Structure and Rheology of Molten Polymers

Our subject is how molecular structure affects melt flow; we will not talk about solid state behavior at all. The science of how materials deform and flow under stress is called rheology. Rheology has been used as a semi-quantitative tool in polymer science and engineering for many years, for example for quality control, but quantitative relationships between structure and measurable properties have been elusive, particularly in the case of commercial polymers. However, catalyst systems have recently been developed that allow greatly improved control of the molecular structure of commercial polymers. This, together with major advances in the modeling of rheological behavior, has brought us much closer to a quantitative treatment of structure-rheology relationships for commercial polymer melts.

This chapter introduces concepts and models that are used in subsequent chapters of this book. A much more thorough treatment of polymer structure can be found in the monograph of Graessley [1].

The treatment of linear viscoelasticity presented in this chapter is sufficient for a full understanding of the models described in subsequent chapters. However, readers wishing to delve more deeply into this subject may wish to consult the monographs by Ferry [1] and Tschoegl [2]. Ferry treats the rheological properties of polymers, while Tschoegl’s book is a compendium of empirical models and relationships between various linear material functions.

In Chapter 4 it was explained that the linear elastic behavior of molten polymers has a strong and detailed dependency on molecular structure. In this chapter, we will review what is known about how molecular structure affects linear viscoelastic properties such as the zero-shear viscosity, the steady-state compliance, and the storage and loss moduli. For linear polymers, linear properties are a rich source of information about molecular structure, rivaling more elaborate techniques such as GPC and NMR. Experiments in the linear regime can also provide information about long-chain branching but are insufficient by themselves and must be supplemented by nonlinear properties, particularly those describing the response to an extensional flow. The experimental techniques and material functions of nonlinear viscoelasticity are described in Chapter 10.

The polymer industry has found two practical uses for polymer melt rheology. The first is to characterize molecular structure; e.g., the molecular weight distribution (MWD) and the long-chain branching (LCB) structure. The second is to characterize the processing behavior of the melt. When used to characterize structure, melt rheology can supplement (or even replace) GPC, NMR, light scattering, and other probes of molecular weight distribution and branching structure. When used to characterize processability, rheological measurements can predict how readily a given melt can be shaped into the desired product. These two uses of rheology are synergistic; if rheology can be used both to predict the processability of a melt having a given MWD and branching structure and to determine the MWD and LCB of that polymer, then one can both inform the polymer chemist what polymer structure he should try to make, and determine whether he has succeeded in making that structure, all using rheological methods.

In Chapter 6, polymer deformation and relaxation in entangled melts were discussed using the “tube” model. Chapter 6 culminated with a discussion of the “double reptation” model, which can predict reasonably well the orientation and stress in the linear viscoelastic regime for some polydisperse linear polymers; i.e., polymers without long side branches. The double reptation model deals with the difficult problem of constraint release using a very simpleminded idea that if a constraint on a “test” chain is released by movement of a surrounding “constraint” chain, then the test chain can relax the stress associated with that constraint immediately. However, at the end of Chapter 6 we noted that there are flaws in the predictions of at least some versions of the double reptation theory. In particular, the theory predicts that at fixed weight average molecular weight, the zero shear viscosity should increase with increasing polydispersity, a prediction not supported by experimental data, at least for modest levels of polydispersity (M_w /M_n < 4 or so). In addition, as discussed in Chapter 6, the simplest version of the double reptation model (with a single exponential kernel) works poorly for monodisperse and bidisperse polymer melts.

The primary methods for obtaining information concerning the molecular weight of a polymer are described in Chapter 2, where it is pointed out that the primary tool for determination of molecular weight distribution (MWD) is gel permeation chromatography (GPC), also known as high-pressure liquid chromatography (HPLC) or size exclusion chromatography (SEC). There are several important reasons why much attention has been directed to using rheology as a substitute for GPC. First, many polymers of commercial importance dissolve either with difficulty or not at all in solvents, so that the chromatography column must be operated at high temperature or is not an option at all. Furthermore, rheological properties are much more sensitive to high molecular weight fractions than GPC elution curves, and these fractions have a very important effect on the ease with which a polymer can be processed in the molten state. Finally, standard rheological properties such as the viscosity and the storage and loss moduli are much easier to measure than GPC elution curves.

In Chapter 6, various versions the “tube” model were presented, which can predict the linear viscoelasticity of monodisperse and polydisperse linear polymers; i.e., polymers without long-chain branching (LCB). To be quantitatively accurate, these models need to include several mechanisms of polymer motion, namely:

In Chapter 4, it was noted that linear viscoelastic behavior is observed only in deformations that are very small or very slow. The response of a polymer to large, rapid deformations is nonlinear, which means that the stress depends on the magnitude, the rate and the kinematics of the deformation. Thus, the Boltzmann superposition principle is no longer valid, and nonlinear viscoelastic behavior cannot be predicted from linear properties. There exists no general model, i.e., no universal constitutive equation or rheological equation of state that describes all nonlinear behavior. The constitutive equations that have been developed are of two basic types; empirical continuum models, and those based on a molecular theory. We will briefly describe several examples of each type in this chapter, but since our primary objective is to relate rheological behavior to molecular structure, we will be most interested in models based on molecular phenomena. The most successful molecular models to date are those based on the concept of a molecule in a tube, which was introduced in Chapter 6. We therefore begin this chapter with a brief exposition of how nonlinear phenomena are represented in tube models. A much more complete discussion of these models will be provided in Chapter 11.

This chapter presents tube-based theories of nonlinear rheology. In principle, a successful theory for the nonlinear rheology of polymer melts must incorporate all the effects described in Chapters 6 and 7 for linear rheology, as well as the relaxation and flow phenomena peculiar to the nonlinear regime described in Chapter 10. The difficulty of the task of developing theories for nonlinear rheology is such that no completely general molecular theory appears to be feasible at present. Thus, nonlinear rheological theories are either largely phenomenological or are restricted to special cases, such as linear polymers, monodisperse star or H polymers, or “pom-pom” polymers. Even for these restricted cases, the theories are usually only semi-quantitative, or are only appropriate for certain types of flow. In Section 11.2, we briefly describe the two nonlinear relaxation mechanisms included in modern rheological constitutive equations, namely chain retraction and convective constraint release (which were also described in Section 10.2). In Section 11.3, we consider the case of monodisperse linear polymers. The nonlinear rheology of bidisperse and polydisperse linear polymers will be taken up in Section 11.4, and comparisons of the theories to experimental data for linear polymers are covered in Section 11.5. Theories for branched polymers are discussed in Section 11.6. Since our interests in this book are in relating molecular properties to polymer rheology, we will not discuss phenomenological models, which are in any case described thoroughly in other books [1–5]. Throughout this chapter, we will restrict our attention to the tube model, and discuss primarily models for which the nonlinear properties can be related to molecular parameters.

In this brief chapter, we give a birds-eye view of the topics we have covered and end with our impression of where the field of polymer rheology and characterization is going, or should go, to achieve its full impact on the development, production., and processing of commercial polymers.