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Rheology of Fiber Filled Polymer Melts: Role of Fiber-Fiber Interactions and Polymer-Fiber Coupling

Posted on: Tuesday, 22 March 2005, 03:00 CST

An experimental study and a numerical modeling analysis are carried out to examine the effects of fiber-fiber interactions and coupling between fiber orientation and polymer chains conformation on the rheological properties of fiber suspensions. The experimental study allowed examination of large fiber volume fractions up to 35% over a range of shear rates that spans eight decades. This study confirmed already known results and led to new ones. In particular, a peak in the steady shear viscosity at the low shear rate region is observed at large volume fractions. Furthermore, new results regarding the applicability of the Cox-Merz rule, the behavior of the damping factor, and the end pressure drops are reported, and physical interpretations are proposed. The results of the numerical modeling showed that it is necessary to account for the polymer- fiber coupling factor to obtain a good fit between the model predictions and the experimental measurements. Comparisons between the model predictions and the experimental measurements allowed study of the variation of the parameters that govern the fiber- fiber interactions and the polymer-fiber coupling with the properties of the suspension and the flow. POLYM. ENG. SCI., 45:385- 399, 2005. 2005 Society of Plastics Engineers

INTRODUCTION

Rigid short fiber polymer composites are widely used in industry because of their low cost and high performance. The physical properties of these polymer composites, such as elastic modulus, strength, thermal expansion, thermal conductivity, and electrical conductivity, depend on the amount, type, size, and orientation of the reinforcing fibers. During processing, fibers move and rotate with the flow of the polymer matrix, which inevitably changes their orientation state and affects the properties of the composite material. The orientation state of the finished composite is of crucial importance to the designer who expects optimum performance from a short-fiber reinforced polymer composite.

The flow-induced fiber orientation greatly affects the rheological properties of filled melts. It is therefore important to understand both the development of the orientation distribution and how rheological properties vary as a function of the fiber orientation distribution. There are many factors that affect flow- induced fiber orientation [1-3]. One of the most important but less understood mechanisms is related to fiber-fiber interaction. In many practical applications, fiber volume fractions in short fiber reinforced composites are as high as 40% [4], while typical fiber aspect ratios range from 10 to 1000. Hence, fibers flow in close proximity to each other. In this case, one may have to deal with many types of fiber interactions that can be associated with hydrodynamic forces, excluded-volume, and friction or mechanical interactions. Such interactions play a dominant role in determining the local orientation of the fibers. Another key factor that may affect the rheology of a fiber suspension and will be investigated in the present study is associated with the effect of the orientation of the fibers on the polymer conformation during the flow. It is indeed expected that in high fiber concentration regimes, the conformation of the polymer chains may be influenced by the orientation of neighboring fibers.

In recent years, a considerable amount of work has been devoted to both experimental characterization and theoretical modeling of short-fiber composites [5-7]. Studies on the coupling of the fiber orientation with the polymer conformation are rare and this topic has not yet received much attention except in the theoretical studies of Fan [8], Azaiez [9], and Ramazani et al. [10]. Among the few studies that exist in the literature and that deal with the above raised issues, one is forced to admit that they are mostly based on empirical correlations and there is little or no understanding of the physics of these interactions. In particular, the meaning and acceptable values of the empirical parameters are still subjects of debate. Our research is motivated by the close relationship of the fiber orientation with these two effects. The objectives are to understand the effects of fiber-fiber interactions and polymer-fiber coupling on the flow induced fiber orientation and rheological properties of fiber suspensions. To achieve these objectives, we carried out a series of experimental studies and analyzed the results in conjunction with the prediction of a mathematical model where coupling between fibers and matrix as well as fiber-fiber interactions are taken into consideration.

FIG. 1. Fiber length distributions.

EXPERIMENTAL PROCEDURE

Materials

The system investigated in this work consists of E-glass fiber filled linear low-density polyethylene (LLDPE). A commercial LLDPE resin (LL8460) supplied by Exxon in pellet form was used as a matrix. The resin has a melt flow index (MFI) of 3.3 and a density of 0.938 kg/m^sup 3^. Two groups of milled E-glass fibers of different initial length and length distributions (1/32-inch and 1/ 16-inch mesh size) were provided by Owens-Corning, Inc., and will be referred to as group A and group B, respectively. The density of the E-glass is 2.6 kg/m^sup 3^. The nominal fiber diameter for these fibers is 16.0 2 m.

The fiber suspensions were prepared in a Haake Rheocord 40 torque rheometer equipped with a Haake 600 series internal batch mixer at 190C. Fiber suspensions with volume fractions of 5%, 10%, 15%, 20%, and 25% were prepared using the two groups of fibers. Samples containing higher fiber content (30% and 35% by volume) were prepared using the group A fibers only.

Equipment and Measurements

Measurements of the rheological properties of the E-glass fiber filled LLDPE were performed over a large range of shear rates (10^sup -4^ to 10^sup 4^ s^sup -1^) using a rotational rheometer as well as a capillary rheometer. The Haake RS150 rotational rheometer with 20 mm parallel plates was used to measure the shear viscosities at low to middle range of shear rate (10^sup -4^ to 10^sup 2^ s^sup - 1^), as well as the viscoelastic properties of the fiber suspensions. Two modes of deformation were used: control stress (CS) and small amplitude oscillatory (OSC). The data collected from steady-state CS test are from low to high shear rate. The frequency sequences are in the range of 0.14 to 277.72 rad/s. The Rosand RH-7- 2 double barrels capillary rheometer was used to determine the shear viscosities at very high shear rates (10 to 10^sup 4^ s^sup -1^). Both dies have a diameter of 1 mm and 180 entrance angle. All experimental measurements were carried out at a temperature of 190C and a fresh sample was used for each test to eliminate effects related to sample history.

Fiber orientations in the suspension after mixing and at the exit of the capillary rheometer were determined. The procedure adopted consisted of cutting a fiber suspension sample sheet using a Leica RM2165 motorized rotary microtome into 100-m thick pieces parallel to the sheet plane. Then images were captured by a CCD camera attached to an Olympus BX60 optical microscope.

EXPERIMENTAL RESULTS AND DISCUSSION

Fiber Length Distribution

Fiber length distributions for the two groups of fibers were determined before and after mixing. Fiber initial length distributions, i.e., fiber lengths before mixing, were measured directly using samples of glass fibers as received from Owens- Corning, Inc. In order to determine the fiber length distributions after mixing, fibers were separated from the polymer matrix by heating the fiber suspension samples in an oven at a temperature of 500C for 90 min to burn out the polymer matrix. For each sample, over 1000 fibers were considered in the measurements. The fibers were disposed on a glass slide and images were captured using a CCD camera coupled to an Olympus BX60 optical microscope. A commercial image analysis software ImagePro was used to measure the length of each fiber.

TABLE 1. The average and standard deviation of the fiber length.

Figure 1 shows the length distributions of the two classes of fibers before and after mixing. It is seen that most of group A fibers before mixing are less than 1 mm long with an average length of 231 m. The highest distribution peak is in the range of 200 to 300 m length. The length distributions of the group B fibers before mixing are bimodal. The largest frequency of fibers occurs at a length around 3 mm, while the average length is above 600 m. The length distributions of these fibers are much broader than those of the group A fibers. The longest fiber observed is about 7 mm. Several other measurements we conducted with different samples led to similar results for the fiber length distributions.

During the mixing process, fibers undergo severe mechanical interactions with the polymer matrix, the blender walls, and the other fibers, resulting in fiber breakage and important changes in the length distribution. Therefore it is important to know the characteristics of the fibers after mixing. The length distribution of group A fibers after mixing, although narrower, does not show important changes compared to their i\nitial distribution. However, the group B fibers sustain dramatic changes after mixing and have only one distribution peak at around 400 m due to the breakage of the long fibers. The characteristics of the two classes of fibers before and after mixing are summarized in Table 1.

FIG. 2. Complex or shear viscosity of group A fiber suspensions for different volume fractions.

Rheological Properties

The volume fractions of the samples we used are as high as 30- 35%, which corresponds to weight fractions of 54-60%. Rheological properties at such high concentrations are hardly seen in the literature. In fact, the highest weight fraction used by most researchers is about 45%, which corresponds to volume fractions of only ~20-25%. Therefore, the results of the present study are more relevant to practical applications in the industry of composite materials where high volume fractions are encountered.

As stated earlier, we measured the shear viscosity η and complex viscosity η* of fiber suspensions in a very wide range of shear rates and frequencies (about eight decades) by combining data measured from both rotational and capillary rheometers. Thus, we are able to present a complete picture of the rheological behavior of fiber suspensions that has not been presented in earlier studies.

Figure 2 shows the viscosity of the pure polymer (0%) with that of group A fiber suspensions at different concentrations. The ranges for each rheometric experiment are also indicated. This figure shows that the change in the viscosity for a volume fraction of 5% is quite small, indicating that such fiber volume load is not large enough to cause important changes. However, the increase in the viscosity becomes more noticeable for larger volume fractions. Furthermore, the effects of fiber volume fraction are more noticeable at low shear rates. For shear rates higher than 200 s^sup -1^, the effects diminish and all data collapse to the same curve as that of the pure polymer melt. Similar trends are observed in the case of the group B fiber suspensions.

The absence of changes in the shear viscosity with an increase in the fiber volume fraction at high shear deformation is well known. This phenomenon can be explained by the fact that at high shear flow deformation, both polymer and glass fibers are oriented in the flow direction; thus, the effect of fiber volume fraction on the shear stresses becomes less significant [11]. This is also confirmed by our observations of fiber orientation before and after the capillary extrusion (Figs. 3 and 4). Figure 3 depicts typical fiber orientation distribution of samples obtained after the mixing process and just before the rheological measurement. It is clear that the fibers are randomly oriented. At low shear rates, covered by the controlled stress (CS) tests, fibers begin to change their orientations to align along the flow direction, though the induced shear deformations are not large enough for them to be fully oriented. In order to keep the same shear rate, extra stress is required by the system to overcome the resistance of fiber movement attributed to the high viscosity of the polymer matrix at low shear rate, and results in an increased viscosity. In the capillary test, the shear deformations are so high that fibers are mostly oriented in the flow direction (see Fig. 4). Resistance to the motion of fibers and polymer matrix is strongly reduced, and at the same time, the viscosity of the polymer matrix has already decreased. The occurrence of slippage may also have contributed to lowering the overall shear stresses and viscosity of the suspensions. For concentrated suspension, apparent slippage is caused by variations in the local concentration of suspended particles near the wall and in the bulk. The importance of slippage for concentrate suspension has been the subject of several studies and can be characterized based on simple shear flow measurements [12-15]. In this present study, preliminary examination of specimens collected from capillary flow tests, however, were not conclusive and further work would be required to quantify the importance of slippage on the flow behavior of polymer-fiber suspensions relative to the contribution that the orientation of fibers has to lowering the overall resistance to flow.

FIG. 3. Fiber orientation of fiber suspensions after mixing, viewed by optical microscopy (magnification 20); 25% of group B fibers.

We should also comment here on the deviation between the complex viscosity and the shear viscosity which clearly increases with the fiber loading. This suggests that the Cox-Merz rule does not apply to the system of fiber suspensions. Indeed, the complex viscosity is lower than the shear viscosity measured from steady-state control stress tests at low shear rates, while it is larger than the shear viscosity measured from capillary tests at high shear rates. It is known that there are deviations between the shear and complex viscosities at large deformation rates. This has been first reported by Mutel [16] for glass fiber-filled polypropylene. However, to the knowledge of the authors, the deviations observed at small deformation rates have not been reported before in the literature. The deviations can be explained based on the changes in the fiber orientation with the coupled rheological properties as well as the different test modes used. It is believed that a change in the fiber orientation during a shear flow gives rise to a pronounced increase of shear stress before reaching a steady value [17].

The fiber orientation changes induced in oscillatory (OSC) tests are quite different from those in steady shear tests. Kim and Song [18] found that both the magnitude of frequency and oscillatory shearing time affect the complex viscosity and the fiber orientation in dynamic shearing tests. Unfortunately, direct experimental investigation of the pattern and the degree of such fiber orientation changes are hardly seen in the literature. Following Mutel's interpretation [16], it is reasonable to assume that for the frequency range covered by OSC tests, the small periodical deformation imposed on the fiber suspension is too weak to induce significant fiber reorientation. Fibers generally maintain their original randomly oriented state and vibrate at the corresponding frequency in equilibrium positions. Because the viscosity of randomly oriented fiber suspensions is higher than that of aligned fiber suspensions for the same fiber aspect ratio and volume fraction [19, 20], the complex viscosity is larger than the shear viscosity measured from capillary tests. Also, we note that an increase in the fiber volume fraction results in a similar increase in the gaps between curves of complex viscosity and shear viscosity from capillary tests. When the fiber volume fraction is lager than 20%, the deviations between both data values and trend of curves are more pronounced. We should finally note that the shear viscosity curves from the rotational rheometer and the capillary rheometer follow the same shape and trend, and show consistent shear-thinning behavior.

FIG. 4. Fiber orientation of extrudate of 25% group B fiber suspensions from the capillary viscometer, viewed by optical microscopy (magnification 5).

Figure 5 shows the shear viscosity and complex viscosity of fiber suspensions at different fiber aspect ratios. The number in parenthesis is the aspect ratio determined from fiber samples after mixing. We note the appearance of a peak in the viscosity curve near the shear thinning transition region for both groups of fiber suspensions. Similar observations have been reported for colloidal particles suspended in polymer solutions [21]. However, our experimental data show that this peak occurs only at high fiber volume fractions (>25%). The appearance of the peak may be attributed to an increase in the mechanical interactions and correspond to the kinetics of formation and breakage of fiber clusters. The fibers may also undergo a reorientation process in the range of shear rates where the peak is observed. At fiber volume fractions as high as 25%, fibers flow so close to each other that their reorientation movements cause strong flow resistance resulting in higher shear viscosity.

Figure 6 shows the storage modulus G', while Fig. 7 shows the loss modulus G'' for the group A fiber suspensions at volume fractions of 0, 10, 20, 30, and 35%. The addition of glass fibers increases both G' and G''. This is in concordance with results presented by Greene and Wilkes [22] for fiber concentrations between 5 wt% and 42 wt%. Like what has been observed in the case of the viscosity, the variations in the values of G' and G'' are not very significant at low concentrations ([straight phi] ≤ 5%) as well as at very high concentrations ([straight phi] >30%). We should finally note that the general shape of the curves of G' and G'' are not changed by the addition of fibers.

In viscoelastic theory, G' represents the solid aspect of the material and G'' the liquid aspect. The ratio, which is often referred to as the damping factor, denotes the relative stiffness of the material. We have used this parameter to investigate the effects of the fiber concentration and oscillatory frequency on the viscoelastic properties of the suspension. Figure 8 depicts changes of tan(δ) with fiber volume fractions at different frequencies in the case of group A fiber suspensions. It is clear that tan(δ) decreases with the fiber volume fraction for very low frequencies (ω = 0.135 rad/s), while for larger frequencies (ω = 277 or 92 rad/s), it is nearly constant around a value of 1.

FIG. 5. Complex or shear viscosity of 25% fiber suspensions for different aspect ratios.

We will now attempt to present an explanation of the above result using the concept of the polymer matrix relaxation. At low frequency, the test time is larger than the typic\al relaxation time of the polymer chain, and therefore the chain can somewhat recover to its original state under the strain. While at high frequency (ω > 10 rad/s), the test time is too small for the polymer chain to recover to its original state. Therefore, it shows more stiffness like a solid. The presence of fibers acts like obstacles to resist the recovery of polymer chain, resulting in an enhancement of the system stiffness.

FIG. 6. Storage modulus G' of group A fiber suspensions for different volume fractions.

The end pressure drops ΔP^sub end^ are the end effects of the capillary viscometry that can be used to further characterize the viscoelastic effects of the fiber suspensions. Figure 9 shows the effect of the fiber volume fraction on ΔP^sub end^. In the case of the pure polymer, ΔP^sub end^ increases with increasing shear rate almost linearly in the log-log coordinates. This result is already well established for polymer melts and solutions [23]. However, changes of ΔP^sub end^ for fiber suspensions are seldom reported except in the studies of Chan et al. [24] and Laun [17]. In the present study, we present the variation ΔP^sub end^ of fiber suspensions with the shear rate rather than shear stress at volume fractions from 5% to 30%. An increase in ΔP^sub end^ is observed with the shear rate and with the fiber volume fraction. However, it is clear that the effects of fiber volume fraction on ΔP^sub end^ are most noticeable at fiber volume fractions less than 25%, and tend to be weaker for larger volume fractions. It is also interesting to note that a 5% fiber suspension leads to major differences in ΔP^sub end^ from what is observed in the pure polymer case, while there is very small effect of this particular volume fraction on other properties such as the shear viscosity and complex viscosity. This indicates that the presence of glass fibers in the polymer melt causes large increases in the end pressure losses ΔP^sub end^ even at low fiber volume loads that otherwise do not affect the material properties. This is due to the fact that in the entrance flow, the large amount of the randomly-oriented glass fibers in the reservoir must be converged into the die and adopts the highly oriented configurations [24].

FIG. 7. Loss modulus G'' of group A fiber suspensions for different volume fractions.

The effects of fiber aspect ratio on ΔP^sub end^ for different fiber suspensions have also been investigated. It was found that an increase in the fiber aspect ratio leads to an increase in ΔP^sub end^. Furthermore, it was noted the effect of the fiber aspect ratio on ΔP^sub end^ is more pronounced than its effect on the shear and complex viscosities. It is believed that the greater fiber breakage at the exit associated with the larger fiber aspect ratio as well as the greater fiber reorientation resistance may cause such important effect.

FIG. 8. Damping factor tan(δ) vs. the fiber volume fraction for group A fiber suspensions for different frequencies.

NUMERICAL MODELLING

In the present study, a modified Theological equation based on the FENE-P model that accounts for fiber interactions and the coupling between polymer chains conformation and fiber orientation, as discussed in Ref. 9, is used. Even though the FENE-P model is, strictly speaking, applicable only to dilute solutions, it has been successfully used to model the rheological behavior of polymer melts [25], and will be adopted here. A brief description of how the model has been adapted to the present fiber polymer-melt suspension will be given. The numerical predictions based on the model are analyzed in conjunction with experimental results to determine the effect of fiber-fiber interactions and polymer-fiber interactions.

FIG. 9. End pressure drops of group A fiber suspensions for different volume fractions.

In Eq. 7, δ is the unit tensor, |γ| is the scalar magnitude of the shear-rate tensor, and χ is equal to (re^sup 2^ - 1)/(re^sup 2^ + 1) where re = UD is the equivalent ellipsoidal aspect ratio of the fibers. The coefficient C^sub I^, known as the Folgar-Tucker constant, is a measure of the intensity of fiber interactions in the suspension, and can be correlated with the fiber aspect ratio re, the volume fraction [straight phi] as well as the fiber orientation [26, 27]. Values of the empirical parameter C^sub I^ have been reported in few studies, and the acceptable range of this parameter is still a subject of debate.

The time evolution equation for a2 cannot be solved without replacing the fourth-order tensor a4 with a function of the second- order tensor. Therefore, a closure approximation is required. Any approximation must satisfy the requirements of the orientation tensor such as symmetry and normalization.

FIG. 10. Effects of the fiber interaction coefficient C^sub I^ on the relative viscosity for the FENE-P model (re = 26, [straight phi] = 15%, and σ = 0.7).

Numerical Results

In the following, the numerical results for the dimensionless shear viscosity η^sub r^ = &951;^sub 0^, and relative fiber contribution to the total viscosity η^sub f^/η, in a two-dimensional shear flow are studied. The effects of the coupling factor σ and the fiber interaction coefficient C^sub I^, will be discussed. The calculations are conducted for an aspect ratio re of 26 corresponding to the group B fibers and 15 for the group A fibers.

Figure 10 depicts the shear viscosity for a coupling factor σ of 0.7 at four values of C^sub I^. The range of the constant C^sub I^ we want to explore is between 10^sup -1^ to 10^sup -4^. Such small values were proposed by Folgar and Tucker [39] and further experimentally tested by Bay [40]. Other authors [41, 42] have used values for C^sub I^ within a similar range for the generation of numerical predictions. Generally, the shear viscosity increases with the increase of C^sub I^ for all ranges of shear rates and the general shape of the curves is unchanged as C^sub I^ is varied. That is to say, C^sub I^ has no effect on the transition shear rate. Moreover, we observed that the increase in η/ η^sub 0^ with increasing C^sub I^ is not linear. We should note here that the values of C^sub I^ tested may not be all necessarily realistic but they allow determining the general trends on the viscosity.

FIG. 11. Effects of the coupling factor σ on the relative viscosity for the FENE-P model (re = 26, [straight phi] = 15%, and C^sub I^ = 0.01).

Figure 11 depicts the effects of σ on the shear viscosity when C^sub I^ = 0.01, re = 26, and [straight phi] = 15% for four different values of σ, which cover the different degrees of coupling between fibers and polymers. It is clear that variations in the values of σ affect both the viscosity plateau and the transition shear rate. The smaller σ, the stronger the coupling between the fibers and polymer chains, and the higher the viscosity plateau. Results presented in Fig. 11 also suggest that the higher the degree of fiber-polymer coupling, the stronger and the earlier the start of the shear-thinning behavior. Furthermore, the shear- thinning slope of the viscosity curve slightly increases when σ decreases. When compared to the effects of C^sub I^ discussed earlier, σ has much greater effects on the shear viscosity. This result shows clearly the importance of accounting for polymer- fiber coupling in order to reproduce the changes in the onset of the shear thinning effects due to fiber additives, as reported in the experimental measurements.

FIG. 12. Effects of the fiber volume fraction φ on the fiber relative contribution to the suspension viscosity for the FENE-P model (σ = 0.5 and re = 15).

Figure 12 shows that the fiber relative contribution to the suspension total viscosity increases dramatically when C^sub I^ increases. Similar results are obtained for the aspect ratio, and it was found that an increase of re increases the fiber contribution to the total stress and the larger the value of C^sub I^, the greater the effect of re. However, it seems that at large values of C^sub I^, the increase in the fiber relative contribution levels off. In addition, the effects of the fiber interaction coefficient C^sub I^ on the components of the fiber orientation tensor a2 were studied. The results indicate that stronger fiber interactions result in a less fully-oriented distribution where more fibers are not oriented in the main flow direction [43].

Model Predictions

The key parameters, C^sub I^ and σ, cannot be measured directly in the experimental study, and as empirical quantities they cannot be related easily to the physics of the flow. Therefore, we opted to determine the appropriate values of these two parameters by matching the numerical predictions of this section with the experimental data presented in the previous one.

In the polymer constitutive equation (Eq. 16), there are two material constants whose numerical values are a priori unknown: the relaxation time λ and the parameter b characterizing the maximal extensibility of the polymer chain. Since these two material constants are intrinsic properties of the polymer chain, they can be determined from the rheological curves of the polymer melt. The other important parameters, namely C^sub I^ and σ, are then determined by fitting the experimental results for the fiber suspensions. Reasonable fit was obtained when selecting b = 50 and λ = 0.2, as shown in Fig. 13. The shear viscosity predicted by the model matches the experimental curve very well for both the viscosity plateau and the shear-thinning slope. For the transition area, the model predicts the start of transition slightly earlier than the experimental data and the shape of the transition is sharper than the experimental one. The slight difference between th\e model prediction and the experimental data is due to the characteristics of the FENE-P model used in the present study.

Figures 14 and 15 show some representative results for the fit between the model predictions and the experimental measurements for group A fibers at low and high fiber loadings. It was found that the polymer-fiber coupling factor σ is a key parameter to get a good fit for the experimental results, and uncoupled results (σ = 1) result always in a very poor fit. However, as discussed earlier, the effect of varying the parameter C^sub I^ on the model predictions are less important than those of the parameter σ. As we can see from the figures, it is difficult to obtain a good fit in the shear thinning transition regions as the volume fraction of fibers increases. This result is symptomatic of a poor fit between the FENE-P model and experimental observations in this shear region. It also points to the more fundamental issue of whether complex polymer-fiber interactions such as those encountered in our system can be accurately represented by a single scalar.

FIG. 13. Experimental data and model predictions for the shear viscosity of LLDPE melt (b = 50 and λ = 0.2).

Similar fits were obtained for other volume fractions of group A and group B fibers, and the values of the parameters σ and C^sub I^ that result in the best fit between the experimental and numerical simulations results are shown in Tables 2 and 3. While it was not possible to draw definite conclusions regarding the effect of the fiber aspect ratio on σ and C^sub I^, results show that the value of σ decreases with increasing volume fraction. This is expected as more fibers can cause stronger effects on the polymer conformation and result in stronger polymer-fiber coupling (i.e., smaller σ). This trend has been consistently observed in the plateau and transition regions for both groups of fibers. The changes in values of σ required to fit the shear thinning region are smaller than those to fit the viscosity plateau. This may be explained by the fact that the effect of fiber volume fraction on the shear viscosity gradually vanishes at high shear rates, with the shear-thinning slopes undergoing very small changes resulting in a narrow range of σ to match them. In addition, the values of σ in the shear thinning region are found to be close to one for group A fibers. This is not surprising since at high shear rates, the polymer conformation is mainly controlled by the deformation rate and the contribution of the fibers on the rheological properties of the suspension becomes less important. However, such trend is not as clear for results obtained with group B fibers, which may be an indication that phenomena not accounted for in the model, such as slippage, may have started to play a more dominant role. We should finally mention that the fitted values of σ and C^sub I^ for the transition region may not be accurate since it was not possible to get an exact fit for the pure polymer in this region of the viscosity curve, which has led to a less satisfactory fit for the polymer suspensions.

FIG. 14. Experimental data and model predictions for the shear viscosity of E-glass fiber filled LLDPE melt (φ = 5% and re = 15).

FIG. 15. Experimental data and model predictions for the shear viscosity of E-glass fiber filled LLDPE melt (φ = 30% and re = 15).

Regarding the effect of the aspect ratio on σ and C^sub I^, it was not possible to draw definite conclusions by comparing the values listed in Tables 2 and 3. Indeed, even though when compared to group A, group B fibers show larger values of σ in the plateau region and smaller values in the transition and shear thinning regions, such variations cannot be easily correlated with the fibers aspect-ratios.

The above analysis shows conclusively that the coupling parameter necessary for improving the fit between the model predictions and the experimental measurements is not a constant but strongly depends on the shear rate and fiber volume fraction. In particular, it is found that σ is a decreasing function of φ at least in the viscosity plateau and transition region. There are also indications that σ decreases with increasing aspect ratio in the shear- thinning region, but an opposite trend is observed in the viscosity plateau.

TABLE 2. The dependance of the coupling factor σ and the fiber interaction coefficient C^sub I^ on the fiber volume fraction for group A fiber suspensions.

TABLE 3. The dependence of the coupling factor σ and the fiber interaction coefficient C^sub I^ on the fiber volume fraction for group B fiber suspensions.

CONCLUSION

The objective of the current research is to get a better understanding of the effects of fiber interactions and polymer- fiber interactions on the flow induced fiber orientation and rheological properties of fiber suspensions. To achieve these objectives, an experimental study was carried out and the results were analyzed in conjunction with the predictions of a mathematical model that accounts for the coupling between fiber orientation and matrix conformation, as well as fiber interactions.

By conducting experiments over a wide range of shear rates varying from 10^sup -4^ s^sup -1^ to 10^sup +4^ s^sup -1^ that has not been explored in earlier studies, we were able to build a full picture of the rheological behavior of fiber-polymer suspensions. Furthermore, the volume fractions we used, which are as high as 35% or 60 wt%, are rarely reported in the literature. Steady and dynamic viscosities as well as the end pressure drops were systematically analyzed to determine the effects of key parameters such as the volume fraction and the aspect ratio. It was found that the rheological properties of fiber suspensions increase with increasing fiber volume fraction and the fiber aspect ratio, the effects being more pronounced at low shear rate (frequency) than high shear rate. The effect of fiber volume fraction on ΔP^sub end^ is prominent even for very small fiber concentration that do not cause major changes in other rheological properties. The analysis led to new conclusions regarding the close interaction between the fiber orientation and rheological properties, and raised important issues about the applicability of the Cox-Merz rule and the behavior of the damping factor. Specifically, it is found that the complex viscosity is lower than the shear viscosity at low shear rates, and larger at high shear rates. The deviations between the two viscosities are more affected by changes in the volume fraction than the aspect ratio. An explanation for this phenomenon based on the changes in the fiber orientation with the coupled rheological properties, as well as the test modes used, was presented. Moreover, an inteipretation on the behavior of the damping factor based on the polymer relaxation time and the role of fibers during the flow was also formulated.

A systematic parametric study of fiber interactions and polymer- fiber coupling has been earned out using a modified FENE-P model to describe the polymer behavior. It was found that a strong fiber interaction increases the steady shear-viscosity and the fiber contribution to the total stress, but does not affect the start and slope of the shear thinning behavior. Results also showed that the fiber orientation distribution becomes more random with the increase of the degree of fiber interaction. On the other hand, the fiberpolymer coupling has a great effect on the steady flow curve of fiber suspensions. A stronger coupling results in a higher viscosity plateau, an earlier shear-thinning transition, and a stronger shear- thinning behavior.

Combining model predictions with experimental results, the effects of fiber-fiber interactions and polymer-fiber coupling were investigated and suitable values of C^sub I^ and σ were determined. It was found that it is necessary to account for the coupling between the polymer conformation and the fiber orientation through the empirical parameter σ in order to obtain a good agreement between the theoretical model and the experimental measurements. When the fiber orientation and polymer conformation are uncoupled, the fitting between the experimental data and the modeling prediction is always poor. Furthermore, this study revealed that the fiber interaction coefficient C^sub I^ and the polymer- fiber coupling parameter σ are not constants, but strongly depend on both the shear rate and the fiber volume fraction.

The results of the present study are interesting and show the great potential of using mathematical and numerical modeling to determine the rheological properties of polymer-fiber suspensions. However, more work needs to be done to further test and improve the model. In particular, transient shear material functions that have already been determined experimentally in some earlier studies [16, 44, 45] can be used to further test the validity of the model. Other improvements to the model may include accounting for the non- isotropic nature of fiber-fiber interactions and the use of new improved models developed for polymer melts.

ACKNOWLEDGMENTS

We would like to acknowledge Owens-Coming, Inc., for providing samples of glass fiber, and Exxon-Mobil Chemical for supplying the polyethylene resin.

NOMENCLATURE

NOMENCLATURE

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Rong Guo, Jalel Azaiez, Celine Bellehumeur

Department of Chemical and Petroleum Engineering, University of Calgary, Calgary, Alberta T2N 1N4, Canada

Correspondence to: J. Azaiez; e-mail: azaiez@ucalgary.ca

Contract grant sponsor: Natural Science and Engineering Research Council of Canada (NSERC).

DOI 10.1002/pen.20285

Published online 27 January 2005 in Wiley InterScience (www.interscience. wiley.com).

2005 Society of Plastics Engineers

Copyright Society of Plastics Engineers Mar 2005

Source: Polymer Engineering and Science

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