Characterization of Polymer Blends: Miscibility, Morphology and Interfaces

By Sabu Thomas

Filling the gap for a reference dedicated to the characterization of polymer blends and their micro and nano morphologies, this book provides comprehensive, systematic coverage in a one-stop, two-volume resource for all those working in the field.

Leading researchers from industry and academia, as well as from government and private research institutions around the world summarize recent technical advances in chapters devoted to their individual contributions. In so doing, they examine a wide range of modern characterization techniques, from microscopy and spectroscopy to diffraction, thermal analysis, rheology, mechanical measurements and chromatography. These methods are compared with each other to assist in determining the best solution for both fundamental and applied problems, paying attention to the characterization of nanoscale miscibility and interfaces, both in blends involving copolymers and in immiscible blends. The thermodynamics, miscibility, phase separation, morphology and interfaces in polymer blends are also discussed in light of new insights involving the nanoscopic scale. Finally, the authors detail the processing-morphology-property relationships of polymer blends, as well as the influence of processing on the generation of micro and nano morphologies, and the dependence of these morphologies on the properties of blends. Hot topics such as compatibilization through nanoparticles, miscibility of new biopolymers and nanoscale investigations of interfaces in blends are also addressed.

With its application-oriented approach, handpicked selection of topics and expert contributors, this is an outstanding survey for anyone involved in the field of polymer blends for advanced technologies.

• Language: English
• Publisher: Wiley
• Release date: Oct 28, 2014
• ISBN: 9783527645619

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1

Polymer Blends: State of the Art, New Challenges, and Opportunities

Jyotishkumar Parameswaranpillai, Sabu Thomas and Yves Grohens

1.1 Introduction

A polymer blend is a mixture of two or more polymers that have been blended together to create a new material with different physical properties. Generally, there are five main types of polymer blend: thermoplastic–thermoplastic blends; thermoplastic–rubber blends; thermoplastic–thermosetting blends; rubber–thermosetting blends; and polymer–filler blends, all of which have been extensively studied. Polymer blending has attracted much attention as an easy and cost-effective method of developing polymeric materials that have versatility for commercial applications. In other words, the properties of the blends can be manipulated according to their end use by correct selection of the component polymers [1]. Today, the market pressure is so high that producers of plastics need to provide better and more economic materials with superior combinations of properties as a replacement for the traditional metals and polymers. Although, plastic raw materials are more costly than metals in terms of weight, they are more economical in terms of the product cost. Moreover, polymers are corrosion-resistant, possess a light weight with good toughness (which is important for good fuel economy in automobiles and aerospace applications), and are used for creating a wide range of goods that include household plastic products, automotive interior and exterior components, biomedical devices, and aerospace applications [2].

The development and commercialization of new polymer usually requires many years and is also extremely costly. However, by employing a polymer blending process – which is also very cheap to operate – it is often possible to reduce the time to commercialization to perhaps two to three years [2]. As part of the replacement of traditional polymers, the production of polymer blends represents half of all plastics produced in 2010. Today, the polymer industry is becoming increasingly sophisticated, with ultra-high-performance injection molding machines and extruders available that allow phase-separations and viscosity changes to be effectively detected or manipulated during the processing stages [3]. Whilst this modern blending technology can also greatly extend the performance capabilities of polymer blends, increasing market pressure now determines that, for specific applications, polymer blends must perform under some specific conditions (e.g., mechanical, chemical, thermal, electrical). This presents a major challenge as the materials must often function at the limit of the properties that can be achieved; consequently, in-depth studies of the properties and performance of polymer blends are essential.

1.2 Miscible and Immiscible Polymer Blends

Generally, polymer blends are classified into either homogeneous (miscible on a molecular level) or heterogeneous (immiscible) blends. For example, poly(styrene) (PS)–poly(phenylene oxide) (PPO) and poly(styrene-acrylonitrile) (SAN)–poly(methyl methacrylate) (PMMA) are miscible blends, while poly(propylene) (PP)–PS and poly(propylene)–poly(ethylene) (PE) are immiscible blends. Miscible (single-phase) blends are usually optically transparent and are homogeneous to the polymer segmental level. Single-phase blends also undergo phase separation that is usually brought about by variations in temperature, pressure, or in the composition of the mixture.

Since, ultimately, the properties of a polymer blend will depend on the final morphology, various research groups have recently undertaken extensive studies of the miscibility and phase behavior of polymer blends. In practice, the physical properties of interest are found either by miscible pairs or by a heterogeneous system, depending on the type of application. Generally, polymer blends can be completely miscible, partially miscible or immiscible, depending on the value of ΔGm [4].

The free energy of mixing is given by

(1.1) equation

For miscibility (binary blend), the following two conditions must be satisfied: the first condition ΔGm < 0; and the second condition

(1.2) equation

where ΔGm is the Gibbs energy of mixing, ϕ is the composition, where ϕ is usually taken as the volume fraction of one of the components.

ΔSm is the entropy factor and is a measure of disorder or randomness, is always positive and, therefore, is favorable for mixing or miscibility especially for low-molecular-weight solutions. In contrast, polymer solutions have monomers with a high molecular weight and hence the enthalpy of mixing (ΔHm) is also a deciding factor for miscibility. ΔHm is the heat that is either consumed (endothermic) or generated (exothermic) during mixing. If the mixing is exothermic then the system is driven towards miscibility. The mixing is exothermic only when strong specific interactions occur between the blend components. The most common specific interactions found in polymer blends are hydrogen bonding, dipole–dipole, and ionic interactions. Several techniques that can be used to understand the specific interaction in polymers, such as Fourier transform infrared (FTIR) spectroscopy, small-angle neutron scattering, ellipsometry, neutron reflectivity, and nuclear magnetic resonance (NMR) spectroscopy.

Experimentally observed phase diagrams in polymer blend systems may be lower critical solution temperature (LCST), upper critical solution temperature (UCST), combined UCST and LCST, hourglass-, and/or closed-loop-shaped. The most commonly observed phase diagrams are LCST (phase separation of a miscible blend during heating) and UCST (phase separation of a miscible blend during cooling). Phase separation in polymer solutions may proceed either by nucleation and growth (NG) or by spinodal decomposition (SD), or by the combination of both [5]. Experimentally, phase separation can be followed by a number of experimental techniques that include light scattering, neutron scattering, ellipsometry, and rheology. The generated morphology can be characterized using scanning electron microscopy (SEM), atomic force microscopy (AFM), transmission electron microscopy (TEM) fluorescence microscopy, infrared, near-infrared and Raman imaging, and confocal microscopy.

1.3 Compatibility in Polymer Blends

In general, the compatibility between the polymer phases decides the properties of a heterogeneous polymer blend [6,7]. The interface between the polymer phases in a polymer system is characterized by the interfacial tension which, when approaching zero, causes the blend to become miscible. In other words, if there are strong interactions between the phases then the polymer blend will be miscible in nature. Large interfacial tensions lead to phase separation, with the phase-separated particles perhaps undergoing coalescence; this will result in an increased particle size and, in turn, decreased mechanical properties. The interfacial tension can be reduced by the addition of interfacial agents known as compatabilizers [8]; these are generally molecules with hydrophobic and hydrophilic regions that can be aligned along the interfaces between the two polymer phases, causing the interfacial tension to be reduced and the compatibility of the polymer blends to be increased. Compatibility results in a reduction of the dispersed particle size, an enhanced phase stability, and increased mechanical properties [8]. The physical properties of miscible, compatabilized and uncompatabilized blends can be characterized using techniques such as thermogravimetric analysis, dynamic mechanical thermal analysis, and universal testing machines.

1.4 Topics Covered in this Book

The following chapters in this book provide a comprehensive overview on the miscibility, phase separation, morphology and other fundamental properties of polymer blends, using a wide range of state-of-the-art techniques. For example, Chapter 2 relates to the miscibility of polymer blends, and provides an overview of the theory behind the phase separation of polymer blends, with specific examples. Chapter 3 describes the compatibility of polymer blends and discusses the influence of compatabilizers on phase morphology and structural properties. Chapter 4 provides a comprehensive review of the rheological properties of thermosetting blends and composites, while Chapter 5 provides details of the light scattering of polymer blends and outlines the theory and applications of light-scattering techniques for studying the phase behavior of polymer blends. In Chapter 6, a survey is provided of the characterization of polymer blends, using X-ray scattering techniques, while Chapter 7 details the basis for using neutron scattering when studying the phase behavior of polymer blends and block copolymer systems. In Chapter 8, the applications of ultrasound on polymer blends are reviewed; these include the characterization of polymer blends, polymer modification, property enhancement and the monitoring of polymer processing. The theories and applications of ellipsometry on polymer blends are discussed in Chapter 9, applications include phase separation, morphology, nanoporosity, adhesive and adsorption properties. Chapter 10 discusses the theories and applications of inverse gas chromatography on polymer blend systems; the advantages and drawbacks of inverse gas chromatography are also discussed in detail. In Chapter 11, the thermal stability of polymer blends is discussed, outlining the changes in thermal stability by blending, together with some specific examples. The application of dynamic mechanical thermal analysis (DMTA) for the study of polymer blends is described in Chapter 12, together with details of the theoretical bases underlying viscoelastic theory, the study of segmental dynamics, free volume and dynamic fragility, and the study of miscibility or the effect of adding plasticizers and chemical or physical crosslinkers. An overview of the thermomechanical properties of polymer blends is provided in Chapter 13, where the importance of particle size on polymer toughening is stressed. Chapter 14 provides a comprehensive review on water sorption and solvent sorption behavior that can be expected from miscible or immiscible but compatible polymer blends. The authors of this chapter concluded that miscibility depends on the molecular structures of the both polymers, morphologies and blend composition, as well as the processes of the blends. Numerical simulation models and methods for polymer blends are covered in Chapter 15, while Chapter 16 deals with the latest microscopic techniques that have the capability to observe the three-dimensional structural details in polymer blend films. In Chapter 17, the morphologies of different types of polymer blends investigated by various microscopic techniques are discussed, with special reference to sample preparation and comparison among SEM, TEM and AFM. In Chapter 18, the most sensitive surface technique, namely secondary ion mass spectrometry (SIMS), is described, which allows molecular and highly spatially resolved lateral information to be obtained. In Chapter 19, the authors discuss the morphology of different types of polymer blends, using fluorescence microscopy to determine the quantitative spatial distribution and molecular information of the polymer components. In Chapter 20, the potential of FTIR techniques for determining the properties of polymers blends both quantitatively and qualitatively is highlighted, while in Chapter 21 hydrogen-bonding interactions between the polymer components, determined using NMR spectroscopy, are discussed. Chapter 22 provides details of the phase morphology of polymer blends using infrared, near-infrared and Raman imaging, while in Chapter 23 the application of, and recent advances in, the characterization of polymer blends using electron spin resonance (ESR) spectroscopy and forward recoil spectroscopy, are discussed. In Chapter 24, attention is focused on the optical properties of polymer blends, in both ultraviolet and visible ranges of light, to investigate the miscibility of polymer blends, while Chapter 25 focuses on the driving forces that produce miscibility in polymer blends, using fluorescence spectroscopy. Chapter 26 provides a comprehensive overview of recent developments and progress in the molecular dynamics, miscibility, nature of interaction, crystallization behavior, and curing kinetics of representative examples of polymer blends using dielectric relaxation spectroscopy (DRS) and thermally stimulated depolarization current (TSDC) techniques. Finally, in Chapter 27 attention is focused on the miscibility of polymer blends, using positron annihilation spectroscopy.

References

1. Paul, D.R. (1989) Control of phase structure in polymer blends, in Functional Polymers (eds D.E. Bergbreiter and C.R. Martin), Plenum Press, New York, p. 1–18.

2. Scobbo, J.J, Jr and Goettler, L.A. (2003) Applications of polymer alloys and blends, in Polymer Blends Handbook (ed. L.A. Utracki), Kluwer Academic Publishers, pp. 951–976.

3. White, J.L. and Bumm, S.H. (2011) Polymer blend compounding and processing, in Encyclopedia of Polymer Blends (eds A.I. Isayev and S. Palsule), Wiley-VCH, Weinheim, pp. 1–26.

4. Flory, P.J. (1953) Principles of Polymer Chemistry, Cornell University Press, New York.

5. Jyotishkumar, P., Ozdilek, C., Moldenaers, P., Sinturel, C., Janke, A., Pionteck, P., and Thomas, S. (2010) Dynamics of phase separation in poly(acrylonitrile-butadiene-styrene)-modified epoxy/DDS system: kinetics and viscoelastic effects. J. Phys. Chem. B, 114, 13271–13281.

6. Jiang, R., Quirk, R.P., White, J.L., and Min, K. (1991) Polycarbonate-polystyrene block copolymers and their application as compatibilizing agents in polymer blends. Polym. Eng. Sci., 31, 1545.

7. George, S.M., Puglia, D., Kenny, J.M., Causin, V., Parameswaranpillai, J., and Thomas, S. (2013) Morphological and mechanical characterization of nanostructured thermosets from epoxy and styrene-block-butadiene-block-styrene triblock copolymer. Ind. Eng. Chem. Res., 52 (26), 9121–9129.

8. Chen, C.C. and White, J.L. (1993) Compatibilizing agents in polymer blends: Interfacial tension, phase morphology, and mechanical properties. Polym. Eng. Sci., 33, 923–930.

2

Miscible Blends Based on Biodegradable Polymers

Emilio Meaurio, Natalia Hernandez-Montero, Ester Zuza, and Jose-Ramon Sarasua

2.1 Introduction

This review reports the advances in the field of biodegradable polymer blends with both natural and synthetic polymers. These materials have attracted industrial and academic attention because they can be tailored to improve certain properties, such as biocompatibility and biodegradability, to the specific requirements of the different applications in the biomedical, pharmaceutical and packaging fields. In addition, they can be produced using simple techniques. The present chapter is divided in two main sections following this brief introduction. The first section provides the necessary theoretical knowledge to understand the overall miscibility behaviors observed in real polymer blends. The thermodynamic theory presented here is based on the simple but highly flexible Flory–Huggins theory, incorporating the regular solution theory to account for the enthalpic contribution. This simple theoretical model is extended here with the appropriate approaches to systems presenting specific interactions, providing an overall general picture of the miscibility behavior of polymer blends. The second section reviews the structure, preparation, miscibility and properties of different biodegradable polymer blends investigated up to now. Particularly, the biodegradable blends based on polylactides have been thoroughly revisited and discussed in great detail. In addition, the miscibility behavior of other commercially available biopolymers of great interest such as poly( -caprolactone) (PCL), poly(3-hydroxybutyrate) (PHB), poly(p-dioxanone) (PPDO) and polyglycolide (PGA) is also briefly reviewed. Finally, Appendix 2.A shows a short outline of the research works used to develop the miscibility study presented here, recapping the chemical structures of the polymers, their solubility parameters and brief comments summarizing the research works.

2.2 Thermodynamic Approach to the Miscibility of Polymer Blends

2.2.1 Introduction

Both in case of low-molecular-weight mixtures or polymer blends, the equilibrium phase behavior is determined by the free energy of mixing of the system, ΔGm:

(2.1) equation

where ΔHm is the enthalpy of mixing, ΔSm is the entropy of mixing and T is the temperature [1–3]. To observe a single-phase system, the following two conditions must be fulfilled: first ΔGm < 0 and second:

(2.2) equation

where ϕi represents the volume fraction of component i. The last condition assures a concave upward ΔGm versus ϕi curve at any composition, assuring the stability of the system and preventing phase separation [1–3].

2.2.2 Molecular Size and Entropy

Historically, the simplest miscible system has been the mixture of ideal gases, consisting on noninteracting point particles. This system can be analyzed using simple arguments based on classical thermodynamics [1]. The enthalpy of mixing is zero (ΔHm = 0) and the entropy of mixing (ΔSm) is given by:

(2.3) equation

where ni is the number of moles of the i-th component, xi is the molar fraction, and R is the ideal gas constant [1].

Equation (2.3) also works properly for mixtures of nonpolar solvents of similar molecular volume, and is considered the classical expression for the entropy of mixing of ideal solutions [1]. These solutions present random molecular mixing, and Eq. (2.3) can actually be derived from the statistical thermodynamic analysis of a lattice model randomly filled with spheres of identical size [4,5]. In addition, in an ideal solution the average strength of the intermolecular interactions between the components of the mixture is identical to the average strength of the interactions occurring between the pure components [1]. Therefore, the enthalpy of mixing in these systems is zero (ΔHm = 0) and miscibility arises from the entropic contribution to the free energy of mixing (ΔGm = −TΔSm).

Investigations dealing with the thermodynamic properties of polymer solutions revealed that Eq. (2.3) underestimates ΔSm in these systems, and is not valid for mixtures of compounds with grossly different molar volumes [6]. The statistical thermodynamic analysis of such solutions was carried out by Flory–Huggins, using a simple lattice representation for the polymer solution and calculating the total number of ways the lattice can be occupied by small molecules and by connected polymer segments, Ω (see Figure 2.1b) [2,6,7]. In this model, each lattice site accounts for a solvent molecule or a polymer segment with the same volume as the solvent molecule (not necessarily the repeat unit of the polymer). The entropy of mixing and the total number of arrangements Ω are related according to Boltzmann's law:

(2.4) equation

where k is the Boltzmann constant. The Flory–Huggins analysis assumes a random arrangement for the chain segments (equal probabilities for all the configurations). The final expression for the entropy of mixing is [2,6]:

(2.5) equation

Figure 2.1 (a) Lattice with random mixing of two types of sphere (10³⁰ possible combinations); (b) Lattice with random mixing of polymer and solvent (10¹⁶ combinations); (c) Lattice with random mixing of two types of polymer chain (10³ combinations).

Equation (2.5) is apparently very similar to the classical ideal Eq. (2.3). The only difference is that volume fractions, ϕi, replace mole fractions, xi. This apparently subtle change is actually crucial to account for the effect on the entropy of mixing associated to the large difference of molecular size between the molecules of the solvent and the polymer. This effect is best understood with the aid of examples based on the lattice models in Figure 2.1. Figure 2.1a–c represent a mixture of solvent molecules, a polymer solution, and a polymer blend respectively, in all cases being ϕ1 = ϕ2 = 0.5. According to Eqs (2.4) and (2.5), in the Flory–Huggins treatment the number of microstates for the lattice models in Figure 2.1 can be obtained from lnΩ = −(N1lnϕ1 + N2lnϕ2), where Ni is the number of molecules. In case of the lattice representing the solvent mixture (Figure 2.1a), N1 = N2 = 50, and Ω ≈ 10³⁰. In case of the lattice representing the polymer solution (Figure 2.1b), N1 = 50, N2 = 5; therefore, Ω ≈ 10¹⁶. The reduction in the number of microstates is a consequence of reducing the total amount of molecules in the system (ΔSm ∝ Ni) as a consequence of introducing larger molecules. Note that if molar fractions (present in the classical Eq. (2.3)) were used instead of volume fractions, the number of microstates calculated would be noticeably reduced (x1 = 50/55, x2 = 5/55, Ω ≈ 10⁷). The main reason for this discrepancy is that in the lattice models of the Flory–Huggins theory, solvent molecules can be allocated in a total of 100 lattice sites (even though only 50 sites are occupied at the same time); the large number of cells available being due to the large size of the polymer molecules. In this respect, the classical calculation actually assumes that the solvent molecules can only be distributed among a total of 55 sites. These simple calculations indicate, in agreement with experimental measurements, that adding solvent molecules to big molecules adds more entropy to the system than that predicted by the classical model [6]. Finally, Figure 2.1c represents a polymer blend with N1 = N2 = 5, therefore Ω ≈ 10³. For this last system the number of microstates is further reduced, and because the blend contains molecules of identical size, volume fractions and molar fractions are equivalent and the classical and the Flory–Huggins models predict the same number of microstates. Hence, the Flory–Huggins model predicts the same Ω for the system in Figure 2.1c as the one predicted by the classical equation for one blend containing spherical molecules of the same size as the polymer molecules. This implies that the Flory–Huggins model does not account for any conformational contribution to the entropy (arising from the different conformations accessible to the flexible polymer chains in Figure 2.1c); it only accounts for the so-called combinatorial entropy (also referred to as translational entropy), that is, the entropy resulting from all the possible combinations for the centers of masses of the molecules.

2.2.3 The Regular Solution

In real solutions, the thermodynamic magnitudes ΔGm, ΔHm and ΔSm, deviate from the ideal values, and the excess functions can be defined as the difference between the real values and the ideal values. For example, the excess enthalpy is [1]:

(2.6) equation

A case of particular interest is the so-called regular solution, a mixture in which the excess entropy and the excess volume are zero, but the average interactions in the blend are different from those in the pure components. Since SE = 0, the regular solution maintains the random molecular order of the ideal solution. In addition, VE = 0 implies that the volume of mixing is also zero (ΔVm = 0) and for a mixing process at constant pressure enthalpies can be equated to internal energies; hence, the enthalpy of mixing, ΔHm, can be calculated according to:

(2.7) equation

where ΔEm is the internal energy of mixing, Em is the internal energy of the mixture, and Ei are the internal energies of the pure components. Considering an isothermal process, the internal energy contributions associated to internal motions (such as vibrational or rotational motions) can be excluded from the calculation as they should be unaffected by the mixing process [1]. The only relevant contributions are therefore those associated to the intermolecular interactions. For simplicity, consider the lattice in Figure 2.1a representing a mixture of low-molecular-weight compounds [2–5]. In this lattice model, each solute segment is surrounded on average by ϕ1z neighbors of solvent molecules (where z is the coordination number of the lattice and solvent is component-1), and by ϕ2z neighbors of solute molecules (solute is component-2). Since the total number of sites in the lattice is n0, the lattice contains n0ϕ2 solute segments each interacting in this manner, hence, the total number of 1–2 contacts is n0ϕ2ϕ1z and the total number of 2–2 contacts is ½n0ϕ2ϕ2z (the factor ½ enters here to avoid counting each contact twice; one time per molecule participating in the 2–2 contact). Similarly, the total number of 1–1 contacts is ½n0ϕ1ϕ1z. Regarding the initial pure solutions, the lattice representing the pure solvent contains n0ϕ1 molecules and ½n0ϕ1z 1–1 contacts (the factor ½ avoids again counting each contact twice), while the lattice representing the pure solute contains similarly ½n0ϕ2z 2–2 contacts. Defining w as the attractive energy corresponding to each contact, ΔHm is therefore:

(2.8)

equation

and recalling that ϕ1 + ϕ2 = 1, Eq. (2.8) can be simplified to:

(2.9)

equation

The simplification step from Eq. (2.8) to Eq. 2.9 eliminates the contributions due to the autoassociation contacts occurring in the mixture (second and third terms in the right-hand side of Eq. (2.8)) with identical terms arising from the pure components. Therefore, only the contacts that are actually modified (broken or formed) contribute to ΔHm in Eq. (2.9); and the enthalpy of mixing is the energetic balance corresponding to the process of breaking identical amounts of 1–1 and 2–2 contacts and replacing them by 1–2 contacts [2–5].

According to Eq. (2.9), the formation of each 1–2 contact involves breaking half 1–1 and half 2–2 contacts on average. The interaction energy corresponding to the formation of each 1–2 contact, Δw12, is defined according to [2–5]:

(2.10) equation

and the enthalpy of mixing corresponding to the regular solution is therefore:

(2.11) equation

where n0 = V/Vr (V is the total volume of the lattice and Vr is the molar volume of the lattice sites) has been considered. All the constant terms in Eq. (2.11) can be included in a new constant term, B:

(2.12) equation

B is the product of the interaction energy corresponding to each 1–2 contact (Δw12) by the density of contacts (the quotient z/Vr), and is termed the interaction energy density. B represents the interaction energy density achieved by the solute lattice sites at infinite dilution (when they are exclusively surrounded by solvent molecules and can establish z 1–2 contact). It is therefore a property of the system independent of the lattice parameters. This can be rationalized considering for example the case of n-alkanes in solution: choosing two methylene units (–CH2–CH2–) as reference instead of one (–CH2–) reduces the density of lateral contacts by half, but the interaction energy of each contact is doubled, and therefore B remains constant. In systems with dispersive interactions exclusively, and if the surface-to-volume ratios of the mixed molecules are similar (as they usually are in many organic systems), B is nearly constant with composition [8]. Finally, note that the product BVr represents the interaction energy per lattice site, and can be used to compare the strength of the dispersive interactions between different systems, as long as the same (arbitrarily chosen) Vr value is used in all cases. The enthalpy of mixing, ΔHm, written in terms of B is:

(2.13) equation

This form of equation for the enthalpy of mixing is known as van Laar-type equation. In spite of being derived for a lattice filled with molecules of equal size, Eq. (2.13) also works correctly for mixtures of molecules of different size. For example, the enthalpy of mixing at infinite dilution calculated per mole of solute is ΔH∞ = BV02 (since Bϕ1 ≈ B and Vϕ2 = V2 in Eq. (2.13); in addition V2 = V02 considering one mole of solute), indicating that ΔH∞ scales linearly with the molar volume of the solute, V02. Following with the n-alkanes as an example, doubling the length of the solute doubles ΔH∞, as expected.

Many nonelectrolyte solutions of small molecules obey approximately Eq. (2.13) [8]. Experimental studies have shown that the regular solution model is appropriate for binary mixtures without specific interactions, including systems containing polymer chains [8]. In general, specific interactions (hydrogen bond, charge transfer, etc.) modify molecular order and the free volume of the blend, affecting the dependence of the excess enthalpy with composition ((randomness was assumed in Eq. (2.13)), and modifying the excess entropy (no longer zero). Temperature dependences are also modified, as they become governed by the evolution of the interactions with temperature [8,9].

2.2.4 The Flory–Huggins Model

This theoretical model adopts the Flory–Huggins treatment for the entropy of mixing, Eq. (2.5), and the regular solution treatment for the enthalpy of mixing, Eq. (2.13), leading to the following expression for the Gibbs free energy of mixing [2]:

(2.14) equation

Equation (2.14) can be considered a general expression valid for regular solutions, regardless of the size of the components (solvents or polymers). Note that the entropic term scales with the number of molecules, and the enthalpic term with the volume of the system; hence, both terms scale differently with the size of the system, and implicitly with the size of the molecules comprising the system. To explore the effect of these size-related aspects on ΔGm, Eq. (2.14) is rewritten as an intensive property, usually referenced to the volume corresponding to one mole of lattice sites, Vr. The number of lattice sites occupied by component i is given by its molar volume ratio, ri, defined as:

(2.15) equation

where Vi is the molar volume of component i, mi is the polymerization degree (mi = 1 in the case of a solvent), V0i is the molar volume of the repeat unit, M0i is the molar mass of the repeat unit, and ρi is the density in the amorphous state. The total number of lattice cells occupied by the ni moles of component i is rini, and its volume fraction is therefore:

(2.16) equation

where n0 stands for the total number of lattice sites (n0 = Σrini). Using Eqs (2.15) and (2.16), and recalling that n0 = V/Vr, Eq. (2.14) can be rewritten as:

(2.17) equation

Equation (2.17) provides ΔGm per mole of lattice sites, or per Vr volume units of mixture. It is valid for any type of regular solution, regardless of the difference in size between the components of the mixture. In Eq. (2.17), the entropic term is always negative (lnϕi < 0, since ϕi < 1) and therefore favorable to the miscibility. The plot of this term versus composition is concave upwards for any system at any temperature. Miscibility depends therefore on the sign and magnitude of the enthalpic term. If B < 0, the enthalpic term is also concave upwards and the plot of ΔGm versus composition maintains the same curvature, indicating complete miscibility (one single phase at any composition). However, in systems with B > 0, the enthalpic contribution is always positive and adopts a concave downwards curvature. As long as B is small enough, the curvature corresponding to the entropic term still dominates and the plot of ΔGm versus ϕ still indicates complete miscibility. However, when B exceeds a certain critical value, the plot of ΔGm versus composition shows two minima, with one concave downwards central region, where phase separation occurs.

Phase separation begins at the critical conditions, when the two minima of the ΔGm versus ϕ plot meet at the same point, and also meet the two inflection points limiting the concavity regions. Therefore, the following conditions apply [1–5]:

(2.18) equation

Applying Eqs (2.18) to (2.17), the following relationship is obtained for the critical conditions [2–5]:

(2.19) equation

Equation (2.19) relates the critical interaction energy (BVr)c, the critical temperature, Tc, and the critical molecular sizes, ric ((related to the critical polymerization degrees according to Eq. (2.15)). As can be seen, the critical interaction energy decreases with increasing molecular size, and Eq. (2.19) can be used to compare the ability of solvent solutions and polymer blends to form miscible systems. Assuming, for simplicity, that Vr = V01 = V02, in case of a mixture of solvents at room temperature r1 = r2 = 1 (since m1 = m2 = 1 in the case of solvents), and (BVr)c = 2RT ≈ 5 kJ mol−1 according to Eq. (2.19). In the case of a solution of a high-molecular-weight polymer, m1 = 1 and 1/(m2)½ ≈ 0, hence (BVr)c ≈ (RT)/2 ≈ 1.2 kJ mol−1. In this case, the unfavorable enthalpic contribution necessary to begin the phase separation is smaller, which explains why the amount of solvents available for high-molecular-weight polymers is reduced compared to low-molecular-weight substances. Finally, for a blend of high-molecular-weight polymers of identical degrees of polymerization (m1 = m2 = m), Eq. (2.16) simplifies to (BVr)c = (2RT)/m. Assuming a typical degree of polymerization of about 10³, then (BVr)c = 0.005 kJ mol−1, and hence phase separation at room temperature already starts at extremely small repulsive interactions. Note that (BVr)c is reduced four times on passing from a solvent mixture to a polymer solution, but on passing to a polymer blend the critical interaction energy is reduced r times (10³ in this example). Hence, the occurrence of attractive specific interactions is usually considered a requirement to observe miscibility in polymer blends [8].

There is, however, another important difference between the low- and high-molecular-weight mixtures seldom discussed in the literature: according to the Flory–Huggins theory, partial miscibility is also very improbable in polymer blends compared to low-molecular-weight mixtures. Figure 2.2 shows the Gibbs free energy of mixing versus composition for mixtures of two solvents, A and B; and for polymer blends with different interaction energies. As before, the systems are assumed to consist of components of equal V0 values. As can be seen in Figure 2.2a, solvent mixtures with BVr = 6 kJ mol−1 and solvent feed compositions in the range 16–84 vol.% will phase separate in one A-rich phase containing about 84 vol.% A and in one B-rich phase containing about 16 vol.% A. However, if the interaction energy increases to BVr = 10 kJ mol−1, then the amount of the minority component in the two phases reduces to about 2 vol.%. There is a relatively wide range of molar interaction energies (from 5 kJ mol−1 to ca. 10 kJ mol−1) for which partial miscibility can be claimed, and a number of systems can be found with BVr values within this range;, hence, partial miscibility is a rather frequent behavior. However, in the case of polymer blends, and assuming again m1 = m2 = 10³, partial miscibility with phases containing the minority component in significant amounts can be observed only within a very narrow BVr range (see Figure 2.2b), from BVr = 0.005 kJ mol−1 to about 0.010 kJ mol−1. Increasing BVr from 0.010 kJ mol−1 to 0.020 kJ mol−1 reduces the volume fraction of the minority component in the separated phases from two-hundredths to below one- thousandth, where immiscibility should be claimed. Therefore, according to the Flory–Huggins model, partial miscibility in polymer blends only occurs within an extremely narrow range of interaction energies located next to the critical value (for the particular example considered here, BVr ranges from 0.005 to about 0.010 kJ mol−1). Partial miscibility is therefore also a very rare phenomenon in polymer blends compared to low-molecular-weight compounds [10].

Figure 2.2 Gibbs free energies of mixing versus composition for systems of different interaction energies (BVr, kJ mol−1) calculated using the Flory–Huggins model (Eq. (2.17)). (a) Mixtures of low-molecular-weight solvents of identical molar volume; (b) Blends of polymers of degree of polymerization m = 10³ and identical repeat unit molar volume.

These ideas can be clarified by using the simple graphs in Figure 2.3, that classify the miscibility behavior of both low-molecular-weight mixtures and polymer blends according to their molar interaction energies (BVr). The red-colored regions indicate miscible systems, green-colored regions indicate partial miscibility, while gray regions indicate phase-separated systems. The graph heights have only qualitative validity, and are related to the probability of finding systems with the BVr values in the abscissa (or to the number of systems actually possible). As can be seen, similar distributions of BVr values have been assumed in both cases, since both polymers and their low-molecular-weight analogs exhibit intermolecular interactions of similar strength. Negative BVr values can only be observed when the systems contain specific chemical groups capable of establishing attractive interactions that overcome the ubiquitous dispersive interactions (of repulsive nature) [8]. This is the principle of the so-called complementary dissimilarity [8], a fairly uncommon feature, and therefore the heights are small when BVr < 0. Positive interaction energies will be observed in most cases. The areas of the red and green regions are related (qualitatively) to the probability of finding completely miscible and partially miscible systems, respectively. As can be seen, both regions are strongly reduced in case of blends of homopolymers of high molecular weight (in this numeric example, to about one-thousandth). Therefore, not only miscibility but also partial miscibility can be considered the exception rather than the rule in the case of polymer blends [8,10].

Figure 2.3 Classification of the miscibility regions according to the interaction energy (BVr) intervals: miscible (red), partially miscible (green) and immiscible (gray) systems. The upper graph shows low-molecular-weight mixtures; the lower graph shows polymer blends (see text for details).

Finally, in the Flory–Huggins treatment the strength of the interactions is frequently expressed in terms of the interaction parameter, χ, defined by Flory as the ratio between the interaction energy and the thermal energy:

(2.20) equation

The Flory–Huggins equation is therefore:

(2.21) equation

Note that both the interaction energy, BVr, and the interaction parameter, χ, are proportional to the size of the system, or Vr. For example, for blends with polyethylene, Vr can be adopted as the molar volume of one methylene unit or as the molar volume of one dimethylene unit. Doubling Vr implies doubling BVr and χ. To allow comparisons in the strength of the interactions between different systems, some authors prefer to work with an arbitrarily chosen constant value for Vr, instead of changing its value from one system to another [11].

2.2.5 The Hildebrand Approach

This approach estimates the strength of the dispersive interactions in mixtures of solvents, and is straightforwardly extended to mixtures involving polymers [8,9]. The cohesive energy, Ecoh, is defined as the internal energy of vaporization (ΔEv) from the liquid state to the ideal gas state (where intermolecular forces are suppressed). The energy necessary for this transformation is mainly spent pulling the molecules apart, since other molecular contributions to the total internal energy (such as vibrational energies, etc.) should remain unaffected because of the isothermal nature of the process. Therefore, Ecoh is a measure of the intermolecular attractive energy, and for low-molecular-weight compounds it can be obtained from the enthalpy of vaporization (ΔHv) according to:

(2.22) equation

Cohesive energies are therefore positive since ΔEv > 0. The cohesive energy density (CED; Eq. (2.23)) is defined as the cohesive energy per unit volume, and measures the intermolecular attraction energy per unit volume in pure substances. In addition, the solubility parameter, δ, is defined as the square-root of the cohesive energy density, (δ = (CED)¹/²):

(2.23) equation

Cohesive energy densities can be used to estimate the value of the interaction energy density, B, using the Hildebrand approach [9]. The total number of contacts in a lattice containing n0 lattice sites is ½n0z (the factor ½ avoids counting each contact twice), and the total attractive energy in the case of identical contacts is ½n0zw. Therefore, the CED of a pure substance i is related to the lattice parameters as follows:

(2.24) equation

The negative sign is introduced because the interaction energies are negative but the CEDs are positive (see Eq. (2.23)). As can be seen, the CED of a pure component provides the energy density of the autoassociation contacts, terms that already appeared when calculating the interaction energy density, B (see Eq. (2.12)). Unfortunately, the energy density for the interassociation contacts cannot be measured since, in addition to the 1–2 contacts, real mixtures always contain 1–1 and 2–2 contacts. This issue is overcome using the Hildebrand approach, assuming that the 1–2 attractive energy can be estimated as the geometric mean of the attractive energies of the pure components (w12 = (w11w22)½) [9]. Therefore, the CED that would be obtained from a hypothetical mixture containing exclusively 1–2 contacts (denoted as CED12) can be estimated according to:

(2.25)

equation

Substituting the values of the attractive energy densities in terms of the CEDs of the pure components in the definition of B (Eq. (2.12)), and recalling that δ² = CED:

(2.26)

equation

Substituting in Eq. (2.13), the enthalpy of mixing can be estimated according to:

(2.27) equation

This equation indicates that the contribution of the dispersive interactions to the enthalpy of mixing is always positive, and therefore unfavorable to miscibility [8,9].

Unfortunately, polymers are nonvolatile compounds, and solubility parameters cannot be obtained directly from the enthalpies of vaporization. Several indirect methods, such as swelling measurements, inverse gas chromatography and solubility testing, can be used to measure the δ values [12]. In addition, different group contribution methods have been developed to estimate their solubility parameters [13]. Due to the wide range of methods available, and to the errors associated with each method, values found in the literature usually extend over a rather wide range [12]. Among the different experimental methods available to obtain δ values, solubility testing is usually considered the most reliable [14].

2.2.6 Extension of the Flory–Huggins Model to Systems with Specific Interactions

In the Flory–Huggins model, randomness is assumed in the derivations of both the entropic term and the Van Laar-type enthalpic term. However, in systems containing specific interactions (such as hydrogen bonding, dipole–dipole, charge transfer, etc.) the number of specific contacts usually exceeds the value expected when considering random mixing [8]. These systems are said to maximize the number of specific interactions, and randomness is therefore lost. In addition, the energetic contribution associated with the establishment of the specific contacts is usually much larger than the energetic contribution associated with the establishment of the dispersive contacts, which means that even slight deviations from randomness can be accompanied by dramatic changes in the dependence of ΔGm with composition. Equation (2.17) is therefore no longer appropriate for these systems.

The simplest approaches to extend the Flory–Huggins model to systems with specific interactions seek the introduction of corrections to Eq. (2.17) that account for the particularities of these systems [8]. Assuming a quasi-random mixture and neglecting free-volume effects, the entropic term can be maintained and the product of volume fractions in the enthalpic term can be retained as a reasonable approach to count the number of interassociating contacts. However, the interaction energy density, B, is no longer independent of the composition in systems with specific interactions. This can be rationalized by considering a mixture of low-molecular-weight compounds of identical molar volume in which the two components of the blend contain respectively one and two interacting groups. When the compound at infinite dilution is the one establishing double interactions, the lattice cells containing the dilute compound will interact with about double strength than in the opposite case (when the compound at infinite dilution is the one that is only able to establish single contacts).

To account for the composition dependence, B can be modified by adding correction terms defined in a parallel manner to the original dispersive term. At infinite dilution of component-2 (when ϕ1 → 1), the correction term should be the product of the density of specific contacts achieved by the dilute component, z02/V02, (where z02 is the number of specific contacts per molecule, or per repeat-unit-2 in the case of a polymer, and V02 is the molar volume of component-2, or of the repeat-unit-2) by the interaction energy of the specific contacts, Δwsi. A similar correction term can be proposed when ϕ2 → 1. At intermediate concentrations, the rule of mixtures can be assumed. Hence, B can be redefined according to:

(2.28) equation

Equation (2.28) has been written assuming the occurrence of only one type of specific interaction in the system (characterized by Δwsi), but in the case of the coexistence of different types of specific interaction it can be easily extended. In addition, the dispersive part in Equation (2.28) (the first term on the right-hand side) can be estimated from (δd1 − δd2)², where δdi are the dispersive solubility parameters, calculated using specially developed group contribution methods that account only for the dispersive part of the interactions [8,15].

Testing Eq. (2.28) against real polymer blends is not a trivial task, however, because there is no simple experimental procedure to measure B (or χ) at different compositions in these systems. Molecular modeling techniques represent an alternative approach to investigate the miscibility of polymer blends and to calculate the values of B or χ as a function of composition (the fundamentals of this procedure are briefly discussed in Section 2.1.9 of this chapter) [16,17]. Molecular Dynamics (MD) simulations of blends of poly(L-lactide) (PLLA) with poly(vinyl phenol) (PVPh) were performed using the Materials Studio (v. 4.0) software package from Accelrys [17]. This software suite includes the COMPASS (Condensed-phase Optimized Molecular Potentials for Atomistic Simulation Studies) force-field, specially optimized to provide an accurate condensed-phase equation of state and cohesive properties for molecules containing a wide range of functional groups. Figure 2.4 presents the values calculated for χ using MD simulations [17]. For PLLA-rich systems, the calculated data are also in good agreement with the value obtained from melting point depression measurements, χ = −0.4 (note, however, that the experimental value is actually valid at the melting temperature of PLLA, about 190 °C, but MD calculations were performed at 25 °C) [18]. The calculated data have been fitted to Eq. (2.29), obtained from Eqs (2.28) and (2.20), letting Δwsi as the only fitting parameter. The remaining constant parameters present in Eq. (2.29) are listed in Table 2.1 [17]. As can be seen in Figure 2.4, Eq. (2.29) fits properly the calculated data using Δwsi, = −1.7 kJ mol−1. The interaction parameter is larger when PLLA is the dilute component because the interaction energy density is larger because of the smaller molar volume of PLLA:

(2.29) equation

Figure 2.4 Values of the interaction parameter, χ, obtained from MD simulations for PLLA/PVPh blends of different composition. The continuous line is the fit of the calculated data to Eq. (2.29), using the constants in Table 2.1 and Δwsi = −1.7 kJ mol−1.

Table 2.1 Constant parameters used in Figure 2.4 to fit the values of χ obtained from MD simulations for the PLLA/PVPh system to Eq. (2.29).

A brief analysis of the Δwsi parameter can provide additional insight into the nature of the specific interactions in the PLLA/PVPh system. The main specific interactions present in this system and the corresponding hydrogen-bonding enthalpies are hydroxyl–carbonyl interassociation (hA = −3.8 kcal mol−1) and hydroxyl–hydroxyl autoassociation through dimer formation (h2 = −5.6 kcal mol−1) and through multimer formation (hB = −5.2 kcal mol−1) [8]. Recalling that interaction energies were defined as the energy corresponding to the formation of one interassociation bond at the expense of breaking half of the autoassociation bond of each component, in a hypothetical scenario where pure PVPh was assumed to be exclusively autoassociated through dimer formation, the interaction energy would be Δwsi = −3.8 − ½ × (−5.6) = −1.0 kcal mol−1 (= −4.0 kJ mol−1). This (actually unrealistic) scenario predicts a favorable contribution to miscibility arising from specific interactions, even though the stronger O–H O–H interactions are replaced by the weaker O–H O=C interactions. The actual reason is that, in this hypothetical situation, the number of formed bonds doubles the number of broken bonds. Nevertheless, this simple calculation explains that weaker interactions can replace stronger interactions if the total number of specific contacts increases. In the infrared (IR) spectrum, this change should be observed as a blue shift of the OH-stretching band [8]. A more realistic estimation should consider that in actual polymer blends, interassociation peaks at about 80% of the hydrogen-bonded C=O groups [8,19], and that in pure PVPh the percentage of autoassociated OH groups is about 60%, mainly in the form of multimers [8,20,21], in which internal OH groups establish two hydrogen bonds (the coordination number is therefore 2 for the nonterminal hydroxyls, but should be in the range of 1.5–2.0 on average [22]). With these assumptions, Δwsi = 0.8 × (−3.8) − ½ × 1.75 × (−5.2) = −0.31 kcal mol−1 (= −1.3 kJ mol−1) is obtained. This more realistic calculation makes evident the complexity of the balance of interactions. Since increasing the hydroxyl density favors the formation of multimers (and the autoassociation in general), a reduced miscibility can be anticipated for polymers such as poly(vinyl alcohol) (PVA) compared to PVPh.

Finally, it is interesting to analyze how the introduction of specific interactions modifies the miscibility behavior in polymer blends. According to the simple treatment presented here, when the density of specific contacts of the two polymers is identical ((z01/V01 = z02/V02; see Eq. (2.28)), B is independent of composition and the ΔGm versus composition diagrams show improved symmetry, resembling those shown in Figure 2.2. However, when the specific contact densities of the components of the blend are different, B depends on composition, and the free energy curves show significant changes. Figure 2.5 shows BVr and ΔGm versus composition plots for blends between two polymers of different specific contact density, calculated assuming varying relative strengths of the specific to dispersive interactions. The parameters used in Eq. (2.28) for polymer-1 were z01 = 1, V01 = 50 cm³, and m1 = 10³; and for polymer-2 z02 = 1, V02 = 100 cm³, and m2 = 500. Hence, both polymeric chains present the same molecular size, but the former doubles the specific contact density of the latter. The same interaction energy as in the PLLA/PVPh system has been assumed, Δwsi = −1.7 kJ mol−1, and the strength of the dispersive interactions has been varied by acting on the difference of the dispersive solubility parameters, Δδd, in Eq. (2.28).

Figure 2.5 Interaction energy (a) and Gibbs free energy (b) of mixing versus composition for systems with specific interactions of fixed strength, Δwsi = −1.7 kJ mol−1, and varying dispersive interactions according to the Δδd values.

As can be seen, the overall shape of the curves changes dramatically; in particular they become extremely asymmetric. Complete miscibility requires (the curve is concave upwards). Calculating the second derivative of Eq. (2.28) and equating to zero reveals that the critical dispersive interaction is Δδd = 0.74 (cal cm−3)½, the critical composition being ϕ1 = 0.98. All of the ΔGm curves with Δδd above the critical value show a concave-downwards region between two concave-upwards regions. When Δδd is in the range 0.74 to 2.9, the concave-upwards region located to the left is actually due to B < 0 (specific interactions prevail over dispersive interactions), while the concave-upwards region to the right (observable only at high magnification) is due to the prevalence of the entropic term over the enthalpic term, occurring always at high dilution levels. Strictly speaking, partial miscibility should be claimed when 0.74 < Δδd < 2.9, but from the experimental point of view partial miscibility should not be easily observed in the range 0.74 < Δδd < 2.0, because negative second derivatives occur within relatively short composition ranges (when Δδd = 1.5, negative values occur in the range 0.82 < ϕ1 < 0.997, and when Δδd = 2.0 the range is 0.68 < ϕ1 < 0.998).

Figure 2.5 predicts some interesting trends. First, partial miscibility – which was considered a very unusual result in systems with dispersive interactions – now becomes a real possibility. Second, the systems in Figure 2.5 showing partial miscibility separate in a partially miscible phase and a nearly pure polymer; hence, one of the polymers can be partially miscible in the second polymer, but not vice versa. Third, the polymer with a higher density of specific contacts (usually the one with the smaller repeat unit) is the one with enhanced miscibility. For example, blends of polylactides (PLAs) with poly(methyl methacrylate) (PMMA) show single glass transition temperatures, of enlarged width for the PMMA-rich blends (see also Section 2.2.1.6) [23]. This behavior is compatible with the curves in Figure 2.5 with Δδd in the range of 1.5–2.0. In addition, poly(ethylene oxide) (PEO) is partially miscible with PLLA up to about 20–50 wt% PEO (the amount is controversial), the second phase being nearly pure PLA (see also Section 2.2.1.2) [24]. This behavior is consistent with the curve with Δδd = 2.6 in Figure 2.5. In addition, both the PLA/PMMA and the PLA/PEO systems show, as expected, a constrained miscibility for the component of higher molar volume. In contrast, for blends of PLA with PVPh, a partial miscibility was reported by Zhang et al., who observed phase separation in a nearly pure PVPh phase and a PLA-rich phase containing up to about 15–20 wt% PVPh [25,26]. These results are at odds with the theoretical model discussed here since, according to Zhang et al., the polymer with the repeat unit of higher molar volume, PVPh, is the one that is partially miscible in the second polymer. The miscibility of the PLA/PVPh system was reexamined a few years later by the present authors' research group, and complete miscibility was actually found to occur in that system [18,27,28]. The results reported by Zhang et al. were attributed to the occurrence of phase separation during the casting step due to the Δχ effect [18,27]. In summary, even though some systems containing specific interactions may not conform to the trends predicted in this section due to the simplistic nature of the theoretical model, in the authors' opinion it provides in most cases valuable intuitive guidelines.

2.2.7 The Dependence of Miscibility on Blend Composition and Temperature

In the original Flory–Huggins model, the interaction parameter χ (or B) reflected exclusively enthalpic contributions to the free energy of mixing [2]. However, this description was soon extended when assuming that χ includes contributions of both enthalpic and entropic nature to ΔGm [2–5]. As discussed in the preceding sections, specific interactions introduce nonrandomness and other effects that need to be considered for a correct description of the actual behavior of polymer blends. Both, theoretical and empirical corrections to χ (or B) have been proposed [8]. Regarding the dependence on composition, the simplest empirical approach assumes a linear dependence of B with composition (B = a + bϕ1). This type of approach has actually been used in Eq. (2.28) and applied to the PLLA/PVPh system in Section 2.1.6 in this chapter.

Regarding the temperature dependence of the interaction parameter χ (or B), in the original Flory–Huggins model the unfavorable enthalpic term is constant with temperature, but the favorable entropic term increases with temperature (see Eq. (2.14) or (2.17)). The model predicts phase diagrams with an Upper Critical Solution Temperature (UCST), or miscibilization upon heating. However, systems with specific interactions usually present Lower Critical Solution Temperature (LCST) phase diagrams. In these systems, thermal agitation decreases the number specific interassociating contacts, and the system shows phase separation upon heating. In general, the temperature dependence of the χ parameter has usually been described (empirically) by [3]:

(2.30) equation

where the constant a represents an entropic contribution to χ, and b represents the enthalpic contribution. For most systems, both a and b will depend on the blend composition. A positive value for the constant b is associated with the mixture exhibiting an UCST, while a negative b corresponds to LCST behavior. Although Eq. (2.30) has proven useful, more complex relationships may also occur. For example, χ may show a parabolic dependence on T, reflecting the simultaneous existence of both an UCST and a LCST [29].

According to Eqs (2.20) and (2.30), the following dependence on temperature can be deduced for the interaction energy density: B = a + bT. In addition, B depends on composition (recall the simple dependence discussed here, B = a + bϕ1). A general expression can be proposed for B combining both dependences: B = A1 + A2T + A3ϕ1 + A4ϕ1T, where the Ai coefficients are now constant coefficients that depend only on the nature of the system. Several expressions of different complexity can be found in the literature [30,31].

2.2.8 The Painter–Coleman Association Model (PCAM)

Coleman and Painter combined an association model with the Flory–Huggins theory to develop a theory that would describe the miscibility behavior of polymer blends with hydrogen bonding [8,32,33]. The theory is based on the assumption that the enthalpic contribution to the Gibbs free energy of mixing ΔGm consists of weak or physical or dispersive interactions, and strong or chemical or specific interactions. The free energy of mixing per mole of lattice sites is given by:

(2.31)

equation

The first two terms represent the combinational entropy of mixing. Since these entropy terms are usually small in polymer blends, ΔGm is dominated by the balance between the third and fourth terms. The third term represents physical forces that can be estimated using Hildebrand's approach from solubility parameters calculated using group contribution methods for a set of carefully chosen groups which are free from association (see also Section 2.1.6) [8,15]. The fourth term represents the favorable hydrogen-bonding contribution to ΔGm. Its magnitude depends on two major factors. One factor is the relative strength of self-association to interassociation; if the strength of interassociation between two dissimilar polymers is greater than that of the self-association of either of them, then miscibility is favored. The second factor is the density of specific interacting sites in the blend; increasing this density can be expected to render the otherwise immiscible blends miscible.

Coleman et al. uses Fourier transform infrared (FTIR) spectroscopy to obtain the association constants, Ki, for all of the hydrogen-bonding equilibria occurring in the pure polymers, and in the blends. Using a van't Hoff-type plot, the enthalpies of hydrogen bonding can be calculated.

(2.32) equation

With these data and the equations developed in their association model, the ΔGm curves can be calculated at any temperature. The phase behavior of the system can be thus quantitatively described (binodal and spinodal calculations, heats of mixing, etc.) [8,32]. However, the original model included an erroneous excess entropy term within the (ΔGH/RT) term of Eq. (2.32) [34]; this erroneous term was based on an incorrect reference state and allowed a nice agreement between theory and experiment. However, when the association constants obtained from FTIR experiments in polymer blends were used in the correct theory, the agreement was poor [34]. Coleman et al. realized that the poor agreement was a consequence of unaccounted effects in the initial theoretical model. Accordingly, in addition to correcting the reference state, the theoretical model was also modified to take into account intramolecular screening and spacing effects [35]. One unanticipated advantage of these investigations was that standard self-association and interassociation equilibrium constants, which are more easily obtained from appropriate model low-molar-mass compounds, can now be used to calculate the hydrogen-bonding contribution to the free energy of mixing in polymer blends. Although phase diagrams have been successfully calculated with the new model, all in all the corrections introduced have increased its complexity so that currently it has been only been tested in a few systems [35].

2.2.9 Analysis of the Miscibility Using Molecular Modeling Calculations

The MD approach is based on Eq. (2.7): recalling that changes of internal energy associated with the mixing process can be estimated as changes in the cohesive energy (with opposite sign, since E < 0 but Ecoh > 0), dividing both sides by the total volume of the mixture, V, and considering also that Vϕ2 = V2 and Vϕ1 = V1, it can be readily shown that [36]:

(2.33) equation

Hence, by building modeling cells for the pure polymers and for the mixture, and by calculating the CEDs for each of those cells, the internal energy of mixing per unit volume of mixture, ΔEm/V, can be obtained. Using also Eqs (2.20) and (2.13) and recalling that ΔHm ≈ ΔEm, the interaction parameter can be obtained from:

(2.34) equation

In principle, MD simulations carried out using accurate force fields and careful procedures should lead to accurate determinations of the interaction parameter, which is the dominant contribution to ΔGm. The entropic contribution is not determined using Eq. (2.33), but this seems a minor drawback [36]. However, MD simulations seem to show a limited accuracy in some calculations. For example, several groups have reported that calculated solubility parameters are usually below the experimental values [16,37,38], with Gestoso et al. obtaining δ = 8.7 ± 1.2 (cal cm−3)½ for PVPh [38], significantly less than the experimental value obtained from solubility testing, δ = 12.0 (cal cm−3)½} [39]. As solubility parameters are obtained from CEDs, MD simulations seem to underestimate the strength of the intermolecular interactions. However, when the CEDs are calculated to obtain ΔHm according to Eq. (2.33), an important error cancellation can be anticipated as the errors associated with the blend and with the pure polymer cells are subtracted. Recently, several polymeric mixtures have been investigated using the MD approach [16,17,21,37], and a very good agreement has been found in all cases with the experimental results with regards to the miscibility behavior. Hence, MD calculations are currently considered very reliable tools in the investigation of the miscibility of polymer blends.

2.2.10 Classification of Miscible Systems

As discussed, miscibility is a rare phenomenon in polymers. However, miscibility conditions can be satisfied in the following cases.

2.2.10.1 Entropically Driven Miscible Systems

This section encompasses the systems with very similar chemical structures, for which ΔHm is positive but very small (smaller than the favorable entropic term). This situation is very rare but can occur in the cases listed below:

In nonpolar mixtures of homopolymers with very similar chemical structures. Traditionally, the 1,2-polybutadiene/cis-1,4-polyisoprene system [39], has been considered to fall within this category, since only van der Waals dispersion forces are expected between the blend