**2. Theoretical Model**

78 Viscoelasticity – From Theory to Biological Applications

long chain branching)[11].

rationalized by theoretical studies.

wall.

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

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

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

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

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

able to relate extrudate swell to the viscoelasticity of polymer melts.
