**2.1. The extrudate swell theory for a long capillary**

Up to now, most of the extrudate swell models have been established on the basis of the analysis of long capillary flow. The swell ratio B has been related to the recoverable shear strain (SR) or elastic strain energy[62]. One most pertinent systematic theories of extrudate swell of entangled polymeric liquids is that of Tanner, who first based his model on the K-BKZ constitutive equation and the free recovery from Poiseuille flow with the aspect ratio of length *L* to diameter *D* being very large[37]. The correlation was later confirmed for a wide class of constitutive equations, including PTT, pom–pom and general network type models for fullydeveloped tube flow[63]. It was found that the extrudate swelling ratio *B* may be expressed

as a function of the first normal difference *N*1 and the shear stress 12 . Regardless of the fact that this clearly shows how the swell is related to the elasticity of viscoelastic polymeric fluids, the viscous heating and the time-dependent nature of swell are not considered.

Die Swell of Complex Polymeric Systems 81

]. (1 ) *W* is the fraction

(4)

(5)

is the shear

, the molecular parameters of the zero shear

From experimental data of

rate-dependent constant.

**2.2. Finite extrudate swell distance** 

introducing two parameters *<sup>t</sup> k* and *<sup>w</sup>*

Here *<sup>w</sup>*

as

viscosity 0 

Song[69]:

2

calculated from the experimental swell ratio data by Eq. (4).

(1 )

length to diameter ratio *L D*/ and the shear rate

 and 1 ( ) 

 <sup>0</sup> 1/ (1 ) 1/2 0

*G* 

*n W B n L D*

, the equilibrium modulus *G*<sup>0</sup> , and the exponents *n* and *a* can be determined.

(3)

or shear stress

The approximate value of the ultimate extrudate swell *B* was obtained as Eq. (3) by

<sup>1</sup> (( ( ) / (1 ( / ) )) 5.098)

*B* depends on both the molecular parameters and the operational variables [the capillary

of the recoverable conformation of the entangled polymeric chain in the flow, which can be

2

*B*

0 1/ 0

Details of the derivation of the above equations (3) and (4) and their application to the extrudate swell of HDPE and PDB are given by Song[69]. Eq.3 may also be expressed in the

*W B*

form as shown in Eq. (5), where *equ B* is the die swell ratio at *L D*/ , and *m*

*f* :

0

*<sup>t</sup> B kn L D G* 

<sup>0</sup> 1/ (1 )

2

*<sup>b</sup> B B*

 

*G*  4

<sup>2</sup> ln(4 5.098)

ln( ( ) / (1 ( / ) )) *n a*

(1 ( / ) ) *mr*

*L D* 

*equ a*

As discussed above, the swelling evolves with time after the polymer is extruded from the capillary, during which time the molecules continue to exhibit similar disentanglementreentanglement transitions and uncoil-recoil transitions to those occurring whilst it is still inside the capillary. To describe this phase, Eq. (3) can be reformulated as Eq. (6) by

<sup>1</sup> ( ( ( ) / (1 ( / ) )) 5.098)

the coefficient *<sup>t</sup> k* describes the evolution of the extrudate swell with time. *<sup>t</sup> k* may be written

<sup>0</sup> 0 0 1 ln( / ) / (1 ( / ) ) *<sup>t</sup> k A tt G*

*f* is a variable which replaces the fixed value of 0.5 used by Song[69] in Eq. (3) and

*<sup>w</sup> n W f*

(6)

(7)

 

*n L D*

More recently, Wang et al.[64,65] extensively exploited the variations of extrudate swell with both polymer characteristics ( *Mn* , / *Mn W M* and *Me* ) and the operational processing conditions. A double molecular mechanism of disentanglement-reentanglement and decoilrecoil was proposed to rationalize variations of polymer elasticity during flow with chain conformation. The die swell behavior essentially results from the molecular dynamics of the system. Thus, it is desirable to establish a single primary theoretical framework to relate the extrudate swell to the intrinsic viscoelasticity and external conditions.

There are three kinds of polymer segments and chains in the extrudates, which have been defined by Song[66] as extending chain, coil chain and entangled polymeric chain. Besides the change in chain conformation and the degree of reorganization of the constituent chains, the extension and flow can also induce the dynamic and reversible disentanglement and reentanglement between polymeric chains, such that the polymeric melts then undergo a partial stress relaxation leading to extrudate swelling at the die exit. The swell ratio is affected by the length to diameter ratio and the residence time. Based on such a viewpoint of dynamics, Song developed a novel molecular theory of such multiple entangled constituent chains in order to analyze non-linear viscoelasticity for the polymeric melts. His derived constitutive equations and material functions in a multiple-flow field were subsequently quantitatively verified[67]. Based on the O-W-F constitutive equation and the multiple transient-network model as well as the double relaxation dynamics of the reentanglement-disentanglement transition (RE-DT) and recoil-uncoil transition (RC-UCT) from the Poiseuille flow, Song proposed that swell evolves in three stages (instantaneous swelling, delayed swelling and ultimate extrudate swelling)[68]. A new set of swelling equations incorporating molecular parameters, operational parameters and growth time in both the steady state and dynamic state were developed. Song's model successfully described the die swell through a long capillary of both linear polyethylene (HDPE) and linear polybutadiene (PBD) with the different molecular weights and different processing variables[69]. In this paper, Song's extrudate swell theory will be extended in order to analyze both the die swell out of a short capillary and the swell of polymer composites.

In the steady shear flow, the shear viscosity can be written as

$$\eta(\dot{\boldsymbol{\gamma}}) = \eta\_0 \left/ \left[ 1 + (\frac{\eta\_0 \dot{\boldsymbol{\gamma}}}{\mathbf{G}\_0})^a \right]^n \right. \tag{1}$$

The above expression is verified by experimental data, and can also be deduced from the O-W-F constitutive equation together with molecular dynamics[69]. The coefficient of the first normal-stress difference 1 in the steady shear flow is

$$\Psi\_1(\dot{\gamma}) = 2n\eta\_0^{1+1/n} \left/ (G\_0)^{1/n} \left/ \left[ 1 + \left( \frac{\eta\_0 \dot{\gamma}}{G\_0} \right)^a \right]^{n+1} \right. \tag{2}$$

From experimental data of and 1 ( ) , the molecular parameters of the zero shear viscosity 0 , the equilibrium modulus *G*<sup>0</sup> , and the exponents *n* and *a* can be determined. The approximate value of the ultimate extrudate swell *B* was obtained as Eq. (3) by Song[69]:

$$B = \frac{1}{2} (\langle n (\frac{\eta\_0}{G\_0})^{1/n} \dot{\gamma} / \left(1 + (L / D)^{\alpha} \right))^{(1-\mathcal{W})} + 5.098)^{1/2} \tag{3}$$

*B* depends on both the molecular parameters and the operational variables [the capillary length to diameter ratio *L D*/ and the shear rate or shear stress ]. (1 ) *W* is the fraction of the recoverable conformation of the entangled polymeric chain in the flow, which can be calculated from the experimental swell ratio data by Eq. (4).

$$\ln(1 - W) = \frac{\ln(4B^2 + \frac{2}{B^4} - 5.098)}{\ln(n(\frac{\eta\_0}{G\_0})^{1/n} \circ / \{1 + (L / D)^d\})}\tag{4}$$

Details of the derivation of the above equations (3) and (4) and their application to the extrudate swell of HDPE and PDB are given by Song[69]. Eq.3 may also be expressed in the form as shown in Eq. (5), where *equ B* is the die swell ratio at *L D*/ , and *m* is the shear rate-dependent constant.

$$B = B\_{eq\mu} + \frac{b\dot{\nu}^{m\_r}}{\left(1 + \left(L/D\right)^a\right)}\tag{5}$$

#### **2.2. Finite extrudate swell distance**

80 Viscoelasticity – From Theory to Biological Applications

as a function of the first normal difference *N*1 and the shear stress 12

extrudate swell to the intrinsic viscoelasticity and external conditions.

In the steady shear flow, the shear viscosity can be written as

normal-stress difference 1 in the steady shear flow is

the viscous heating and the time-dependent nature of swell are not considered.

that this clearly shows how the swell is related to the elasticity of viscoelastic polymeric fluids,

More recently, Wang et al.[64,65] extensively exploited the variations of extrudate swell with both polymer characteristics ( *Mn* , / *Mn W M* and *Me* ) and the operational processing conditions. A double molecular mechanism of disentanglement-reentanglement and decoilrecoil was proposed to rationalize variations of polymer elasticity during flow with chain conformation. The die swell behavior essentially results from the molecular dynamics of the system. Thus, it is desirable to establish a single primary theoretical framework to relate the

There are three kinds of polymer segments and chains in the extrudates, which have been defined by Song[66] as extending chain, coil chain and entangled polymeric chain. Besides the change in chain conformation and the degree of reorganization of the constituent chains, the extension and flow can also induce the dynamic and reversible disentanglement and reentanglement between polymeric chains, such that the polymeric melts then undergo a partial stress relaxation leading to extrudate swelling at the die exit. The swell ratio is affected by the length to diameter ratio and the residence time. Based on such a viewpoint of dynamics, Song developed a novel molecular theory of such multiple entangled constituent chains in order to analyze non-linear viscoelasticity for the polymeric melts. His derived constitutive equations and material functions in a multiple-flow field were subsequently quantitatively verified[67]. Based on the O-W-F constitutive equation and the multiple transient-network model as well as the double relaxation dynamics of the reentanglement-disentanglement transition (RE-DT) and recoil-uncoil transition (RC-UCT) from the Poiseuille flow, Song proposed that swell evolves in three stages (instantaneous swelling, delayed swelling and ultimate extrudate swelling)[68]. A new set of swelling equations incorporating molecular parameters, operational parameters and growth time in both the steady state and dynamic state were developed. Song's model successfully described the die swell through a long capillary of both linear polyethylene (HDPE) and linear polybutadiene (PBD) with the different molecular weights and different processing variables[69]. In this paper, Song's extrudate swell theory will be extended in order to analyze both the die swell out of a short capillary and the swell of polymer composites.

0 n

(1)

1

*<sup>n</sup> <sup>a</sup>*

(2)

0

*G* 

0

0

 

1 00

 

( ) 2 /( ) / 1

*n n n G*

( ) / [1 ( ) ] <sup>G</sup> *a*

The above expression is verified by experimental data, and can also be deduced from the O-W-F constitutive equation together with molecular dynamics[69]. The coefficient of the first

1 1/ 1/ 0

. Regardless of the fact

As discussed above, the swelling evolves with time after the polymer is extruded from the capillary, during which time the molecules continue to exhibit similar disentanglementreentanglement transitions and uncoil-recoil transitions to those occurring whilst it is still inside the capillary. To describe this phase, Eq. (3) can be reformulated as Eq. (6) by introducing two parameters *<sup>t</sup> k* and *<sup>w</sup> f* :

$$B = \frac{1}{2} (k\_t (\eta(\frac{\eta\_0}{G\_0})^{1/n} \dot{\gamma} / (1 + (L / D)^\alpha))^{(1 - \mathbb{W})} + 5.098)^{f\_w} \tag{6}$$

Here *<sup>w</sup> f* is a variable which replaces the fixed value of 0.5 used by Song[69] in Eq. (3) and the coefficient *<sup>t</sup> k* describes the evolution of the extrudate swell with time. *<sup>t</sup> k* may be written as

$$k\_t = 1 + A\_0 \ln(t/t\_\alpha) / \left(1 + (\eta\_0 \dot{\gamma}/G\_0)^\alpha\right) \tag{7}$$

As the swell approaches a maximum at *t t* , *<sup>t</sup> k* approaches unity corresponding to the model used by Song[69]. However in contrast to Song's model, in practice *t* is not infinite and actually has a finite value which can be determined experimentally[70], i.e., the maximum ultimate swelling will be realized at time *t* along the extrusion distance. If the extrusion distance is expressed as *Z D* , where D is the capillary diameter and Z is a numerical factor, *t* and *Z* can be correlated as follows from the shear rate,

$$t\_{\alpha\gamma} = 8k\_n Z / \dot{\gamma} \tag{8}$$

Die Swell of Complex Polymeric Systems 83

<sup>0</sup> is the half-entry

shows the elastic strain induced by the stored

(15)

*<sup>R</sup> B S* (17)

(18)

 and 1 ( ) 

for a very short capillary as

for filled

shows the effects of capillary length, i.e., the degree of

(16)

where *RS* is the recoverable shear strain; *<sup>l</sup>*

0

expressed in terms of the Bagley entrance correction factor ( *e* ).

Eq. (7) and <sup>0</sup> 1( ) (1 )

*k*

[ ] 1(/ )

*L D*

*L a*

 

*a*

**2.4. Extrudate swell theory of filled composites** 

theory may be modified for use with filled composites.

viscoelasticity; *<sup>l</sup>*

as Eq. (12).

Eq. (18),

Eq. (19),

energy in the capillary reservoir, which is related to the capillary geometry and the fluid

convergence angle of a viscoelastic fluid, which is given by a function of the constitute exponent *n* of the fluid, the entry pressure drop *P* and the ratio of the capillary diameter *D* to its reservoir diameter *Dr* . The pressure loss in the entry region can be approximately

<sup>1</sup> 4(1 ( / ) ) 2 (( / ) 1) tan [ ] 3( 1) 3(1 )

2 *P*

The total die swell ratio in Eq. (3) can be approximated as Eq. (17), which has the same form

1 1/2 (5.098 / 4 / 4) *<sup>w</sup>*

Eq. (17) can be modified to describe the extrudate swell out of a short capillary, as shown in

<sup>1</sup> (1 ) ( ( ) 5.098 / 4)

where *<sup>t</sup> k* is the time-dependent coefficient reflecting the swelling evolution as defined by

relaxation. *M* / 4 is the recoverable effect from the stored energy in the capillary reservoir. It

defined by Eq. (14). Thus, Eq. (18) may be used for both long and short capillaries since it

In particle-filled composites, the filler is distributed in an entangled matrix network and the filling affects the network relaxation[74]. Regardless of the variation in viscoelastic

composites are similar to those of the pure polymer matrix, i.e., the above extrudate swell

However, experiments have shown that the die swell is usually weakened with increasing filler concentration [75]. This suggests that the die swell of a composite can be expressed by

includes sufficient variables, whereas Eq. (3) is only appropriate for long capillaries.

*<sup>w</sup> W f*

2 4

*w*

may be *ca.* 1 for a sufficiently long capillary while it may be *<sup>l</sup>*

properties, it has been found experimentally that the values of

*tL R <sup>M</sup> B kk S* 

*e n n*

1

*e*

1.5( 1) 1.5( 1)

*n n D Dr r D D*

is a function of the entry converging flow parameter;

where *nk* is a constant for a given material, whose approximate value is

$$k\_n = (1 + 3n) / 4n \tag{9}$$

where *n* is the constitute exponent of the fluid. From Eqs. (7)–(9), it can be shown that

$$\ln t\_{\alpha} = \ln k\_{\alpha} + \ln Z - \ln \dot{\gamma} - \frac{k\_{\dot{t}}}{1 + \left(\tau \dot{\gamma}\right)^{\alpha}} \tag{10}$$

Eq. (10) can be rewritten as Eq. (11).

$$\ln \text{ln}(t\_{\text{eq}} \,\,^\*\dot{\nu}) - \ln Z = \ln k\_{\text{n}} - \frac{k\_t}{1 + (\tau \dot{\gamma})^\alpha} \tag{11}$$

Thus, either Eq. (10) or Eq. (11) can be used to predict the time of maximum swelling and the corresponding swell distance.

#### **2.3. Extrudate swell theory for a short capillary**

The above model is based on the assumption that the chain elongation incurred on reservoir entry is fully relaxed in the capillary. This is only approximately true for extrusion in a long capillary and a very poor assumption for a short capillary where the the entry flow is more complicated and the entry effect is more prominent[71]. Liang[72] reported the results of many swelling experiments using a short capillary and some quantitative empirical relations between the swell ratio and the material characteristics and operational parameters have been obtained[37,70,73]. In this paper, we attempt to modify Song's model in order to describe the swelling behavior in short capillary extrusion.

Liang[74] described the die swell ratio as follows,

$$B = \left(1 + \mathcal{Z}\_l S\_R\right)^{1/2} \tag{12}$$

$$S\_R = \left(N\_1 / 2\pi\right)^{1/2} \tag{13}$$

$$
\lambda\_l = 0.5 \tan \alpha\_0 \tag{14}
$$

where *RS* is the recoverable shear strain; *<sup>l</sup>* shows the elastic strain induced by the stored energy in the capillary reservoir, which is related to the capillary geometry and the fluid viscoelasticity; *<sup>l</sup>* is a function of the entry converging flow parameter; <sup>0</sup> is the half-entry convergence angle of a viscoelastic fluid, which is given by a function of the constitute exponent *n* of the fluid, the entry pressure drop *P* and the ratio of the capillary diameter *D* to its reservoir diameter *Dr* . The pressure loss in the entry region can be approximately expressed in terms of the Bagley entrance correction factor ( *e* ).

$$\tan \alpha\_0 = \frac{1}{e} [\frac{4(1 - \left(D \,/\, D\_r\right)^{1.5(n+1)})}{\Im(n+1)} + \frac{2\,\sharp(\left(D\_r \,/\, D\right)^{1.5(n+1)} - 1)}{\Im(1 - n)}] \tag{15}$$

$$\frac{1}{e} = \frac{\Delta P}{2\pi} \tag{16}$$

The total die swell ratio in Eq. (3) can be approximated as Eq. (17), which has the same form as Eq. (12).

$$B = \left(5.098 \,/\, 4 + S\_R^{1-w} \,/\, 4\right)^{1/2} \tag{17}$$

Eq. (17) can be modified to describe the extrudate swell out of a short capillary, as shown in Eq. (18),

$$B = \frac{1}{2} (k\_{\! \! \! } k\_{L \! \! \! \! }\_{L \! \! \! } \frac{M}{4} (\\$\_{R})^{(1-W)} + 5.098 \ / \ 4 \}^{\! \! \! \! \! } \text{u} \tag{18}$$

where *<sup>t</sup> k* is the time-dependent coefficient reflecting the swelling evolution as defined by

Eq. (7) and <sup>0</sup> 1( ) (1 ) [ ] 1(/ ) *a L a w k L D* shows the effects of capillary length, i.e., the degree of

relaxation. *M* / 4 is the recoverable effect from the stored energy in the capillary reservoir. It may be *ca.* 1 for a sufficiently long capillary while it may be *<sup>l</sup>* for a very short capillary as defined by Eq. (14). Thus, Eq. (18) may be used for both long and short capillaries since it includes sufficient variables, whereas Eq. (3) is only appropriate for long capillaries.

#### **2.4. Extrudate swell theory of filled composites**

82 Viscoelasticity – From Theory to Biological Applications

8/ *<sup>n</sup> t kZ*

Eq. (10) can be rewritten as Eq. (11).

the corresponding swell distance.

**2.3. Extrudate swell theory for a short capillary** 

describe the swelling behavior in short capillary extrusion.

Liang[74] described the die swell ratio as follows,

As the swell approaches a maximum at *t t* , *<sup>t</sup> k* approaches unity corresponding to the model used by Song[69]. However in contrast to Song's model, in practice *t* is not infinite and actually has a finite value which can be determined experimentally[70], i.e., the maximum ultimate swelling will be realized at time *t* along the extrusion distance. If the extrusion distance is expressed as *Z D* , where D is the capillary diameter and Z is a

*n*

1( ) *t*

*t*

(11)

(1 3 ) / 4 *nk nn* (9)

where *n* is the constitute exponent of the fluid. From Eqs. (7)–(9), it can be shown that

*<sup>k</sup> t kZ*

ln( \* ) ln ln 1( )

Thus, either Eq. (10) or Eq. (11) can be used to predict the time of maximum swelling and

The above model is based on the assumption that the chain elongation incurred on reservoir entry is fully relaxed in the capillary. This is only approximately true for extrusion in a long capillary and a very poor assumption for a short capillary where the the entry flow is more complicated and the entry effect is more prominent[71]. Liang[72] reported the results of many swelling experiments using a short capillary and some quantitative empirical relations between the swell ratio and the material characteristics and operational parameters have been obtained[37,70,73]. In this paper, we attempt to modify Song's model in order to

> 1/2 (1 ) *l R B S*

<sup>1</sup> ( /2 ) *RS N*

<sup>0</sup> 0.5tan *<sup>l</sup>*

1/2

*<sup>k</sup> t Zk*

ln ln ln ln

*n*

(8)

(10)

(12)

(13)

(14)

numerical factor, *t* and *Z* can be correlated as follows from the shear rate,

where *nk* is a constant for a given material, whose approximate value is

In particle-filled composites, the filler is distributed in an entangled matrix network and the filling affects the network relaxation[74]. Regardless of the variation in viscoelastic properties, it has been found experimentally that the values of and 1 ( ) for filled composites are similar to those of the pure polymer matrix, i.e., the above extrudate swell theory may be modified for use with filled composites.

However, experiments have shown that the die swell is usually weakened with increasing filler concentration [75]. This suggests that the die swell of a composite can be expressed by Eq. (19),

$$B(\dot{\gamma}, \phi) = B(\dot{\gamma}, \phi = 0)F(\phi) \tag{19}$$

Die Swell of Complex Polymeric Systems 85

The swell data for a rubber compound[73] were used to verify the effectiveness of the suggested equations for polymer extrusion in a short capillary. Sample SI was a calcium carbonate (CaCO3) filled natural rubber (NR) compound, in which the content of CaCO3 was 20 phr. Sample SII was a carbon-black (CB)-filled NR/styrene-butadiene rubber SBR) /cis-1,4-butadiene rubber (CBR) compound. The blending ratio of NR/SBR/CBR was 45/10/45, and the content of CB was 56 phr. Both the rubber compounds included some additives. An Instron capillary rheometer (model 3211) was used in the tests. Two capillary dies with different lengths were selected in order to measure the rheological properties of the sample materials. The length of the long die was 40 mm, the length of the short die was 0.2 mm, and

Another set of data was for a polypropylene/glass bead composite[75]. The matrix resin was a common polypropylene (Pro-fax 6331). The melt flow rate (2.16 kg, 230 oC) and density were 12 g/10 min and 0.9 g/cm3, respectively. The glass bead filler had a mean diameter of 219 nm and density of 2.5 g/cm3. Rheological experiments with these samples were carried out using a Rosand capillary rheometer with twin cylinders. Two dies of different length with the same diameter (1 mm) were selected in order to measure the entry pressure losses. The length/diameter of the short die was << 1 and the other was 16. The test temperature

> and 1 1 ln( / ln(( / )) / )

> > .

 

 *d d* versus ln

from the

was 190 oC, and apparent shear rates were varied from 50 to10–4s–1.

the diameter of both dies was 1 mm.

**Figure 1.** Plots of ln

experimental data for LDPE[77] of

 versus 1 ln

> ~ and 1 *N* ~

where *B*(,) and ( , 0) *<sup>m</sup> B B* are the die swell of the composite with a filler content of and the die swell of the pure matrix, respectively, and *F*( ) is the filling effect. Eq. (19) implies that the viscoelastic behavior of the filled composites is dominated by the elasticity of the composite matrix in the high shear rate range. Liang[75] found that (1 ) *<sup>w</sup> B* , where *<sup>w</sup>* was the wall shear stress, and and were constants related to the elasticity of the matrix melt and the geometry of the filler particles, respectively. The function *F*( ) is analogous to those used for the viscosity of a suspension of spheres, and may be called a 'concentration shift factor'[76]. There are several forms for *F*( ) . Here we use the form shown in Eq. (20),

( ) [1 ( / ) ] *<sup>k</sup> <sup>p</sup> <sup>c</sup> F* (20)

$$
\phi\_c = (\dot{\gamma} \, / \dot{\gamma}\_c)^q \tag{21}
$$

where *k* , *p* and *q* are constants which depend on the filler-polymer system and the flow field; *<sup>c</sup>* is the percentage decrease of the network elasticity caused by filling with the particles, which is directly related to the shear rate as described by Eq. (21); *<sup>c</sup>* is the limiting shear rate when the network is almost completely destroyed, in accordance with percolation theory. An approximation to Eq. (21) may be rewritten in stress form as Eq. (22).

$$\phi\_c = (\boldsymbol{\pi} / \boldsymbol{\pi}\_c)^{q^\circ} \tag{22}$$

#### **3. Comparison of die swell given by the model with experiment**

#### **3.1. Experimental data**

For comparison of the predicted swell ratios using the model described above with experimental data, several sets of raw swell data were taken from the literature and replotted in appropriate formats.

In testing the validity of the model for a system with large L/D, the rheometric and swell data of a pure IUPAC-LDPE standard was used as shown in Figures 1 and 2 [77,78]. The capillary had die diameter 3.00 *D mm* and experiments were carried out at 150℃.

To show whether the model is valid in analyzing the maximum extrusion distance during swelling, we used the data for a semi-dilute polymer solution of a partially hydrolyzed polyacrylamide manufactured by Rhone Poulenc (Rhodoflood AD37, Mw= 6 x 106, degree of hydrolysis = 24%)[70]. Purified water containing 20 g/l of NaCl was employed as a solvent for the polymer. The polymer concentration was 3000 ppm. The swell tests were conducted in a stainless steel capillary with length of 51 mm and inner diameter of 0.60 mm.

The swell data for a rubber compound[73] were used to verify the effectiveness of the suggested equations for polymer extrusion in a short capillary. Sample SI was a calcium carbonate (CaCO3) filled natural rubber (NR) compound, in which the content of CaCO3 was 20 phr. Sample SII was a carbon-black (CB)-filled NR/styrene-butadiene rubber SBR) /cis-1,4-butadiene rubber (CBR) compound. The blending ratio of NR/SBR/CBR was 45/10/45, and the content of CB was 56 phr. Both the rubber compounds included some additives. An Instron capillary rheometer (model 3211) was used in the tests. Two capillary dies with different lengths were selected in order to measure the rheological properties of the sample materials. The length of the long die was 40 mm, the length of the short die was 0.2 mm, and the diameter of both dies was 1 mm.

84 Viscoelasticity – From Theory to Biological Applications

 and ( , 0) *<sup>m</sup> B B* 

was the wall shear stress, and

particles, which is directly related to the shear rate

where *B*(,) 

where *<sup>w</sup>*

field; *<sup>c</sup>* 

shown in Eq. (20),

**3.1. Experimental data** 

diameter of 0.60 mm.

replotted in appropriate formats.

*BB F* ( , ) ( , 0) ( )

implies that the viscoelastic behavior of the filled composites is dominated by the elasticity of the composite matrix in the high shear rate range. Liang[75] found that (1 ) *<sup>w</sup> B*

the matrix melt and the geometry of the filler particles, respectively. The function *F*( )

analogous to those used for the viscosity of a suspension of spheres, and may be called a

( ) [1 ( / ) ] *<sup>k</sup> <sup>p</sup> <sup>c</sup> F* 

where *k* , *p* and *q* are constants which depend on the filler-polymer system and the flow

limiting shear rate when the network is almost completely destroyed, in accordance with percolation theory. An approximation to Eq. (21) may be rewritten in stress form as Eq. (22).

For comparison of the predicted swell ratios using the model described above with experimental data, several sets of raw swell data were taken from the literature and

In testing the validity of the model for a system with large L/D, the rheometric and swell data of a pure IUPAC-LDPE standard was used as shown in Figures 1 and 2 [77,78]. The

To show whether the model is valid in analyzing the maximum extrusion distance during swelling, we used the data for a semi-dilute polymer solution of a partially hydrolyzed polyacrylamide manufactured by Rhone Poulenc (Rhodoflood AD37, Mw= 6 x 106, degree of hydrolysis = 24%)[70]. Purified water containing 20 g/l of NaCl was employed as a solvent for the polymer. The polymer concentration was 3000 ppm. The swell tests were conducted in a stainless steel capillary with length of 51 mm and inner

capillary had die diameter 3.00 *D mm* and experiments were carried out at 150℃.

**3. Comparison of die swell given by the model with experiment** 

 

(/ )*<sup>q</sup> c c* 

is the percentage decrease of the network elasticity caused by filling with the

' (/ )*<sup>q</sup> c c* 

 and 

 

are the die swell of the composite with a filler content of

(20)

(21)

as described by Eq. (21); *<sup>c</sup>*

(22)

(19)

were constants related to the elasticity of

is the filling effect. Eq. (19)

. Here we use the form

is the

 ,

is

 

 

and the die swell of the pure matrix, respectively, and *F*( )

'concentration shift factor'[76]. There are several forms for *F*( )

Another set of data was for a polypropylene/glass bead composite[75]. The matrix resin was a common polypropylene (Pro-fax 6331). The melt flow rate (2.16 kg, 230 oC) and density were 12 g/10 min and 0.9 g/cm3, respectively. The glass bead filler had a mean diameter of 219 nm and density of 2.5 g/cm3. Rheological experiments with these samples were carried out using a Rosand capillary rheometer with twin cylinders. Two dies of different length with the same diameter (1 mm) were selected in order to measure the entry pressure losses. The length/diameter of the short die was << 1 and the other was 16. The test temperature was 190 oC, and apparent shear rates were varied from 50 to10–4s–1.

**Figure 1.** Plots of ln versus 1 ln and 1 1 ln( / ln(( / )) / ) *d d* versus ln from the experimental data for LDPE[77] of ~ and 1 *N* ~ .

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

Die Swell of Complex Polymeric Systems 87

~ curve[70] using Eqs. (1) and (2).

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).

were obtained by fitting the experimental

experimental data in Figure 4a, as does *Z* as shown in Figure 4b.

**Figure 3.** Calculated plots of *B* versus

*a* and 0 

*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* ,

*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

 

from the experimental data for a semi-dilute polymer solution
