Rheology of Molten Polymers with Solubilized Gaseous Component

Polymer melts (or polymer solutions) with a solubilized gaseous component (which occur under sufficiently high pressures, thus forming homogeneous mixtures), and polymer melts (or polymer solutions) with dispersed gas bubbles (thus forming heterogeneous mixtures of polymeric fluid and gas bubbles) are encountered in thermoplastic foam processing and polymer devolatilization. Thus, a good understanding of the rheological behavior of such mixtures is very important to the design of processing equipment and successful optimization of such polymer processing operations. From the 1950s through the 1970s, the dynamics of a single, spherical gas bubble dispersed in a stationary Newtonian or viscoelastic medium was extensively reported in the literature (Barlow and Langlois 1962; Duda and Vrentas 1969; Epstein and Plesset 1950; Folger and Goddard 1970; Marique and Houghton 1962; Plessst and Zwick 1952; Rosner and Epstein 1972; Ruckenstein and Davis 1970; Scriven 1959; Street 1968; Street et al. 1971; Tanasawa and Yang 1970; Ting 1975; Yang and Yeh 1966; Yoo and Han 1982; Zana and Leal 1975). While such investigations are of fundamental importance in their own right, they are not much help to describe bubble dynamics in thermoplastic foam extrusion or structural foam injection molding, for instance. There is no question that an investigation of bubble dynamics in a flowing molten polymer with dispersed gas bubbles is a very difficult subject by any measure. Thus, understandably, a relatively small number of research publications on bubble dynamics in a flowing molten polymer have been reported (Han and Villamizar 1978; Han et al. 1976; Yoo and Han 1981). The complexity of the problem arises from other related issues, such as the solubility and diffusivity of gaseous component(s) in a flowing molten polymer, which in turn depend on temperature and pressure of the system. Further, a gaseous component solubilized in molten polymer in the upstream side of a die, for instance, may nucleate as the pressure of the fluid stream decreases along the die axis, after which they could grow continuously as the molten polymer with dispersed gas bubbles flows through the rest of the die.

Rheology of Thermoplastic Polyurethanes

Thermoplastic polyurethane (TPU) has received considerable attention from both the scientific and industrial communities (Hepburn 1982; Oertel 1985; Saunders and Frish 1962). Applications for TPUs include automotive exterior body panels, medical implants such as the artificial heart, membranes, ski boots, and flexible tubing. Figure 10.1 gives a schematic that shows the architecture of TPU, consisting of hard and soft segments. Hard segments, which form a crystalline phase at service temperature, are composed of diisocyanate and short-chain diols as a chain extender, while soft segments, which control low-temperature properties, are composed of difunctional long-chain polydiols with molecular weights ranging from 500 to 5000. The soft segments form a flexible matrix between the hard domains. TPUs are synthesized by reacting difunctional long-chain diol with diisocyanate to form a prepolymer, which is then extended by a chain extender via one of two routes: (1) by a dihydric glycol chain extender or (2) by a diamine chain extender. The most commonly used diisocyanate is 4,4’-diphenylmethane diisocyanate (MDI), which reacts with a difunctional polyol forming soft segments, such as poly(tetramethylene adipate) (PTMA) or poly(oxytetramethylene) (POTM), to produce TPU, in which 1,4-butanediol (BDO) is used as a chain extender. There are two methods widely used to produce TPU: (1) one-shot reaction sequence and (2) two-stage reaction sequence. The reaction sequences for both methods are well documented in the literature (Hepburn 1982). It should be mentioned that MDI/BDO/PTMA produces ester-based TPU. One can also produce ether-based TPU when MDI reacts with POTM using BDO as a chain extender. TPUs are often referred to as “multiblock copolymers.” In order to have a better understanding of the rheological behavior of TPUs, one must first understand the relationships between the chemical structure and the morphology; thus, a complete characterization of the materials must be conducted. The rheological behavior of TPU depends, among many factors, on (1) the composition of the soft and hard segments, (2) the lengths of the soft and hard segments and the sequence length distribution, (3) anomalous linkages (branching, cross-linking), and (4) molecular weight.

Rheology of Block Copolymers

Block copolymer consists of two or more long blocks with dissimilar chemical structures which are chemically connected. There are different architectures of block copolymers, namely, AB-type diblock, ABA-type triblock, ABC-type triblock, and AmBn radial or star-shaped block copolymers, as shown schematically in Figure 8.1. The majority of block copolymers has long been synthesized by sequential anionic polymerization, which gives rise to narrow molecular weight distribution, although other synthesis methods (e.g., cationic polymerization, atom transfer radical polymerization) have also been developed in the more recent past. Owing to immiscibility between the constituent blocks, block copolymers above a certain threshold molecular weight form microdomains (10–50 nm in size), the structure of which depends primarily on block composition (or block length ratio). The presence of microdomains confers unique mechanical properties to block copolymers. There are many papers that have dealt with the synthesis and physical/mechanical properties of block copolymers, too many to cite them all here. There are monographs describing the synthesis and physical properties of block copolymers (Aggarwal 1970; Burke and Weiss 1973; Hamley 1998; Holden et al. 1996; Hsieh and Quirk 1996; Noshay and McGrath 1977). Figure 8.2 shows schematically four types of equilibrium microdomain structures observed in block copolymers. Referring to Figure 8.2, it is well established (Helfand and Wasserman 1982; Leibler 1980) that in microphase-separated block copolymers, spherical microdomains are observed when the volume fraction f of one of the blocks is less than approximately 0.15, hexagonally packed cylindrical microdomains are observed when the value of f is between approximately 0.15 and 0.44, and lamellar microdomains are observed when the value of f is between approximately 0.44 and 0.50. Some investigators have observed ordered bicontinuous double-diamonds (OBDD) (Thomas et al. 1986; Hasegawa et al. 1987) or bicontinuous gyroids (Hajduk et al. 1994) at a very narrow range of f (say, between approximately 0.35 and 0.40) for certain block copolymers. Figure 8.2 shows only one half of the symmetricity about f = 0.5. Transmission electron microscopy (TEM), small-angle X-ray scattering (SAXS), and small-angle neutron scattering (SANS) have long been used to investigate the types of microdomain structures in block copolymers.

Chemorheology of Thermosets

Thermosets (e.g., unsaturated polyester, epoxy, urethane) are small molecules containing functional groups, which undergo chemical reactions (commonly referred to as “cure”) in the presence of an initiator(s) or a catalyst(s). In a broader sense, thermosets can be regarded as being parts of reactive polymer systems, which include pairs of polymers (e.g., blends of maleated polyolefin and nylon 6, as presented in Chapter 11) that undergo chemical reactions during compounding, and mixtures of an elastomer and a vulcanizing agent that undergo cross-link reactions (commonly referred to as vulcanization) at an elevated temperature. The subject of investigating the rheological behavior of reactive polymer systems is referred to as “chemorheology.” Since chemorheology is such a very broad field of investigation, one must specify the polymer system under consideration, classifying as chemorheology of thermosets, chemorheology of reactive polymer blends, chemorheology of elastomer vulcanization, and so on. In this chapter, for a number of reasons we restrict our presentation to the chemorheology of thermosets only. These reasons include (1) the limited space available here, meaning that it is not possible to present the chemorheology of every reactive polymer system, (2) thermosets play a very important role in polymer processing from an industrial point of view, and (3) the presentation of the chemorheology of thermosets in this chapter lays the foundation for the presentation of processing of thermosets in Chapters 11–13 of Volume 2. In the 1970s and 1980s, considerable amounts of effort were spent on investigating the chemorheology of thermosets. There are many experimental techniques that have been used to investigate the cure kinetics of thermosets: differential scanning calorimetry (DSC), Fourier transform infrared (FTIR) spectroscopy, dielectric measurements, and rheokinetic measurements. There are monographs (Kock 1977; May 1983; Turi 1981) and a comprehensive review article (Halley and Mackay 1996) on the subject. A better understanding of the chemorheology of thermosets requires an understanding of the kinetics of chemical reactions during cure. It can then easily be surmised that an understanding of the chemorheology of thermosets is much more complex than the rheology of thermoplastics presented in Chapter 6 through Chapter 12.

Rheology of Immiscible Polymer Blends

The polymer industry has been challenged to produce new polymeric materials by blending two or more homopolymers or random copolymers or by synthesizing graft copolymers. To meet the challenge, various methods have been explored, namely, (1) by synthesizing a new monomer, polymerizing it, and then blending it with an existing homopolymer or random copolymer, (2) by copolymerizing existing monomers and then blending it with an existing homopolymer or random copolymer, (3) by chemically modifying an existing homopolymer or random copolymer and then blending it with other homopolymers or copolymers already available, or (4) by synthesizing new compatibilizer(s) to improve the mechanical properties of two immiscible homopolymers or random copolymers that otherwise have unacceptable mechanical properties. There are numerous monographs (Cooper and Estes 1979; Han 1984; Paul and Newman 1978; Platzer 1971, 1975; Sperling 1974; Utracki 1990) describing various aspects of polymer blends. In the 1970s, Han and coworkers (Han 1971, 1974; Han and Kim 1975; Han and Yu 1971a, 1971b, 1972; Han et al. 1973, 1975; Kim and Han 1976) conducted seminal experimental studies on the rheology of immiscible polymer blends and related the observed rheological behavior to blend morphology. Independently, in the same period, Vinogradov and coworkers (Ablazova et al. 1975; Brizitsky et al. 1978; Tsebrenko et al. 1974, 1976; Vinogradov et al. 1975) conducted a series of experimental studies relating the blend rheology to blend morphology. Van Oene (1972, 1978) also pursued, independently, experimental studies for a better understanding of rheology–morphology relationships in immiscible polymer blends. Since then, using different polymer pairs, numerous researchers have conducted experimental studies, which were essentially the same as, or very similar to, the previous experimental studies of Han and coworkers, Vinogradov and coworkers, and van Oene in the 1970s. It is fair to state that those studies in the 1980s and 1990s have not revealed any significant new findings.

Rheology of Liquid-Crystalline Polymers

Liquid crystals (LCs) may be divided into two subgroups: (1) lyotropic LCs, formed by mixing rigid rodlike molecules with a solvent, and (2) thermotropic LCs, formed by heating. One finds in the literature such terms as mesomorphs, mesoforms, mesomorphic states, and anisotropic liquids. The molecules in LCs have an orderly arrangement, and different orders of structures (nematic, smectic, or cholesteric structure) have been observed, as schematically shown in Figure 9.1. The kinds of molecules that form LCs generally possess certain common molecular features. The structural characteristics that determine the type of mesomorphism exhibited by various molecules have been reviewed. At present, our understanding of polymeric liquid crystals, often referred to as liquid-crystalline polymers (LCPs), is largely derived from studies of monomeric liquid crystals. However, LCPs may exhibit intrinsic differences from their monomeric counterparts because of the concatenation of monomers to form the chainlike macromolecules. The linkage of monomers inevitably means a loss of their translational and orientational independence, which in turn profoundly affects the dynamics of polymers in the liquid state. These intramolecular structural constraints are expressed in the flexibility of the polymer chain. Generally speaking, the chemical constitution of the monomer determines the flexibility and equilibrium dimensions of the polymer chain (Gray 1962). Figure 9.2 illustrates the variability of chain conformation (flexible chain, semiflexible chain, and rigid rodlike chain) forming macromolecules. Across this spectrum of chain flexibility, the persistence in the orientation of successive monomer units varies from the extreme of random orientation (flexible chains) to perfect order (the rigid rod). Hence, efforts have been made to synthesize LCPs that consist of rigid segments contributing to the formation of a mesophase and flexible segments contributing to the mobility of the entire macromolecule in the liquid state (Ober et al. 1984). From the point of view of molecular architecture, as schematically shown in Figure 9.3, two types of LCP have been developed: (1) main-chain LCPs (MCLCPs), having the monomeric liquid crystals (i.e., mesogenic group) in the main chain of flexible links, and (2) side-chain LCPs (SCLCPs), having the monomeric liquid crystals attached, as a pendent side chain, to the main chain.

Rheology of Flexible Homopolymers

Numerous flexible homopolymers and flexible random copolymers are commercially available. Thus, understandably, a number of research groups have reported on the rheological behavior of flexible homopolymers and flexible random copolymers in the bulk and solution states. There are too many studies to cite them all here. In Chapters 3 to 5 we presented the rheological behavior, in general terms, of linear flexible homopolymers in steady-state shear flow, elongational flow, and/or oscillatory shear flow. In this chapter, we present the effects of temperature, molecular weight (although in Chapter 4 we presented theoretical predictions of the effect of molecular weight), and molecular weight distribution on the rheological behavior of linear flexible homopolymers, and also flexible homopolymers with long-chain branching. The rheological behavior of much more complex polymer systems is presented in other chapters of this volume. From the point of view of polymer processing, temperature is one of the most important variables that greatly affect the rheological behavior of polymeric liquids. Therefore, it is very important to present the effect of temperature on rheological behavior, placing emphasis on the methods that enable one to obtain temperature-independent correlations for rheological properties. Such correlations, when available, will help one to estimate the rheological properties of the same polymer without conducting additional experiments. With respect to polymer synthesis and polymer processing, a better understanding of the effects of molecular weight and molecular weight distribution on the rheological behavior of a polymer is of fundamental importance. In Chapter 4 we have presented molecular theory, demonstrating that the molecular weight of a linear flexible homopolymer has a profound influence on its rheological properties. Thus, information on the relationships between molecular weight and rheology, when available, will help one to choose, with little waste of time and effort, optimum processing conditions. One of the common features of all commercial homopolymers is that they are polydisperse and, therefore, it is not difficult to surmise that the molecular weight distribution of a polymer also has a profound influence on its rheological properties.

Experimental Methods for Measurement of the Rheological Properties of Polymeric Fluids

There has been a continuing interest in developing experimental techniques for the measurement of the rheological properties of viscoelastic fluids. As discussed in Chapter 3, reliable experimental data are needed in order to evaluate the effectiveness of a constitutive equation in its ability to predict the rheological properties of viscoelastic fluids. Also, as is presented in later chapters, a better understanding of the rheological properties of polymers is very important for the determination of optimum processing conditions, as well as for the attainment of desired physical/mechanical properties in the finished product. Further, reliable measurement of the rheological properties of polymers can be used to control polymerization reactors in industry and also to control polymer processing operations. In this chapter, we present experimental methods for measurement of the rheological properties of polymeric fluids. For this, we discuss experimental methods to determine (1) steady-state simple shear flow and oscillatory shear flow properties using cone-and-plate rheometry, (2) steady-state shear flow properties using capillary/slit rheometry, and (3) elongational flow properties of polymeric fluids. There are other rotational types of rheological instruments, such as those with concentric-cylinder and eccentric-parallel plates. However, such rheological instruments are not widely used today and thus in this chapter we do not present the principles and applications of such rheological instruments. In presenting the experimental methods for rheological measurements we refer to the fundamentals presented in Chapters 2 and 3. For further details of the experimental methods, there are monographs (Collyer and Clegg 1998; Dealy 1982; Ferry 1980; Walter 1975) that are devoted entirely to the discussion of rheological measurements. The primary purpose of this chapter is to demonstrate how the fundamentals presented in Chapters 2–4 can be used in the measurement of the rheological properties of polymeric fluids. Optical rheometry is an important experimental technique for investigation of the relationship between any microphase morphology dynamics and the rheological behavior of complex polymeric fluids (e.g., liquid-crystalline polymers), which exhibit strong chain orientation during flow (Fuller 1995).

Continuum Theories for the Viscoelasticity of Flexible Homogeneous Polymeric Liquids

There are two primary reasons for seeking a precise mathematical description of the constitutive equations for viscoelastic fluids, which relate the state of stress to the state of deformation or deformation history. The first reason is that the constitutive equations are needed to predict the rheological behavior of viscoelastic fluids for a given flow field. The second reason is that constitutive equations are needed to solve the equations of motion (momentum balance equations), energy balance equations, and/or mass balance equations in order to describe the velocity, stress, temperature, and/or concentration profiles in a given flow field that is often encountered in polymer processing operations. There are two approaches to developing constitutive equations for viscoelastic fluids: one is a continuum (phenomenological) approach and the other is a molecular approach. Depending upon the chemical structure of a polymer (e.g., flexible homopolymer, rigid rodlike polymer, microphase-separated block copolymer, segmented multicomponent polymers, highly filled polymer, miscible polymer blend, immiscible polymer blend), one may take a different approach to the formulation of the constitutive equation. In this chapter we present some representative constitutive equations for flexible, homogeneous viscoelastic liquids that have been formulated on the basis of the phenomenological approach. In the next chapter we present the molecular approach to the formulation of constitutive equations for flexible, homogeneous viscoelastic fluids. In the formulation of the constitutive equations using a phenomenological approach, emphasis is placed on the relationship between the components of stress and the components of the rate of deformation (or strain) or deformation (or strain) history, such that the responses of a fluid to a specified flow field or stress can adequately be described. The parameters appearing in a constitutive equation are supposed to represent the characteristics of the fluid under consideration. More often than not, the parameters appearing in a phenomenological constitutive equation are determined by curve fitting to experimental results. Thus phenomenological constitutive equations shed little light on the effect of the molecular parameters of the fluid under investigation to the rheological responses of the fluid.

Molecular Theories for the Viscoelasticity of Flexible Homogeneous Polymeric Liquids

The fact that a polymer consists of a number of chains of different lengths, each in turn consisting of a series of monomer units, means that the motion of one part of the polymer chain will profoundly affect the motion of other parts. Hence, for a given polymer, a description of microscopic processes occurring under a given flow field depends on hypotheses regarding the molecular structure and mechanisms of flow in the polymer. Today, it is well-known, gained from practical experience, that the molecular weight, the molecular weight distribution, and the degree of long-chain branching influence the rheological properties of polymeric liquids. Therefore, a better understanding of the relationship between molecular parameters and rheological properties is very important from the standpoints of both polymer synthesis and polymer processing. However, the theoretical development of this aspect of the problem is far from complete, although some important progress has been made. In the preceding chapter, we discussed the viscoelastic behavior of polymeric liquids from the phenomenological point of view, without associating the significance of theoretical predictions to molecular origin(s). Specifically, we have seen that the rheological equations of state contain parameters that vary from polymer to polymer. Since it has amply been demonstrated by experiment that the extent of a particular viscoelastic behavior is greatly influenced by the molecular parameters, such as molecular weight, molecular weight distribution, and the degree of long-chain branching, predictions of any viscoelastic behavior of polymers on the basis of phenomenological theory is of very limited use to either control the quality of polymers produced or improve the performance of polymers, unless the parameters appearing in various continuum constitutive equations are related to molecular parameters.