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The SPE Extrusion Division

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Michigan Technological University Plastic Flow, LLC

Houghton, MI 49931 Hancock, MI 49930

Unfortunately, the root cause behind the encapsulation phenomena, whether it is caused by the difference in viscosity, or in viscoelasticity, or in some other property of the two polymers, is still not completely understood. Simulations of coextrusion using a purely viscous formulation available in the literature [3], including our earlier work [4], have not been successful in capturing the encapsulation. This may lead one to believe that the encapsulation must be caused by viscoelastic effects, which was supported by some papers [5] on viscoelastic simulation of a bi-layer flow in the literature. In contrast, the simulation results presented in this paper indicate that the encapsulation is not caused by viscoelastic effects. Therefore, at this point we believe that the unbalanced force at the contact line where the two polymers together meet the die wall is the main driving force which results in the observed encapsulation. This unbalanced force at the contact line originates from the difference in the wettability and surface tension of the two polymers.

Besides the constitutive equation, simulation of viscoelastic flow requires solving the mass and momentum conservation equations. Assuming a steady, inertia-less, isothermal, incompressible flow with no body force, the conservation equations for momentum and mass are simplified to

where the total stress, σ is given by the following equation

with p being the pressure, n

In a finite element simulation of viscoelastic flow, strainrate tensor is discontinuous across the element boundaries, whereas a continuous interpolation is used for e

u

where u is the velocity, and n is the unit vector perpendicular to the interface.

In the present work, a three-dimensional mesh of tetrahedral finite elements was generated over the complete flow channel in the die. This finite element mesh is not modified or regenerated at any stage during coextrusion simulation. Thereby, allowing simulation of even highly complex coextrusion systems.

In the mesh partitioning technique which is employed in this work, the interface between adjacent layers of different polymers is represented by a surface mesh of linear triangular finite elements. However, the surface mesh of triangular elements on the interface and the threedimensional mesh of tetrahedral elements in the coextrusion die are completely independent of each other. This decoupling between the two finite-element meshes is possible because in the mesh partitioning technique for coextrusion simulation, the interface between adjacent polymer layers is not required to match with the interelement boundaries in the three-dimensional mesh of tetrahedral finite elements. Instead, in the software used in this work, the interface is allowed to pass through the interior of the tetrahedral finite elements in the threedimensional mesh.

In the mesh partitioning technique for coextrusion simulation the tetrahedral elements which are intersected by the mesh of triangular elements on the interface are partitioned into two tetrahedral, pyramidal, or prismatic finite elements. Further details of the mesh partitioning technique are available in our earlier publications [4, 11].

The experimental data for the viscosities of the two polystyrenes [3], along with the fit to the experimental data using the Giesekus model parameters given above [5], is shown in Fig. 1. It is evident from Fig. 1 that the viscosity of Styron 472 is much higher than that of Styron 678E.

It was reported by Sunwoo et al. [5] that they could capture the encapsulation of Styron 472 by Styron 678E when they included viscoelastic effects in the simulation, which was the main motivation for including the viscoelastic effects in coextrusion simulation in this work. Sunwoo et al. [5] reported that the predicted encapsulation in their simulation of the bi-layer flow was the largest and the closest to the experiments when a large value of the parameter α

To characterize the flow, a non-dimensional shear rate (Deborah number, De) is defined as

where U is the average velocity in the square portion of the flow channel, b is the half of the square cross-section side length, and λ

To define the Deborah number for the bi-layer flow in the square channel, the average value of the characteristic relaxation time, λ

For De = 1, with Styron 472 in the lower layer and Styron 678E in the upper layer, the axial velocity distribution for the bi-layer flow in the square channel is shown in Fig. 4. Beyond De = 1 simulation of the bilayer flow did not converge. Starting with the same uniform velocity at the entrances of the two polymers, after the two polymers meet the velocity slowly increases as the two polymers go through the converging portion of the channel, and the highest velocity is obtained once the two polymers reach the square portion of the channel. The velocity then remains relatively unchanged as the two polymers flow through the square channel

The transverse velocity distribution due to secondary flow in the bi-layer coextrusion in the square channel for De = 1 is shown in Fig. 5. For the bi-layer flow, in Fig. 5 there are eight recirculating vortices. However, in contrast to the eight vortices in flow of a single polymer, which have the same size, the vortices for the bi-layer flow in Fig. 5 have different sizes. For instance, in Fig. 5 the two vortices at the bottom are significantly smaller than the two vortices just above the two bottom vortices. Also, in contrast to the vortices for flow of a single polymer, for the secondary flow in Fig. 5 there is no stagnation region in the middle of the square cross-section. Instead, near the center of the square cross-section the fluid from the bottom vortices is entering into the flow in the two vortices on top.

The shape of the interface between the two polymer layers for De = 1 is shown in Fig. 6 (a). The corresponding final shape of interface at the exit of the square channel is shown in Fig. 6 (c). Starting with a straight line interface shape at the contact line, where the two polymers meet for the first time, the interface in Fig. 6 (a) starts to wave upwards near the middle and near the two ends with troughs in between, resulting in the W-shaped interface at the die exit in Fig. 6 (c). In contrast to the wavy interface shape in Fig. 6 (a) and 6 (c), for De = 0.1 in Figs. 6 (b) and 6 (d), the predicted interface shape just has a slight curvature but does not have a wavy shape. For De = 0.1 the elastic effects in the flow, and hence, the secondary flow vortices are negligible, resulting in the simple shape of the interface which is similar to the interface shape obtained in a purely viscous simulation by Karagiannis et al. [3].

The magnitude of the second normal stress difference, |τ

The pressure variation for the bi-layer flow in square channel is shown in Fig. 8. As expected, the pressure is zero at the exit and increases towards the two entrances. Also, for the same velocity at the entrances of the two polymers, the pressure gradient is higher in the narrower feed channel of the upper layer.

At this point, we believe that polymer encapsulation in coextrusion is a surface phenomenon, which is caused by the difference in the wettability (spreading tendency of the polymer on die surface) and surface tension of the two polymers used. The polymer with higher wettability is expected to encapsulate the other polymer during coextrusion.

2. J. Dooley and L. Rudolph, J. of Plastic Film and Sheeting, Vol. 19, 111 – 122 (2003).

3. A. Karagiannis, A. N. Hyrmak, and J. Vlachopoulos, Rheologica Acta, Vol. 29, 71 – 87 (1990).

4. M. Gupta, SPE ANTEC Technical Papers, Vol. 54, 217 – 222 (2008).

5. K. B. Sunwoo, S. J. Park, S. J. Lee, K. H. Ahn, and S. J. Lee, Rheologica Acta, Vol. 41, 144 – 153 (2002).

6. R. G. Larson, “Constitutive Equations for Polymeric Melts and Solutions”, Butterworths, Boston, MA, (1988).

7. J. N. Reddy, “An Introduction to Finite Element Method”, 3rd Ed., McGraw Hill, (2006).

8. A. N. Brooks and T. J. R. Hughes, Computer Methods in Applied Mechanics and Engineering, Vol. 32, 199 – 259 (1982).

9. R. Guienette and M. Fortin, J. of Non-Newtonian Fluid Mechanics, Vol. 60, 27 – 52 (1995).

10. T. Coupez and S. Marie, International Journal for Supercomputer Applications and High Performance Computing, Vol. 11, 277 – 285 (1997).

11. M. Gupta, SPE ANTEC Technical Papers, Vol. 56, 2032 – 2036 (2010).

Fig. 1 Viscosity of the two polystyrenes.

Fig. 2 Interface shape in the experiments. The distance from the square entrance is (a) 0.4 L, (b) 2.42 L, and (c) 6.71 L, with L being the length of the side of the square [3].

Fig. 3 Interface shape predicted by a purely viscous simulation. The distance from the square entrance is (a) 0, (b) 0.42 L, and (c) 2.42 L, with L being the length of the side of the square [3].

Fig. 4 Axial velocity distribution in a bi-layer flow in a square channel for De=1.

Fig. 5 Secondary flow vortices in a bi-layer flow in a square channel for De=1.

￼Fig. 6 Shape on the interface in a bi-layer coextrusion in a square channel. Interface shape for (a) De=1, (b) De=0.1. Interface shape at channel exit for (c) De=1, (d) De=0.1.

Fig. 7 Magnitude of the second normal stress difference in a bi-layer flow in a square channel for De=1.

Fig. 8 Pressure distribution in a bi-layer flow in a square channel for De=1.

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