Polymer rheology constitutive equations

The reality, however, is not as simple as that. There are several possibilities to describe viscosity, 77, and first normal stress difference coefficient, P1. The first one originates from Lodge s rheological constitutive equation (Lodge 1964) for polymer melts and the second one from substitution of a sum of N Maxwell elements, the so-called Maxwell-Wiechert model (see Chap. 13), in this equation (see General references Te Nijenhuis, 2005). 

The uniaxial extensiometers described so far are suitable for use with viscous materials only. They cannot, for example, be used to measure the steady extensional viscosity of such commercially important polymers as nylons and polyesters used in the textile industry, and which may have shear viscosities as low as 100 Pa sec at processing temperatures. As a consequence, other techniques are needed but these invariably involve nonuniform stretching. Here one cannot require that the stress or the stretch rate be constant. Also, the material is usually not in a virgin (stress-free) state to begin with. One can therefore not obtain the extensional viscosity directly from these measurements. Nonetheless, data from properly designed non-uniform stretching experiments can be profitably analyzed with the help of rheological constitutive equations. In addition, such data provide a simple measure of resistance that polymeric fluids offer to extensional deformation. 

As mentioned in Sections 1 and 3, a rheological constitutive equation relates the components of the three-dimensional extra-stress (matrix) to the components of the strain (matrix) or the rate of strain (matrix) in any given flow field. Extensional viscosity data of the kind shown in Figure 5 should therefore be explainable on the basis of a proper constitutive equation. Unfortunately, there is no simple constitutive equation that accurately predicts the behavior of a polymer melt in all the commonly encountered flow situations. Some equations do a remarkably good job... 

Viscoelasticity is the property of materials that exhibit both viscous and elastic characteristics when undergoing deformation (http //en.wikipedia.org/wiki/Visco-elasticity). Linear viscoelastic behavior is exhibited by a material when it is subjected to a very small or very slow deformation. So when a viscoelastic material is subjected to a deformation that is neither very small nor very slow, its behavior in no longer linear, and there is no universal rheological constitutive equation that can predict the response of the material to such a deformation [18]. Nonhnear viscoelastic behavior is more important than linear properties of mbber/polymer nanocomposites as the industrial processing of viscoelastic materials (mbbers/polymers) always involves large and rapid deformations in which the behavior is nonlinear. 

In the above equation the superscript p labels the property value per unit mass of polymer, while Yi is the number of moles of low molecular weight penetrant in the mixture per unit mass of polymer. As already said, the free energy of the system is assumed to depend only on the present value of temperature, pressure and volume at given mixture composition. No further dependence on the past histories of the independent variables is included beyond that required by the rheological constitutive equations. 

Hence, the effective diffusion coefficient depends on the choice of ayy and azz. Rheological constitutive equations are only enable the stress tensor to be determined to within an isotropic constant. DO argued briefly that, in the tube model, if the length of a polymer is equal to its equihbrium value, so that stress arises from orientation, Tr a = 0. For example, if the diagonal components of the solution to the constitutive equation for the stress tensor are given by a = Ni,a = N2,a = 0, then the appropriate components of the stress tensor should be written, a = Ni + c, a = N2 + c, a = c, so that. 

Each link in this chain of relationships is now nearing a state of development sufficient to make it possible to predict a priori the reaction conditions required to produce a polymer having a prescribed melt processing behavior. This book contributes little to step 1 of this chain, but focuses instead on step 2, and, to the extent currently possible, step 3. The book contributes also to step 4 by presenting some of the rheological constitutive equations that can be used in the simulation of flows and stresses in polymer processing operations. 

Doi, M. and Edwards, S.F., 1978. Dynamics of concentrated polymer systems 1. Brownian motion in equilibrium state, 2. Molecular motion under flow, 3. Constitutive equation and 4. Rheological properties. J. Cheni. Soc., Faraday Trans. 2 74, 1789, 1802, 1818-18.32. 

M. Doi and S. F. Edwards, Dynamics of concentrated polymer systems. Part I. Brownian motion in the equilibrium state, J. Chem. Soc. Faraday Trans. II, 74, 1789 (1978) M. Doi and S. F. Edwards, Dynamics of concentrated polymer systems. Part 2. Molecular motion under flow, J. Chem. Soc. Faraday Trans.II, 74, 1802 (1978) M. Doi and S. F. Edwards, Dynamics of concentrated polymer systems. Part 3. The constitutive equation, J. Chem. Soc. Faraday Trans. II, 74,1818 (1978) M. Doi and S. F. Edwards, Dynamics of concentrated polymer systems. Part 4. Rheological properties, J. Chem. Soc. Faraday Trans. II, 75,38 (1979). 

The nonlinear viscoelastic models (VE), which utilize continuum mechanics arguments to cast constitutive equations in coordinate frame-invariant form, thus enabling them to describe all flows steady and dynamic shear as well as extensional. The objective of the polymer scientists researching these nonlinear VE empirical models is to develop constitutive equations that predict all the observed rheological phenomena. 

We have tried to give a quick glimpse of the interrelationships among some commonly used constitutive equations for polymer melts and solutions. None predicts quantitatively the entire spectrum of the rheological behavior of these materials. Some are better than others, becoming more powerful by utilizing more detailed and realistic molecular models. These, however, are more complex to use in connection with the equation of motion. Table 3.1 summarizes the predictive abilities of some of the foregoing, as well as other constitutive equations. 

There are numerous other GNF models, such as the Casson model (used in food rheology), the Ellis, the Powell-Eyring model, and the Reiner-Pillippoff model. These are reviewed in the literature. In Appendix A we list the parameters of the Power Law, the Carreau, and the Cross constitutive equations for common polymers evaluated using oscillatory and capillary flow viscometry. 

Wagner et al. (63-66) have recently developed another family of reptation-based molecular theory constitutive equations, named molecular stress function (MSF) models, which are quite successful in closely accounting for all the start-up rheological functions in both shear and extensional flows (see Fig. 3.7). It is noteworthy that the latest MSF model (66) is capable of very good predictions for monodispersed, polydispersed and branched polymers. In their model, the reptation tube diameter is allowed not only to stretch, but also to reduce from its original value. The molecular stress function/(f), which is the ratio of the reduction to the original diameter and the MSF constitutive equation, is related to the Doi-Edwards reptation model integral-form equation as follows ... 

Thus, adequate determination of nonlinear rheological parameters can be obtained, using industrial polymer processing-relevant flows, albeit with very substantial computational efforts, virtually assuring the relevance of the use of the constitutive equation for solving other complex processing flows. 

M. H. Wagner and J. Schaeffer, Constitutive Equations from Gaussian Slip-link Network Theories in Polymer Melt Rheology, Rheol. Acta, 31, 22-31 (1992). 

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. 

Flaw activity and the resultant stress concentration factors can also be expected to depend on the material s rheology. The sample loading history and path play a major role in determining the behavior of a given flaw as described elsewhere (6,11,12,13), and these ideas are currently being extended to account for recent developments in constitutive equation theory for solid polymers and the idea of a flaw spectrum. In this paper, time and path dependence are not considered further, and the calculations are based on elastic stress concentration factors associated with elliptic flaw geometries. 

The theoretical foundation of this software is provided by principles of continuum mechanics, together with improved numerical methods for the solution of the mathematical equations and by the use of pertinent constitutive equations for the description of the rheological behaviour of molten polymers. 

M.H.Wagner, J.Schaeffer, Constitutive equations from Gaussian slip-link network theories in polymer melt rheology, Rheol. Acta 31 (1992), 22-31. 

In the story of numerical flow simulation, the ability to predict observed and significant viscoelastic flow phenomena of polymer melts and solutions in an abrupt contraction has been unsuccessful for many years, in relation to the incomplete rheological characterization of materials, especially in elongation. The numerical treatments have often been confined to flow of elastic fluids with constant viscosity, described by differential constitutive equations as the Upper Convected Maxwell and Oldroyd-B models. Fortunately, the recent possibility to use real elastic fluids with constant viscosity, the so-called Boger fluids [10], has narrowed the gap between experimental observation and numerical prediction [11]. 

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. 

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