**1. Introduction**

Die swell is a common phenomenon in polymer extrusion. When a viscoelastic fluid flows out of a die, the extrudate diameter is usually greater than the channel size. This is called die-swell, extrudate swell or the Barus effect. The degree of extrudate swell is usually expressed by the die-swell ratio (B) of extrudate diameter versus die diameter. A better understanding of such flow behavior will be beneficial for the optimization of processing parameters and the design of extrusion equipment, both of which affect product quality and production cost.

Innumerable valuable experimental data of die swell have been published, each focusing on different aspects affecting extrudate swell. There have been various interpretations of the extrusion swell of viscoelastic fluids from the macroscopic view of polymer rheology, such as a normal stress effect, an elastic energy effect, an entropy enlargement effect, an orientation effect and a memory effect. In fact, these interpretations are all related to each other[1,2]. It is generally believed that die swell is an important characteristic of the fluid elasticity during flow. The most common technique used to study rheological properties of polymer melts is capillary rheometry. In a capillary, polymer flows from the reservoir through a die and finally swells out of the exit. Under the action of extension, shear and compression, some molecular chains become disentangled, uncoiled or oriented in the convergent region resulting in an entry effect. During die flow, the resultant stress and strain cannot be relaxed completely. Simultaneously, some chains continue to be sheared and elongated during extrusion. On emerging from the die exit, the molecules are relaxed in elastic deformation by reentanglement and recoiling. The extrudate tends to contract in the flow direction and to grow in the normal direction, leading to extrudate swell[3]. As it does inside the capillary, the swelling evolves with time outside the capillary to reach a maximum at a certain extrusion distance. Graessley et al.[4] have reported that die swell occurs in two steps: (1) a very rapid swelling with a relatively large swell ratio very close to the die exit, which is known as running die swell; (2) a subsequent slow expansion to give

© 2012 Wang, licensee InTech. This is an open access chapter distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. © 2012 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

an equilibrium die swell. Thus, the extent of the swelling will depend on both external factors as well as the intrinsic characteristics of the polymer[5]. The former include the geometry (contraction ratio and angle, die length and diameter as well as their ratio L/D) of the extrusion system (reservior, entry and die) [6,7] and the capillary operating conditions (the applied shear rate, stress and temperature) [8,9] and the external environment medium[10]. Extrudate swell can be used to assess the polymer viscoelasticity upon melt extrusion. Through rheological characterization, extrudate swell can also be correlated with the molecular structure of the polymer (molecular weight, and extents of crosslinking and long chain branching)[11].

Die Swell of Complex Polymeric Systems 79

at low Reynolds number, in which the swell ratio is greater than one. However, at higher Reynolds numbers, the dimensions of the extrudate are possibly smaller than those of the die. Few data have been published concerning the effect of Reynolds number on the die swell[41,42]. Han[29] suggested that the equilibrium die swell measurements are independent of rheometer geometry for common extrusion at relatively low flow rate through a capillary where the ratio of reservoir diameter to capillary die diameter is less than 10, and length to diameter ratio of the capillary die is greater than 20. Such a capillary is termed a long capillary, in which the effects of the entry flow from the reservoir on the die exit region flow can be neglected. Most theoretical models have been derived for a long capillary[43–47]. However, there have been few quantitative predictions of die swell in actual polymer processing operations in which the ratio of width to length in the die flow channel is relatively short[48-58]. The die swell behavior of polymeric materials should be very important in short capillary flow. However, entrance effects in the short tube flow of polymer melts are relatively complicated. Thus, it is necessary to extend the theoretical models of the die swell in a long capillary to that in a short capillary. This is one purpose of

On the other hand, particulate-filled polymer composites are widely used in engineering to improve polymer properties. Their flow properties are important when molding various products. Some swell experiments have been conducted on filled polymers in which it was noticed that the die swell was related to the filler shape, size, dispersion, concentration and interfacial modification of the inclusions in the matrix[59,60]. However, there are few quantitative theories of the swelling in composites, in contrast to the situation for pure polymer melts[37,61]. Thus, it is necessary to explore how the swell behaviors of composites differ from those of a pure polymer. A second aim of this paper is to develop a single unified quantitative extrudate swell model for both pure polymers and their composites.

The following sections first present a unified theoretical equation to describe extrusion swell in both long capillaries and short capillaries. The evolution of the finite distance of extrudate swelling and the particular features of the swelling of filled composites are also detailed. Finally, comparisons are made between the predictions of the model and experimental data.

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

this paper.

**2. Theoretical Model** 

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

Precise polymer processing requires an adequate quantitative description of flow. The viscous behavior determines the throughput, while the elastic properties are important for dimensional stability. Many theoretical studies have been conducted to improve our understanding and prediction of extrudate swell[12], in which a very critical aspect is the choice of an appropriate constitutive model[13,14]. In simulations, specific values of the model parameters—especially material parameters—must be defined. Some of these parameters can be measured by rheological experiments. Thus, quantitative analyses are able to relate extrudate swell to the viscoelasticity of polymer melts.

Extrudate swell was initially exploited to study the effects of die swell on processability and its correlation with the relevant rheological properties of polymer melts and the early work has been reviewed by Kontos and White[15] and Graessley[4]. In these early studies[16–18], a wide range of experiments on polymer melt swelling were performed. Some workers have reported that die swell varies with temperature[19], pressure during processing[20], molecular weight and its distribution[21–23], as well as molecular structure[24–27] and compounding ingredients[28]. Studies of die swell as a function of length to diameter ratio of die (L/D), entry speed and shape of the die have been reported by Han [29], Cotten[30], Lenk [31] and Vinogradov and Malkin[32]. Recently, the die swell of particle-filled polymers has attracted more attention because of their wider engineering applications[33,34]. These experiments have generated various valuable data of extrudate swell which need to be rationalized by theoretical studies.

Previous examinations of the existing data have identified some semi-empirical correlations relating the swell ratio to rheological parameters. In the 1970s, Bagley and Duffey[35], Graessley et al.[4], Han[36] and Tanner[37] proposed expressions for the relationship between the swell ratio B and the first normal stress difference or shear stress, on the supposition that polymer melt shear flow obeyed a simple law. One of the most famous approaches is that of Tanner[38] based on elastic recovery theory. In this model, the maximum diameter of the extrudate is related to the recoverable shear strain at the capillary wall.

Such continuum mechanics studies have been regularly reported from the early 1980s onwards. Investigations of the swelling phenomenon have been carried out using numerical simulations, especially by using the finite element method when considering complicated boundary conditions[39,40]. Most of the analyses have been conducted for laminar flow or at low Reynolds number, in which the swell ratio is greater than one. However, at higher Reynolds numbers, the dimensions of the extrudate are possibly smaller than those of the die. Few data have been published concerning the effect of Reynolds number on the die swell[41,42]. Han[29] suggested that the equilibrium die swell measurements are independent of rheometer geometry for common extrusion at relatively low flow rate through a capillary where the ratio of reservoir diameter to capillary die diameter is less than 10, and length to diameter ratio of the capillary die is greater than 20. Such a capillary is termed a long capillary, in which the effects of the entry flow from the reservoir on the die exit region flow can be neglected. Most theoretical models have been derived for a long capillary[43–47]. However, there have been few quantitative predictions of die swell in actual polymer processing operations in which the ratio of width to length in the die flow channel is relatively short[48-58]. The die swell behavior of polymeric materials should be very important in short capillary flow. However, entrance effects in the short tube flow of polymer melts are relatively complicated. Thus, it is necessary to extend the theoretical models of the die swell in a long capillary to that in a short capillary. This is one purpose of this paper.

On the other hand, particulate-filled polymer composites are widely used in engineering to improve polymer properties. Their flow properties are important when molding various products. Some swell experiments have been conducted on filled polymers in which it was noticed that the die swell was related to the filler shape, size, dispersion, concentration and interfacial modification of the inclusions in the matrix[59,60]. However, there are few quantitative theories of the swelling in composites, in contrast to the situation for pure polymer melts[37,61]. Thus, it is necessary to explore how the swell behaviors of composites differ from those of a pure polymer. A second aim of this paper is to develop a single unified quantitative extrudate swell model for both pure polymers and their composites.

The following sections first present a unified theoretical equation to describe extrusion swell in both long capillaries and short capillaries. The evolution of the finite distance of extrudate swelling and the particular features of the swelling of filled composites are also detailed. Finally, comparisons are made between the predictions of the model and experimental data.
