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This chapter discusses methods for converting the many body problem in interacting polymer fluids to a statistical field theory. Auxiliary field transforms are introduced, that decouple interactions among polymer species and reduce the problem to that of individual polymers interacting with one or more fluctuating fields — the subject of Chapter 3. Examples of models for a variety of systems are enumerated, including polymer solutions, blends, block and graft copolymers, polyelectrolytes, liquid crystalline polymers, and polymer brushes.

Broadly classified, there are three types of polymer blends, namely, (1) miscible polymer blends, (2) immiscible polymer blends, and (3) partially miscible polymer blends. There are many different experimental methods that can be used to investigate the miscibility of polymer blends, such as differential scanning calorimetry (DSC), dynamic mechanical thermal analysis (DMTA), dielectric measurement, cloud point measurement, microscopy, light scattering, small-angle X-ray scattering, small-angle neutron scattering, fluorescence technique, and nuclear magnetic resonance (NMR) spectroscopy. Each of these experimental methods can only probe the homogeneity (or heterogeneity) in a polymer blend at a certain scale range. Thus, the determination of the miscibility in polymer blends depends on the resolution limit of the experimental method(s) employed. For instance, DSC and DMTA have frequently been used to determine the miscibility in polymer blends by determining glass transition temperature Tg. When a single Tg value is observed in a polymer blend, the blend can be considered miscible. However, there is a general consensus among researchers that such an experimental criterion, while very useful, cannot guarantee that a polymer blend is miscible on a segmental level. Therefore, a serious question may be raised as to whether a polymer pair can be regarded as being miscible on the segmental level (say, less than approximately 5 nm). It has been reported that DMTA can resolve the size of domains (or separated phases) on the order of 5–10 nm (Molnar and Eisenberg 1992) and DSC is not as sensitive as DMTA for determining the Tg of a polymer blend (Stoelting et al. 1970). In the use of DSC to investigate the miscibility of polymer blends, one often encounters the situation where a very broad (say, 40–60 ◦C) single glass transition appears for certain blend compositions, such as polystyrene/poly(α-methyl styrene) (PS/PαMS) blends (Kim et al. 1998; Lin and Roe 1988; Saeki 1983) and polystyrene/poly(vinyl methyl ether) (PS/PVME) blends (Kim et al. 1998; Schneider and Wirbser 1990; Schneider et al. 1990). Under such circumstances, it is not clear how an unambiguous, single value of Tg can be read off from a DSC thermogram.

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.

Polymer researchers have had a long-standing interest in understanding the evolution of blend morphology when two (or more) incompatible homopolymers or copolymers are melt blended in mixing equipment. In industry, melt blending is conducted using either an internal (batch) mixer (e.g., a Banbury mixer or a Brabender mixer) or a continuous mixer (e.g., a twin-screw extruder or a Buss kneader). There are many factors that control the evolution of blend morphology during compounding, the five primary ones being (1) blend composition, (2) rheological properties (e.g., viscosity ratio) of the constituent components, (3) mixing temperature, which in turn affects the rheological properties of the constituent components, (4) the duration of mixing in a batch mixer or residence time in a continuous mixer, and (5) rotor speed in a batch mixer or screw speed in a continuous mixer (i.e., local shear rate or shear stress). When two immiscible polymers are compounded in mixing equipment, two types of blend morphology are often observed: dispersed morphology and co-continuous morphology. Numerous investigators have reported on blend morphology of immiscible polymers, and there are too many papers to cite them all here. Some investigators (Han 1976, 1981; Han and Kim 1975; Han and Yu 1972; Nelson et al. 1977; van Oene 1978) examined blend morphology to explain the seemingly very complicated rheological behavior of two-phase polymer blends, and others (Favis and Therrien 1991; He et al. 1997; Ho et al. 1990; Miles and Zurek 1988; Scott and Macosko 1995; Shih 1995; Sundararaj et al. 1992, 1996) investigated blend morphology as affected by processing conditions. Today, it is fairly well understood from experimental studies under what conditions a dispersed morphology or a co-continuous morphology may be formed, and whether a co-continuous morphology is stable, giving rise to an equilibrium morphology, or whether it is an unstable intermediate morphology that eventually is transformed into a dispersed morphology (Lee and Han 1999a, 1999b, 2000). Let us consider the morphology evolution in an immiscible blend consisting of two semicrystalline polymers, A and B, in a compounding machine, and let us assume that the melting point (Tm,A) of polymer A is lower than the melting point (Tm,B) of polymer B.