**4. Conclusions**

92 Viscoelasticity – From Theory to Biological Applications

polypropylene. To a certain degree, *<sup>c</sup>*

Filler content *<sup>c</sup>*

increasing shear rate the critical content *<sup>c</sup>*

numbers[41,42].

Figure 8. *<sup>c</sup>*

elastic energy is stored.

*B BF* ( , 0) ( , ) / ( )

 

<sup>0</sup> *B F BF k* ( , )/ ( ) ( / )

 

  where 0 *B*

 

i.e., *B*( , 0) 

in the PP melt.

at the interfacial region around the particles. The dispersed particles perturb the flow of the melt. It is interesting that this is consistent with previous publications showing that the dimensions of the extrudate were possibly smaller than those of the die at higher Reynolds

Plots of die swell versus filler concentration[75] are shown in Figure 9. Eq. (20) well demonstrates this correlation. In fitting the data with Eq. (20), it is found that with

increases. Larger die swell ratios are observed for pure PP at higher shear rates, since more

**Figure 9.** Extrudate swell ratio of PP/glass bead composites with different filler contents.

and *k*

the composite as shown in Figure 10. In this case, in the expression ' (/ )*<sup>q</sup>*

increases while *q*' decreases with increasing filler content. More importantly, *B F* ( , )/ ( )

is also appropriate for correlating shear stress and filler content for

is almost a linear function of shear stress. It can be written as

increases while *q* decreases when adding more glass beads to the

*q*

represents a critical shear rate when the melt

decreases, while the die swell ratio at

 against

as

0

*c c <sup>c</sup>*

decrease with increasing amount of filler

   ,

In addition, Eq. (21) is also successful in fitting the plots of *B B* ( , ) / ( , 0)

swelling is completely offset by the shrinkage of the mesoscopic network.

5% 1.249E10 0.2616 10% 2.945E13 0.1694 15% 6.188E15 0.1482

**Table 5.** Parameters in Eq. (19) (p=1) for the composite in Figure 5

Two limitations of Song's polymer extrudate swell theory have been identified for the first time. Song's model has been modified in order to predict the finite distance at which the swelling reaches a maximum. Furthermore, the model was extended to describe the die swell on extrusion from a short capillary by considering the entry effect in Song's molecular dynamics model and incorporating Liang's expression. The resulting modified model can be applied to extrusion swelling for both long and short capillaries, whereas Song's model is only appropriate for long capillaries. More importantly, the modified model is also suitable for analysis of the swelling of particle-filled composites which cannot be treated by Song's model. The composite swell ratio can be separated into the product of the matrix swell ratio and the concentration shift factor. The excellent agreements between the values predicted by the modified model and experimental data reported in the literature for a variety of different systems demonstrate its viability for a wide range of materials and experimental conditions.
