Fig. 8 and they seem to indicate that the material rheology does not affect the shape of the residence time distributions.
The results obtained in this work can be generalized in order to obtain windows of processing conditions for resins with different properties and/or blends of homologous materials. Taking the example of an extruder with a given die attached, one can draw a window of the working points for materials with different average molecular weights or degrees of polydispersity. Such a window is displayed in Fig. 9 for PS resins with molecular weights in the range 1.5 x [10.sup.5] - 4.0 x [10.sup.5] and degrees of polydispersity [Mathematical Expression Omitted] between 1 and 8. The example presented is for the single screw extruder operated at 60 rpm with an annular die of [D.sub.i]/[D.sub.0] = 0.75 attached. Similar windows can be obtained for other processing conditions.
On a final note, we looked at the curves of operating points when blending resins with different molecular parameters. We choose two examples. In the first example we blend 50% by weight a resin with [Mathematical Expression Omitted] and a degree of polydispersity I = 2, with resins with average molecular weight in a range between 5 x [10.sup.4] - 5.5 x [10.sup.5]. All polymers have the same degree of polydispersity, I = 2. The working points for the different blends are located on curve A within the processing window [ILLUSTRATION FOR FIGURE 10 OMITTED].
In the second example, we blend in various proportions two resins of the same degree of polydispersity (I = 2) but different average molecular weights (1.5 x [10.sup.5] and 4.0 x [10.sup.5]). The working points are located on the curve B within the processing window. Details for the procedure used in calculating the rheology of the blends are given in Appendix A. Using a similar approach, one can define the working point for any blend obtained by mixing homologous polymers, as long as one can determine the average molecular weight and degree of polydispersity for the mixture.
SUMMARY AND CONCLUSIONS
In this work we analyzed the influence of material parameters such as molecular weight and molecular weight distribution on resin processability in extrusion processes. The example taken does not make reference to the solid-conveying or melting sections of the extruder and considers only the metering zone of a single screw extruder with an annular die attached. The rheological behavior of the fluid was described using a model which explicitly correlates the system viscosity with molecular weight and degree of polydispersity. A fluid dynamics analysis package, FIDAP, based on the finite element method was employed in the isothermal flow simulations.
The results obtained demonstrate that the molecular weight and material polydispersity impact on the working point and power consumption in processing. Materials with broader molecular weight distributions require less pressure and consequently less power consumption for processing. On the other hand, for the same rotational speed, samples with narrower molecular weight distributions will generate higher output values. Increasing the average molecular weight, while maintaining the same degree of material polydispersity, will increase output and power consumption. We found that power consumption increases linearly with molecular weight and decreases exponentially with degree of polydispersity. Residence time distributions, when expressed in terms of a dimensionless time (time made dimensionless on the average residence time in the system) are not affected by material properties.
The model presented in this work can be used to develop windows of processing conditions for resins of different material properties. Within such windows, one can also analyze the effect of blending homologous polymers on material processability. The work herein uses the example of polystyrene resins processed in single screw extruder. A similar approach can be employed for other materials, such as polyolefins, and other processing equipment, for example, twin-screw extruders.
The molecular weight distribution for a blend can be easily obtained by using a linear combination of two monomodal Log-Normal distribution functions, i.e.:
[Mathematical Expression Omitted] (A1)
where a is a superposition parameter and [Omega].sub.1] and are the distributions for the two fractions. The average molecular weight and the polydispersity for a blend can be written as:
[Mathematical Expression Omitted] (A2)
[Mathematical Expression Omitted] (A3)
When the W(x) function is used to characterize the sample MWD, the shear viscosity for the blend becomes:
[Mathematical Expression Omitted] (A4)
where [[Eta].sub.1] is the viscosity associated with the first distribution and [[Eta].sub.2] with the second one.
The authors would like to acknowledge the use of computing services from the Ohio Supercomputer Center.
1. Z. Tadmor and I. Klein, Engineering Principles of Plasticating Extrusion, Reinhold, New York (1970).
2. Z. Tadmor and C. E. Cogos, Principles of Polymer Processing, John Wiley & Sons, New York (1979).
3. C. Rauwendaal, Polymer Extrusion, Hanser Verlag, Munich (1986).
4. D. Nichetti and I. Manas-Zloczower, "Viscosity Model for Polydisperse Polymer Melts," J. Rheol., 42, 951 (1998).
5. G. V. Vinogradov and A. Ya. Malkin, Rheology of Polymers, Springer Verlag, Heidelberg-Berlin (1980).
6. FIDAP Package, Fluid Dynamics International, Inc., Evanston, Ill.
7. R. E. Christensen and C. Y. Cheng, SPE ANTEC Tech. Papers, 37, 74 (1991).
8. L. F. Laroche, presentation at the Society of Rheology Meeting, Columbus, Ohio, 1997.
|Printer friendly Cite/link Email Feedback|
|Author:||Nichetti, D.; Manas-Zloczower, I.|
|Publication:||Polymer Engineering and Science|
|Date:||May 1, 1999|
|Previous Article:||In-situ generation of polyamide-6 fibrils in polypropylene processed with a single screw extruder.|
|Next Article:||A new method for estimating the cellular structure of plastic foams based on dielectric anisotropy.|