*3.2.1. Swelling simulations for pure LDPE extruded from capillaries with different L/D ratios*

The dashed curves in Figure 2 present the experimental data[77] of extrudate swell ratio for LDPE for different capillary *L D*/ ratio and shear rates from which (1 ) *W* can be calculated assuming 1 *<sup>t</sup> k* using Eq. (4), as listed in Table 1. *<sup>w</sup> f* is determined by fitting *B* against *L D*/ under different shear rates.

In addition, the swell ratio was also fitted by 0 (/ ) *<sup>B</sup> B B k LD* as suggested by Liang[75] (as shown by the dotted curves in Figure 2) and by Eq. (5) (as shown by the solid curves in Figure 2). The resulting values of key parameters for the three cases are compared in Table 1.


**Table 1.** Values of parameters for LDPE obtained from the plots in Figures 1 and 2

**Figure 2.** Predicted curves and experimental data of extrudate swell ratio for LDPE with different capillary L/D ratio and shear rates.

The comparisons of swell ratio predicted by Eq. (3), 0 (/ ) *<sup>B</sup> B B k LD* and Eq. (5) are also compared with experimental data (points) in Figure 2. All three models predict that the swell ratio decreases with increasing capillary L/D ratio and increases with increasing shear rate. It is also found that all three equations describe well the approximate linear relationship between the experimental swell ratio and L/D at larger values of L/D, i.e., in long capillaries. However, below a certain critical value of L/D, B increases rapidly with decreasing L/D. This trend is well predicted by Eq. 5 only. Liang's model[77] 0 (/ ) *<sup>B</sup> B B k LD* gives a completely different picture to experiment. Song's model Eq. (3) underestimates the swell ratio compared with experiment, showing that Eqs. (3) and (4) are only appropriate for swelling on extrusion from a long capillary, because of the assumption in Song's theory[69] that the entry effect is almost relaxed in capillary flow. In a short capillary, the chain elongation on entry cannot be neglected. The critical L/D ratio, which determines whether a capillary behaves as 'long' or 'short', increases with shear rate. This may be explained by the more incomplete relaxation of the elongated chains in the entry region at higher shear rate. The degree of relaxation depends on both the capillary length and the strength of the flow field as described by Eq. (5).

86 Viscoelasticity – From Theory to Biological Applications

against *L D*/ under different shear rates.

Dashed curves By Eq. (3) ( 1 *<sup>t</sup> k* )

1*W <sup>w</sup>*

capillary L/D ratio and shear rates.

*ratios* 

**3.2. Comparison of the predicted values of swell ratio with the experimental data** 

The dashed curves in Figure 2 present the experimental data[77] of extrudate swell ratio for LDPE for different capillary *L D*/ ratio and shear rates from which (1 ) *W* can be

In addition, the swell ratio was also fitted by 0 (/ ) *<sup>B</sup> B B k LD* as suggested by Liang[75] (as shown by the dotted curves in Figure 2) and by Eq. (5) (as shown by the solid curves in Figure

*n G*0 /Pa *a* <sup>0</sup>

0.57243 3784728 0.74712 6811860 IUPAC-LDPE

*f* <sup>0</sup> *B Bk equ B b*

2). The resulting values of key parameters for the three cases are compared in Table 1.

Dotted curves by <sup>0</sup> (/ ) *<sup>B</sup> B B k LD*

10 0.74 0.36 1.83 0.0021 1.573 2.4906 1 0.64 0.39 1.70 0.00185 1.561 1.9614 0.63 0.58 0.43 1.69 0.00125 1.557 1.6841 0.1 0.45 0.47 1.38 0.00030 1.316 0.7806

**Figure 2.** Predicted curves and experimental data of extrudate swell ratio for LDPE with different

**Table 1.** Values of parameters for LDPE obtained from the plots in Figures 1 and 2

*f* is determined by fitting *B*

Solid curves by / (1 ( / ) ) *<sup>a</sup> equ B B b LD* 

/Pa.s

*3.2.1. Swelling simulations for pure LDPE extruded from capillaries with different L/D* 

calculated assuming 1 *<sup>t</sup> k* using Eq. (4), as listed in Table 1. *<sup>w</sup>*

#### *3.2.2. Extrusion distance giving the maximum swell ratio for semi-dilute polymer solutions*

For a capillary with a L/D ratio of 85, the swelling of polyacrylamide solution was observed to reach a maximum at a given time *t* at a given distance (expressed as a multiple *Z* of capillary diameter *D* ) from the capillary[70], i.e., the swelling is finite. In the simulations, *n* , *a* and 0 were obtained by fitting the experimental ~ curve[70] using Eqs. (1) and (2). *Kt* and (1 ) *W* can be fitted using Eq. (3). *t* and *Z* were subsequently calculated from Eq. (11) and compared with the experimental data as shown in Figure 3. The calculated swell ratio B is close to the experimental data for 3.5468 *Kt* and (1 ) 0.5964 *W* . The simulated value of *t* decreases as a power law function of shear rate as observed from the experimental data in Figure 4a, as does *Z* as shown in Figure 4b.

**Figure 3.** Calculated plots of *B* versus from the experimental data for a semi-dilute polymer solution

**Figure 4.** Curves of ln( / ) *t Z* and *Z* versus calculated from the experimental data.

#### *3.2.3. Verification of the swelling equation for rubber in a short capillary*

The recoverable strain can be calculated by Eq. (23) due to Tanner[37].

$$S\_R = \left[ \mathcal{Z}(B^\delta - 1) - \mathcal{Z} \right]^{0.5} \tag{23}$$

Die Swell of Complex Polymeric Systems 89

**Figure 5.** Plots of fitted swell ratio versus shear stress compared with experimental data.

SI

SII

containing a higher filler content.

percentages. The fitted values of 0

marked when increasing the filler content.

Samples L/D / 4 *<sup>t</sup> k M*

0.2 0.32

0.2 0.22

*3.2.4. Application of the swelling equations for a PP/glass bead composite* 

 , <sup>0</sup> 

As shown in Figure 6, the viscosity of the PP/glass bead composite can be calculated from the experimental data of stress versus shear rate in the literature[75]. Eq. (1) accurately predicts the variation of viscosity with shear rate for composites with different filler

increase with increasing filler content in the composite while shear thinning becomes more

**Table 2.** Values of the parameters in Eq. (24) for rubber compounds

The filler content of sample SI with a matrix of NR rubber is lower than that of sample SII with a matrix of NR/SBR/CBR blend[73]. As seen from Table 2, the average relaxation time

is shorter, and the values of *a* and (1 ) *W* are lower for sample SI than for sample SII

40 0.18 0.56

40 0.48 0.70

0.12 0.26

0.72 0.53

, *a* and *n* are listed in Table 3; all of these values

*a* (1 ) *W*

0.64

0.72

Eq. (24) can be approximately fitted to give values of / 4 *Mkt* , *a* and 1*W* when 0.5 *<sup>w</sup> f* .

$$B = \frac{1}{2} (\frac{Mk\_t}{4}) \frac{1 + \left(\tau \dot{\gamma}\right)^a}{1 + \left(L/D\right)^a} S\_R \rangle^{\left(1 - W\right)} + \frac{5.098}{4} \rangle^{0.5} \tag{24}$$

The experimental swell ratios[73] are shown in Figure 5, from which the values of the parameters in Eq. (24) can be obtained by fitting and they are listed in Table 2. Figure 5 also illustrates that the swell ratios predicted by Eq. (24) are in good agreement with the experimental data. It should be noted that the swell ratios reported in the literature[73] were measured when the extrudate was naturally cooled to ambient temperature. The swell ratio is considerably larger for extrusion from a very short capillary than from a long capillary. A larger fraction of the relaxation occurs in the melt state at higher temperature in the longer capillary than in the short capillary. The unrelaxed energy is not easily released from the capillary at low temperature. The fitted value of (1 ) *W* is larger for swelling on extrusion from the short capillary than from the long capillary, which reflects a greater amount of retained elasticity in the former case.

**Figure 5.** Plots of fitted swell ratio versus shear stress compared with experimental data.

**Figure 4.** Curves of ln( / ) *t Z*

retained elasticity in the former case.

and *Z* versus

*3.2.3. Verification of the swelling equation for rubber in a short capillary* 

The recoverable strain can be calculated by Eq. (23) due to Tanner[37].

*Mk B S*

(a) (b)

Eq. (24) can be approximately fitted to give values of / 4 *Mkt* , *a* and 1*W* when 0.5 *<sup>w</sup>*

*L D* 

1 1( ) (1 ) 5.098 0.5 (( ) ) 2 4 1(/ ) 4 *<sup>a</sup> <sup>t</sup> <sup>W</sup> a R*

The experimental swell ratios[73] are shown in Figure 5, from which the values of the parameters in Eq. (24) can be obtained by fitting and they are listed in Table 2. Figure 5 also illustrates that the swell ratios predicted by Eq. (24) are in good agreement with the experimental data. It should be noted that the swell ratios reported in the literature[73] were measured when the extrudate was naturally cooled to ambient temperature. The swell ratio is considerably larger for extrusion from a very short capillary than from a long capillary. A larger fraction of the relaxation occurs in the melt state at higher temperature in the longer capillary than in the short capillary. The unrelaxed energy is not easily released from the capillary at low temperature. The fitted value of (1 ) *W* is larger for swelling on extrusion from the short capillary than from the long capillary, which reflects a greater amount of

calculated from the experimental data.

6 0.5 [2( 1) 2] *RS B* (23)

(24)

*f* .

The filler content of sample SI with a matrix of NR rubber is lower than that of sample SII with a matrix of NR/SBR/CBR blend[73]. As seen from Table 2, the average relaxation time is shorter, and the values of *a* and (1 ) *W* are lower for sample SI than for sample SII containing a higher filler content.


**Table 2.** Values of the parameters in Eq. (24) for rubber compounds

#### *3.2.4. Application of the swelling equations for a PP/glass bead composite*

As shown in Figure 6, the viscosity of the PP/glass bead composite can be calculated from the experimental data of stress versus shear rate in the literature[75]. Eq. (1) accurately predicts the variation of viscosity with shear rate for composites with different filler percentages. The fitted values of 0 , <sup>0</sup> , *a* and *n* are listed in Table 3; all of these values increase with increasing filler content in the composite while shear thinning becomes more marked when increasing the filler content.

Die Swell of Complex Polymeric Systems 91

*cb B* 

(here *k* and *p* in

*mB*

*f* for the pure matrix melt) to lower values with increasing filler concentration. This shows that recovery is harder for an entangled network with higher

*f cb*

In addition, it is interesting that the swell is well depicted by one simple equation

sensitive to the shear, reflecting the recovery capability of the composites. When increasing

that the elasticity is weakened while the rigidity becomes stronger, which corresponds to the

may be related to the composite rigidity or the density while *mB* is

versus shear rate with experimental data.

 

against shear rate based on the experimental data in Figures 6 and

Eq. (20) are both unity) illustrates the reinforcement effect while reflecting the weaker swelling capability. In our opinion, there are two levels of network structure. The microscopic one involves molecular entanglement, whilst the other involves mesoscopic particles dispersed in the compound melt. The concentration shift factor shows that the swelling functions of the two levels may be separated. The mesoscopic network probably shrinks, counteracting part of the microscopic melt swelling. The offset effect can be described by the concentration shift factor. However, the filler is far larger than the polymer chains and does not vary the basic molecular relaxation dynamics of the melt matrix except

becomes larger while *mB* becomes smaller, showing

Dashed curves given by Eq. (3) Dotted curves given by (/ )*mb*

0 0.56165 0.5137 127.51 0.16 5% 0.55176 0.5003 138.73 0.1307 10% 0.53287 0.4400 153.45 0.09821 15% 0.49195 0.4217 169.62 0.08876

**Table 4.** Values of model parameters in the equations for the composite in Figure 7

filler content than for the corresponding matrix network.

Filler content 1*W <sup>w</sup>*

the filler content in the composite, *cb*

**Figure 8.** Comparison of plots of *B B* ( , ) / ( , 0)

Plots of *B B* ( , ) / ( , 0) 

  7 are shown in Figure 8. The concentration shift factor ( ) [1 / ] *<sup>c</sup> F*

 

observed decrease in swell ratio.

0.5137 0.5 *<sup>w</sup>*

(/ )*mb cb B* . *cb* 

**Figure 6.** Experimental data for variation of viscosity with shear rate and the simulated values for the composites.


**Table 3.** Structural parameters for PP/glass bead composites

**Figure 7.** Comparison of predicted swell ratio with experimental data for PP/glass bead composites.

Eq. (6) can be used to predict the extrudate swell ratio of the composites with the fitted structural parameters listed in Table 3. Comparison of the predicted data with the experimental values are shown in Figure 7, assuming 1 *<sup>t</sup> k* . The modified theory shows better agreement at higher shear rate which indicates its validity for particulate-filled composites. Table 4 shows that(1 ) *W* , which is the residual fraction of the recoverable entanglement in polymeric chains after flow shear, decreases with increasing filler content. Smaller values of(1 ) *W* correspond to lower elastic recovery, i.e., swell ratios are lower for composites with higher filler content. Furthermore, it can be seen that *<sup>w</sup> f* decreases (from

0.5137 0.5 *<sup>w</sup> f* for the pure matrix melt) to lower values with increasing filler concentration. This shows that recovery is harder for an entangled network with higher filler content than for the corresponding matrix network.


**Table 4.** Values of model parameters in the equations for the composite in Figure 7

90 Viscoelasticity – From Theory to Biological Applications

Filler content *n* <sup>0</sup>

**Table 3.** Structural parameters for PP/glass bead composites

composites.

**Figure 6.** Experimental data for variation of viscosity with shear rate and the simulated values for the

5% 0.2071 1.31785 1.6011 2.2607 10% 0.2175 1.47162 1.6849 2.8496 15% 0.3812 1.85625 1.7210 3.5809

**Figure 7.** Comparison of predicted swell ratio with experimental data for PP/glass bead composites.

composites with higher filler content. Furthermore, it can be seen that *<sup>w</sup>*

Eq. (6) can be used to predict the extrudate swell ratio of the composites with the fitted structural parameters listed in Table 3. Comparison of the predicted data with the experimental values are shown in Figure 7, assuming 1 *<sup>t</sup> k* . The modified theory shows better agreement at higher shear rate which indicates its validity for particulate-filled composites. Table 4 shows that(1 ) *W* , which is the residual fraction of the recoverable entanglement in polymeric chains after flow shear, decreases with increasing filler content. Smaller values of(1 ) *W* correspond to lower elastic recovery, i.e., swell ratios are lower for

/s *a* <sup>0</sup>

/kPa.s

*f* decreases (from

In addition, it is interesting that the swell is well depicted by one simple equation (/ )*mb cb B* . *cb* may be related to the composite rigidity or the density while *mB* is sensitive to the shear, reflecting the recovery capability of the composites. When increasing the filler content in the composite, *cb* becomes larger while *mB* becomes smaller, showing that the elasticity is weakened while the rigidity becomes stronger, which corresponds to the observed decrease in swell ratio.

**Figure 8.** Comparison of plots of *B B* ( , ) / ( , 0) versus shear rate with experimental data.

Plots of *B B* ( , ) / ( , 0) against shear rate based on the experimental data in Figures 6 and 7 are shown in Figure 8. The concentration shift factor ( ) [1 / ] *<sup>c</sup> F* (here *k* and *p* in Eq. (20) are both unity) illustrates the reinforcement effect while reflecting the weaker swelling capability. In our opinion, there are two levels of network structure. The microscopic one involves molecular entanglement, whilst the other involves mesoscopic particles dispersed in the compound melt. The concentration shift factor shows that the swelling functions of the two levels may be separated. The mesoscopic network probably shrinks, counteracting part of the microscopic melt swelling. The offset effect can be described by the concentration shift factor. However, the filler is far larger than the polymer chains and does not vary the basic molecular relaxation dynamics of the melt matrix except

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 numbers[41,42].

Die Swell of Complex Polymeric Systems 93

**Figure 10.** Plots of *B F* ( , )/ ( )

composites.

conditions.

Kejian Wang\*

**Author details** 

**5. References** 

Corresponding Author

 \*

*Institute of Plastics Machinery and Engineering,* 

*Beijing University of Chemical and Technology, Beijing, China* 

[2] Liang J. Z. J Mater Process Technol 1995; 52, 207-215.

[1] Liang J.Z. Plast Rubber Compos Process Appl 1991; 15, 75-79.

[3] Lodge A.S. Elastic Liquids, Academic Press, New York, 1964.

**4. Conclusions**

 

versus shear stress from experimental data for PP/glass bead

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

In addition, Eq. (21) is also successful in fitting the plots of *B B* ( , ) / ( , 0) against as Figure 8. *<sup>c</sup>* increases while *q* decreases when adding more glass beads to the polypropylene. To a certain degree, *<sup>c</sup>* represents a critical shear rate when the melt swelling is completely offset by the shrinkage of the mesoscopic network.


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

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 increasing shear rate the critical content *<sup>c</sup>* decreases, while the die swell ratio at 0 increases. Larger die swell ratios are observed for pure PP at higher shear rates, since more elastic energy is stored.

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

*B BF* ( , 0) ( , ) / ( ) is also appropriate for correlating shear stress and filler content for the composite as shown in Figure 10. In this case, in the expression ' (/ )*<sup>q</sup> c c <sup>c</sup>* increases while *q*' decreases with increasing filler content. More importantly, *B F* ( , )/ ( ) , i.e., *B*( , 0) is almost a linear function of shear stress. It can be written as <sup>0</sup> *B F BF k* ( , )/ ( ) ( / ) where 0 *B* and *k* decrease with increasing amount of filler in the PP melt.

**Figure 10.** Plots of *B F* ( , )/ ( ) versus shear stress from experimental data for PP/glass bead composites.
