Polymer rheology equation

The polymer rheology is modeled by extending the usual power-law equation to include second-order shear-rate effects and temperature dependence assuming Arrhenius type relationship. 

Experimental polymer rheology data obtained in a capillary rheometer at different temperatures is used to determine the unknown coefficients in Equations 11 - 12. Multiple linear regression is used for parameter estimation. The values of these coefficients for three different polymers is shown in Table I. The polymer rheology is shown in Figures 2 - 4. 

Graessley and co-workers have studied the rheological properties of solutions of branched PVAc in diethyl phthalate (178, 188), using polymer concentrations of 0.17, 0.225, and 0.35 g ml-1. At the lowest concentration, the low shear-rate viscosity was simply related to [17], so that it was lower for branched polymers the equation ... 

Cooke BJ, Matheson AJ (1976) Dynamic viscosity of dilute polymer solutions at high frequencies of alternating shear stress. J Chem Soc Faraday Trans II 72(3) 679-685 Curtiss CF, Bird RB (1981a) A kinetic theory for polymer melts. I The equation for the single-link orientational distribution function. J Chem Phys 74 2016—2025 Curtiss CF, Bird RB (1981b) A kinetic theory for polymer melts. II The stress tensor and the rheological equation of state. J Chem Phys 74(3) 2026—2033 Daoud M, de Gennes PG (1979) Some remarks on the dynamics of polymer melts. J Polym Sci Polym Phys Ed 17 1971-1981... 

The two-way arrow between polymer rheology and fluid mechanics has not always been appreciated. Traditionally we look at polymer rheology as input to fluid mechanics, as a way to supply constitutive equations. Gary Leal pointed out the use of fluid mechanics to provide feedback to tell us whether the constitutive equation is satisfactory. In the past, we tested constitutive models by examining polymeric liquids with very simple kinematics, homogeneous flows as a rule, either simple shear or simple shear-free types of flows. We don t actually use polymers in such simple flows, and it s essential to understand whether or not these constitutive equations actually interpolate properly between those simple types of kinematics. So there s a two-way arrow that we have to pay more attention to in the future. 

Polymeric fluids are the most studied of all complex fluids. Their rich rheological behavior is deservedly the topic of numerous books and is much too vast a subject to be covered in detail here. We must therefore limit ourselves to an overview. The interested reader can obtain more thorough presentations in the following references a book by Ferry (1980), which concentrates on the linear viscoelasticity of polymeric fluids, a pair of books by Bird et al. (1987a,b), which cover polymer constitutive equations, molecular models, and elementary fluid mechanics, books by Tanner (1985), by Dealy and Wissbrun (1990), and by Baird and Dimitris (1995), which emphasize kinematics and polymer processing flows, a book by Macosko (1994) focusing on measurement methods and a book by Larson (1988) on polymer constitutive equations. Parts of this present chapter are condensed versions of material from Larson (1988). The static properties of flexible polymer molecules are discussed in Section 2.2.3 their chemistry is described in Flory (1953). 

The development of molecular constitutive equations for commercial melts is still a challenging unsolved problem in polymer rheology. Nevertheless, it has been found that for many melts, especially those without long-chain branching, the rheological behavior can be described by empirical or semiempirical constitutive equations, such as the separable K-BKZ equation, Eq. (3-72), discussed in Section 3.7.4.4 (Larson 1988). To use the separable K-BKZ equation, the memory function m(t) and the strain-energy function U, or its strain derivatives dU/dli and W jdh, must be obtained empirically from rheological data. 

The critical molecular weight (Mcr) was introduced in Section 1 l.B.3.a. Its value is roughly twice the entanglement molecular weight (Me) according to Equation 11.24. Me is related to the structure of the repeat unit by Equation 11.25. Experimental values of Me were listed in Table 11.4. The importance of Mcr in the context of polymer rheology will be discussed further in Chapter 13. Note that Equation 12.14 reduces correctly to Equation 12.10 when a=0.5. 

This standard covers measurement of the rheological properties of polymers with both stable and unstable melt viscosity parameters at various temperatures and shear rates. The test procedure lists typical test temperature conditions for polyethylene 190°C, for polypropylene 230°C, for poly(vinyl chloride) 170-205°C, however, this indicates that the most useful data are generally obtained at temperatures consistent with processing experience. The test method also prescribes using the Rabinowitsch shear rate correction (see above) and indicates that the basic rheology equations (17.10), (17.15) and (17.16) yield true shear rate and true viscosity for Newtonian fluids only for non-Newtonian fluids only the apparent shear rate and viscosity are obtained. 

Another important quantity of general use in fluid mechanics and polymer rheology is the total stress tensor. This tensor is defined in terms of the stress tensor and the hydrostatic pressure, as indicated in the following equation ... 

In practice, the Newtonian behavior is confined to low molecular weight liquids. Polymer melts obey Newton s law only at shear rates close to zero and polymer solutions only at concentrations close to zero. The most general rheological equation is... 

Dimensional analysis is commonly applied to complex materials and processes to assess the significance of some phenomenon or property regime. It enables analysis of situations that cannot be described by an equation however, there is no a priori guarantee that a dimensionless analysis will be physically meaningful. Dimensionless quantities are combinations of variables that lack units (i.e., pure numbers), used to categorize the relationship of physical quantities and their interdependence in order to anticipate the behavior. Several dimensionless quantities relevant to polymer rheology and processing are... 

The following rheological equation describes the flow of a certain polymer melt ... 

Solutions of polymers exhibit a number of unusual effects in flows. Complex mechanical behavior of such liquids is governed by qualitatively different response of the medium to applied forces than low-molecular fluids. In hydrodynamics of polymers this response is described by rheological equation that relates the stress tensor, a, to the velocity field. The latter is described by the rate-of-strain tensor, e... 

In order to characterize polymeric fluids and to test rheological equations of state it is customary to use simple, well defined flows. The two main flows are simple shear and simple elongational. These are shown schematically in Figure 1. In shear flow, material planes (see Figure 1) move relative to each other without being stretched, whereas in extensional flow the material elements are stretched. These two different flow histories generate different responses in not only flexible chain polymers but in liquid crystalline polymers. When these flows are carried... 

Marrucci G, lannirubertok G (2004) Interchain pressure effect in extensional flows of entangled polymer. Macromolecules 37 3934—3942 Morris FA (2001) Understanding rheology. Oxford University Press, Oxford Oldroyd JG (1950) On the formulation of rheological equations of state. Proc R Soc A200 523-541... 

However, for reactive extrusion this relation should be used with considerable reserve. It can be applied to simple fluids, but its applicability in polymeric liquids is at least very questionable. Therefore, a rheological equation describing the polymer-monomer mixture as a single liquid is more useful. 

Delaunay D, Le Bot PH, Fulchiron R, Luye JF, Regnier G (2000b) Nature of contact between polymer and mold in injection molding. Part II Influence of mold deflection on pressure history and shrinkage. Polym Eng Sci 40 1692-1700 Denn MM (2001) Extrusion instabilities and wall slip. Annu Rev Fluid Mech 33 265-287 Denn MM (2004) Fifty years of non-Newtonian fluid dynamics. AIChE J 50 2335-2345 Dinh SH, Armstrong RC (1984) A rheological equation of state for semi-concentrated fiber suspensions. J Rheol 28 207-227... 

Other examples of macroscopic constitutive equations employed for describing polymer rheology and mechanics include conformation tensor models as well as the more recently proposed Pom-Pom and Rolie-Poly models. Most of these approaches have been inspired by simple mechanical models of polymers. 

The fourth chapter presents some constitutive theories and equations for suspensions. Suspension rheology normally deals with the flow behavior of two-phase systems in which one phase is solid particles like fillers but the other phase is water, organic liquids or pol)oner solutions. Literature on suspension rheology does not include flow characteristics of filled polymer systems. Neverttieless, ttiis chapter needs to be included as the foimdations for understanding ttie basics of filled polymer rheology stem from the flow behavior of suspensions. In fact, most of the constitutive theories and equations that are used for filled polymer systems are borrowed firom those that were initially developed for suspension rheology. 

Chapter 4 deals witii constitutive theories and equations for suspensions and lays down the foundations for understanding the basics of filled polymer rheology. Starting from the simplest dilute... 

---