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The role of the interaction coefficient in the prediction of the fiber orientation in planar injection moldings

Posted on: Wednesday, 16 July 2003, 06:00 CDT

The mechanical properties of injection molded parts in glass reinforced materials are sensitive to processing. A successful design requires a good estimate of the product performance before production. Its performance is strongly affected by the fiber orientation field set up during processing. The fiber orientation pattern is complex and varies three-dimensionally in the moldings. Some commercial simulation programs already allow the prediction of the fiber orientation induced during the flow by the associated stress field. The results from the simulations are dependent on a parameter accounting for the interactions between fibers during the flow, known as the fiber interaction coefficient. In this paper the effectiveness of the interaction parameter on controlling the predicted patterns of the fiber orientation is studied. This is done by comparing and analyzing the experimental data and the corresponding predictions.

1. INTRODUCTION

Upon injection molding of fiber reinforced thermoplastic, complex patterns of fiber orientation result through the thickness and along the flow path of the moldings (1). Most of the properties of the composite depend on the distribution of fiber orientation. Thus, a precise determination of the composite mechanical properties requires either the prediction or the measurement of the fiber orientation field developing in injection molding. The measurement of fiber orientation is a cumbersome and labor intensive task; thus the need to explore the capabilities offered by new computer modeling tools follows naturally.

The studies of Jeffery (2), who laid the basis for currently available simulation packages, are valid only for very dilute suspensions. Dilute suspensions are those in which the distance between neighboring particles is, on average, bigger than their length. Most of the commercially interesting composites do not fit to this dilute regime. For other regimes, interactions between neighboring fibers occur during flow and affect the final pattern of fiber orientation. It is intuitive to accept that the neighboring fibers hinder the free rotation of a fiber.

Research work led by Tucker et al. (3-5), based on experiments conducted with chopped fibers immersed in silicone oil, led to a model that enabled accounting for the fiber-fiber interactions during flow. More recently Fan et al (6) developed a promising direct numerical simulation of fiber-fiber interactions in shear flow but requiring too much computation time.

No complete theory or model is yet available to fully predict the fiber orientation resulting from the injection molding process. Some existing models allow the calculation of the rheological and mechanical properties of short fiber reinforced injection molded composites (5) provided the fiber orientation is well characterized.

In modeling the flow of injection molded composites, it is usual to consider the fiber orientation evolution as decoupled from the flow simulation (7). Consequently, the fiber orientation field is not taken into account to determining the flow dynamics or to calculating the material viscosity, which can lead to important prediction errors. The improvement of the models by using coupled solutions was attempted by Kabanemi et al. (8) in order to obtain 3D fiber orientation predictions. Their formulation includes the coupling between the fiber-orientation and the viscosity using an earlier model proposed by Metzner (9).

In commercial flow simulation packages, like C-Mold and Moldflow, it is possible to model the fiber orientation in each element of the model after the definition of the thermomechanical history. Most of these commercial tools are based on the implementation of the Folgar- Tucker model (3). This model includes a fiber interaction coefficient C^sub I^ in an added-in diffusion term. This approach was successful in predicting with reasonable accuracy the fiber orientation in plates and circular discs molded with heavily reinforced composites (10).

In this work a study was conducted to analyze the effect of the fiber interaction coefficient C^sub I^ on the predictions of the fiber orientation developing in 10% by weight glass fiber reinforced polycarbonate discs. The orientation predictions were compared with experimental data obtained at three positions along the flow path of circular discs (radial direction).

2. THEORY

2.1. Definition of the Fiber Interaction Regime

The interaction regime in the composite melt is associated to the aspect ratio of the fibers (L/d), and to its volume fraction, (V^sub [function of]^). Typical long and slender fibers have an aspect ratio between 15 and 40. A dilute solution where the fibers hardly interact with each other is characterized by V^sub [function of]^ < (d/L)^sub 2^ (3). No commercially important composites fall into this dilute regime.

If the volume fraction of fibers is inside the range (d/L)^sub 2^ < V^sub [function of]^ < (d/L), the suspension is semi- concentrated, and the spacing between fibers being smaller than L is larger than d. The interaction between fibers is frequent. The 10% by weight short-glass-fiber poly-carbonate composites used in this study fit into this semi-concentrated regime.

A highly concentrated solution is defined when V^sub [function of]^ > (d/L) and the spacing between the fibers is of the order of magnitude of the fiber diameter, d.

2.2. Tensors of Fiber Orientation

The state of fiber orientation is well described by the second- and the fourth-order tensors of fiber orientation (5, 10). The fourth-order tensors are required to predict both rheological and mechanical properties of fiber suspensions, because these are fourth- order tensor properties.

The use of the second-order tensor is convenient in terms of calculation time and is preferentially applied in most of the commercial codes. The transformation from second- to fourth-order requires the solution of a closure problem (5). A number of closure approximations have been proposed and applied with varying degree of success (5, 11). Moreover, the tensors offer the possibility of representing the statistical distribution of fiber orientation, and may be assigned to a physical meaning. The second-order tensor of fiber orientation has 3 x 3 terms (like a 3 x 3 matrix). The sum of the diagonal terms of the tensor is equal to the unity, and each of the diagonal values is allowed to vary between 0 and 1. The value of each of the diagonal values stands for the relative orientation around one of the co-ordinate axis. The axes are commonly defined as 1-radial, 2-tangential, and 3- the out of plane (or thickness) direction. The tensor being symmetrical by definition implies that only three out-of-diagonal elements need to be computed. Therefore, five calculations are required to define the second-order tensor, without further simplifying assumptions (only two of the diagonal values are independent).

Fig. 1. Random planar fiber orientation.

The elements of the second-order tensor have a physical meaning. The value of each of the diagonal elements of the tensor stands for the relative orientation in one of the co-ordinate axes. The flow of polymer composites in injection moldings is planar, leading to planar fiber orientation states. Thus small values of the a^sub 33^ tensor component are typical obtained. The element a^sub 11^ is close to the unity if high orientation in flow direction exists. In planar and random distributions of fiber orientation the values of a^sub 11^ and a^sub 22^ are close to 0.5, a schematic representation being shown in Fig. 1.

In this case the second-order tensor is:

2.3. Modeling the Fiber Orientation

When the fiber concentration in the composite corresponds to the concentrated or semi-concentrated regimes, it is important to determine the effect of the interactions between neighboring fibers of the liber orientation. The interaction coefficient, C^sub I^, depends on the fiber aspect ratio, L/d, and on the volume fraction. Some authors suggest that the interaction coefficient should be dependent on the state of orientation (12, 13). This statement apparently contradicts the concept of the interaction coefficient being an intrinsic property of the suspension, and brings an additional transient character to the coefficient. However, without a detailed model for describing the transient interactions between fibers, there is no way to predict C^sub I^, and therefore it must be determined experimentally. Tucker and Advani (4) suggested an empirical expression to be used when an experimentally determined interaction coefficient is not available:

This expression is valid for concentrated regimes (V^sub [function of]^.L/d> 1).

High values of the interaction coefficient lead to random orientation states, whereas low values are likely to correspond to highly aligned distributions of fiber orientation in the flow direction (5).

3. EXPERIMENTAL

3.1. Molding

The experimental study was done using a circular center sprue- gated disc, 1.5 mm thick and 114 mm in diameter. This geometry implies a radial divergent flow. A 10% by weight glass fiber reinforced grade of polycarbonate was used (Lexan 500 R from GE Plastics). The manufacturer compounds this standard grade material with a release agent and a flame-retardant additive. The choice for this mat\erial was meant to produce a moderate level of interaction between neighboring fibers during flow, therefore replicating the situation of a semi-concentrated regime.

The molding was done in a molding cell based on a 600 kN Krauss- Maffei 60/210 A injection molding machine. The molding program (Table 1) was devised to study the effect of melt temperature and flow rate on the fiber orientation of the moldings.

3.2. Fiber Orientation Measurement

The measurements of fiber orientation were made on polished cross sections cut from the molded plates. Two specimens per molding condition were randomly selected to measure the fiber orientation distribution along and across the flow direction. For every molding condition the fiber orientation distribution was experimentally characterized at three points along the flow path: at 20 mm, 35 mm and 50 mm from the gate (Fig. 2).

Fig. 2. Sections cut from the disc for experimental fiber orientation characterization.

Table 1. Molding Program.

The surfaces of samples cut from fiber reinforced parts were subjected to successive stages of polishing (sandpaper of progressively smaller roughness) to achieve a microscopically smooth surface. The fiber orientation was measured using image analysis tools in images obtained by reflection microscopy of the polished cross sections, using the method proposed by Bay and Tucker (14). For the measurements, the cross section was divided into three columns of twelve layers each with equal thickness to assess the sampling error.

3.3. From Measurements to the Fiber Orientation Tensor

The sections of the fibers appear in the images (Fig. 3) as circles if the fibers are aligned in a direction perpendicular to the cut section ([theta] = 0[degrees]), rectangles if the fiber axis is parallel to the section ([theta] = 90[degrees]) or ellipses otherwise.

The out of plane orientation angle, [theta], is derived from the major and minor axes of the ellipse, a and b, as

Because the direction of penetration of the fiber in the specimens is not fully defined in a single cross section there is some uncertainty about the angle [straight phi], indistinguishable from the angle [straight phi] + 180[degrees]. This ambiguity that affects only the out-of-diagonal tensor components is present in our data.

The in-plane orientation is determined by the angles defined by the major axis of the ellipse and the preselected reference axes (Fig. 3). These angles, [theta] and [straight phi], can be determined by digitizing the coordinates of the endpoints of the major and minor axes of the ellipses, either manually or by image analysis tools.

The relevant components of the unit vector p for the calculation of the orientation tensor are shown in Eqs 4 to 6 for the specimens cut in the 1-3 sectioning plane. p^sub 1^ = sin [theta] cos [straight phi] (4)

P^sub 2^ = cos [theta] (5)

P^sub 3^ = sin [varphi] sin [straight phi] (6)

The elements of the second-order tensors that describe the fiber orientation can be calculated from the values of the vectors p^sub t^ using Eq 7.

The parameter F^sub n^ is a weighting function that corrects the bias resulting from the lower probability of a fiber lying parallel to the section plane appear in the image (14).

3.4. C-Mold Simulations

For the simulation of the fiber orientation a model with 707 elements was created in the C-Mold package (15). The direct sprue and the impression were included in the FEM model. The mesh used in the C-Mold simulations is shown in Fig. 4.

Fig. 3. Possible forms of fiber sections in a polishing surface, [straight phi] is the out of plane orientation angle.

The information required to perform the simulation includes:

* The mesh of the part;

* The material properties;

* The process conditions;

* The parameter data.

The processing conditions were defined according to the set up for the experimental tests. The required material properties were obtained in the material database of C-Mold, except for the number average fiber length, diameter and fiber volume fraction. These data were determined experimentally after pirolysis of polycarbonate granules, as 238 [mu]m, 14 [mu]m and 5.8%, respectively.

In order to increase the accuracy of the numerical solution, the default number of layers across the thickness was increased from the C-Mold standard number of 12 to 20.

There are two main parameters to be adjusted when fiber orientation is simulated with C-Mold: the interaction coefficient, C^sub I^, and the inlet boundary condition. In this study the boundary condition (at the sprue entrance) was defined considering that the fibers are aligned in the flow direction at the skin, and randomly aligned at the core. This parameter is less important when compared to C^sub I^, since at a short distance after the gate entrance the effect of this parameter on the simulation results vanishes. The inlet boundary condition is very important for large gates in which the orientation at the entrance dominates the flow pattern.

3.5. Analysis of the Influence of C^sub I^

The effect of varying the interaction coefficient, C^sub I^ was determined quantitatively considering the average deviation

Fig. 4. Mesh used to model the fiber orientation.

between the experimented and the predicted values for the a^sub 11^ tensor component. For each set of processing conditions the calculation uses data from three locations along the flow path (20, 35 and 50 mm distance from the gate) and twelve locations across the thickness:

The subscripts exp and pred refer to experimental and predicted results, respectively. The indices i and j refer to the location along the flow path and to the layers across the thickness, respectively.

4. RESULTS AND DISCUSSION

The results of the experimental fiber orientation characterization are presented in terms of the a^sub 11^ second order tensor component. This tensor component characterizes the fraction of fibers aligned in the flow direction. The C-Mold software predicts a predominantly planar fiber orientation pattern with very small values for the out-of-plane component of the tensor of fiber orientation (a^sub 33^ = 0.005). Thus, the transverse-flow- direction fiber orientation at any point in the results may be seen as the difference between all and the unity (a^sub 221 [approximate] 1-a^sub 11^).

4.1. Variation Along the Flow Path

Both experimental data and predictions corresponding to the processing condition code 280.10 considering different C^sub I^ are shown in Fig. 5, at three locations along the flow path. The C-Mold predictions show a skin with fiber orientation predominant in the flow direction, with the a^sub 11^ tensor component of fiber orientation higher than 0.6. At the core, the fiber alignment is high in the across flow direction both in the simulations predictions and in the experimental data, as it is suggested by the large value of a^sub 22^ (small value of a^sub 11^).

The experimental results suggest a less defined layering through- thickness. At most of the data points the a^sub 11^ tensor component is lower than 0.5. These results confirm that the disc geometry, having a divergent flow, induces higher fiber orientation in the across flow direction.

The simulations with C-Mold seem to predict qualitatively well the fiber orientation close or far from the gate. At intermediate locations (in this case, 35 mm from the gate) the degree of fiber orientation in the flow direction is overestimated regardless the molding conditions as it is summarized in the Fig. 6.

4.2. Effect of the Melt Temperature

Increasing the melt temperature (Fig. 7) leads to lower levels of fiber orientation in the flow direction. This is in accordance with the expectations, since the flow-induced stresses acting in the melt are proportional to the viscosity. Lower melt temperature increases and thus the alignment of the fibers. Experimental data shows that molding with higher melt temperatures results in close to random in- plane through-thickness fiber orientation a^sub 11^ [approximate] a^sub 22^ [approximate] 0.5).

The relatively better agreement between predictions and experimental data that are observed at the lower melt temperature conditions (280.10, 280.14 and 280.32) may be attributable to the stronger stress field built up in the melt during the flow. Moreover, it seems that in C-Mold the effect of shear stresses are more effectively predicted than the effect of extensional stresses.

Fig. 5. Prediction and experimental fiber orientation in the flow direction at 280[degrees]C melt temperature and 9.7 cm^sup 3^/s flow rate, at three distances from the gate: a) 20 mm. b) 35 mm and c) 50 mm.

4.3. Effect of the Flow Rate

At the lowest melt temperature (Fig. 8) a clear tendency of the fibers to get more aligned in the across flow direction is observed (a^sub 11^ < 0.5). A slower flow rate seems to favor the fiber orientation in the flow direction. The fiber orientation in the flow direction is overpredicted close to the wall and at the core of the moldings.

4.4. Effect of the Interaction Coefficient in the Simulations

For the different processing conditions the effect of C^sub 1^ in the predictions is shown in Fig. 9.

As can be also seen therein, the deviation varies inversely to C^sub 1^. The lowest deviation is obtained with C^sub 1^ of 0.1. This is the value of C^sub 1^ that leads to an overall minimum deviation between experimental and predicted values of the tensor component a^sub 11^. Lower melt temperatures appear to improve the prediction of the fiber orientation pattern, as suggested by the smaller deviations observed in these cases (280.10, 280.14 and 280.32). This is understandable because the shear flow is better modeled and probably overestimated by the C-Mold software. As a general observation it can be said that the effect of C^sub 1^ on the predictions is qualitatively important, but it is not possible to find a value of C^sub 1^ that might lead to general qualitative and quan\titative close prediction of the fiber orientation in the moldings.

5. CONCLUSIONS

The interaction coefficient, C^sub 1^, which is usually a parameter to be defined by the user when modeling the fiber orientation field in injection moldings, has an important effect on the qualitative prediction of the fiber orientation. It affects significantly the patterns of fiber orientation predicted in the flow simulations. Larger values for C^sub 1^ tend to make the fiber orientation close to a random in-plane distribution whereas lower values predict a clear alignment in the flow direction.

The fiber orientation field determined from experimental data in center gated discs molded using 10% glass fiber polycarbonate seems to be better predicted when a large value of C^sub I^ is used. When the value of this interaction coefficient is decided from using the empirical expression suggested by Tucker (4), poor predictions result for lightly reinforced materials in this molding geometry.

Fig. 6. Deviation between predictions and experimental results for different positions along the flow path: 20, 35 and 50 mm from the gate.

Fig. 7. Prediction and experimental fiber orientation in the flow direction at 13.8 cm^sup 3^/s flow rate and at 20 mm from the gate, at different melt temperatures: a) 280[degrees]C, b) 300[degrees]C and c) 320[degrees]C.

Fig. 8. Prediction and experimental fiber orientation in the flow direction at 280[degrees]C melt temperature and at 20 mm from the gate, at different flow rates: a) 9.7 cm^sup 3^/s, b) 13.8 cm^sub 3^/ s and c) 32.2 cm^sub 3^/s.

Fig. 9. Deviation between C-Mold predictions using different C^sub I^ and experimental results.

The predictions close to the gate or at the end of the flow path are much closer to the experimental results than at intermediate positions in the flow path. This unexpected result suggests that further investigation of the specific flow conditions and fiber orientation in this region is necessary.

For this material and molding geometry, tuning the interaction coefficient does not lend to an overall better fit of predictions to experimental data.

It was shown that the improvement of the prediction of the fiber orientation requires the definition of an adequate interaction coefficient for each case. Unfortunately there is not yet a clear criterion for its specification. Using the direct numerical simulation of fiber-fiber interaction proposed by Fan et al. (6) may overcome the uncertainty in choosing of C^sub I^, but it is not yet a commercially available simulation program.

Nevertheless we may conclude that, for the case of geometries in which extensional stresses are high (such as center gated parts), values of C^sub I^ much higher than those obtained with the Tucker and Advani equation should be used, for getting patterns closer to the random in-plane state.

6. ACKNOWLEDGMENTS

The authors are indebted to IMAT-Institute of Materials, and the Center for Polymer Engineering of the Department of Polymer Engineering of the Universidade do Minho, Portugal. The authors also acknowledge the assistance provided by the C-MOLD Europe office at Enschede, The Netherlands.

REFERENCES

1. N. M. Neves, G. Isdell, A. S. Pouzada, and P. C. Powell, Polym. Comp., 19, 5 (1998).

2. G. B. Jeffery, Proc. Roy. Soc., A102 (1922).

3. F. Folgar and C. L. Tucker III, J. of Reinf. Plast. and Comp., 3 (1984).

4. C. L. Tucker III and S. G. Advani, Processing of Short-Fiber Systems, In: S. G. Advani, (Ed.), Flow and Rheology in Polymer Composites Manufacturing, Comp. Mater. Series, Amsterdam (1994).

5. S. G. Advani and C. L. Tucker III, J. of Rheology, 31, 8 (1987).

6. X. J. Fan. N. Phan-Thien, and R. Zheng, J. Non-Newtonian Fluid Mech., 74 (1998).

7. M. C. Altan, S. Subbiah, S. I. Guceri, and R. B. Pipes, Polym. Eng. Sci., 30, 14 (1990).

8. K. K. Kabanemi, J. F. Hetu, and A. Garcia-Rejon, Intern. Polym. Proc., 12, 2 (1997).

9. M. L. Becraft and A. B. Metzner, J. Rheology, 36 (1992).

10. R. S. Bay and C. L. Tucker III, Polym. Comp., 13, 4 (1992).

11. J. S. Cintra and C. L. Tucker III, J. Rheology, 39 (1995).

12. M. R. Kamal and A. T. Mutel, Polym. Comp., 10 (1989).

13. S. Ranganathan and S. S. G. Advani, J. Rheology, 35 (1991).

14. R. S. Bay and C. L. Tucker III, Polym. Eng. Sci, 32, 4 (1992).

15. C-Mold version 96.10, Advanced CAE Technology, Inc., Ithaca, N.Y.

A. J. PONTES, N. M. NEVES, and A. S. POUZADA

Department of Polymer Engineering Universidade do Minho Campus de Azureem 4800-058 Guimaraes, Portugal

Copyright Society of Plastics Engineers Jun 2003

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