Specific Interactions and the Miscibility of Polymer Blends

Michael M. Coleman, Paul C. Painter, John F. Graf

• Publisher: CRC Press
• Year: 2017
• Topic: Technology & Engineering
• eBook ISBN:
  9781351415156
• Subtopic: Physics
• Index: Physics

This book with software provides powerful tools for the analysis, prediction and creation of new polymer blends, an area of significant commercial potential. The R&D approaches and methods described in the book have attracted the interest of polymer R&D leaders in industry, and have been put into use in several major chemical companies. The companion set of computer programs speeds and facilitates work in this area. FROM THE AUTHORS' PREFACE: During the 1980's a steadily increasing number of compatible systems [polymer blends] have been reported. We believe that miscible mixtures will prove to be fairly common and the purpose of this book is to explore the circumstances in which single phase materials can be obtained. We will also describe a model for the phase behavior of these mixtures which we believe to have a predictive value, or be used as a practical guide to polymer miscibility. Our approach is based on the use of association models which have until recently been largely ignored in treating hydrogen bonding in polymer mixtures. They have most frequently been applied to mixtures of alcohols with simple hydrocarbons, where the equilibrium constants used to describe association have most frequently been determined by a fit to thermodynamic data (e.g., vapor pressures, heat of mixing). In our work we have sought to, first, adapt this approach to a description of the phase behavior of polymer mixtures; second, develop spectroscopic methods that provide an independent measurement of the equilibrium constants. Our purpose in this book is to explore and describe this approach and illustrate its broad utility. We address two overlapping yet different audiences. One would be primarily interested in the broad nature of this approach and the practical applications of a simple model. The second would be more interested in the derivations of the equations and some of the fundamental aspects of the spectroscopy of these systems. Accor

CHAPTER 1

The Thermodynamics of Mixing

An Incomplete and Biased Overview

A. INTRODUCTION TO THEORIES OF MIXING

Guggenheim¹ classified mixtures primarily into two types, “those in which molecular orientation is unimportant and those in which is it all important.” In this latter class are, of course, molecules that interact through the formation of strong specific interactions, particularly hydrogen bonds, but the majority of both theoretical and experimental studies of mixing have been aimed at obtaining an understanding of the behavior of mixtures of the first type and it is these simpler systems we will consider first. For binary mixtures of small (i.e., non-polymeric) molecules a particularly simple expression for the Gibbs free energy of mixing is obtained if the energy of interaction between the unlike components is the same as the energy of interaction between like molecules (i.e., the interchange energy is zero):

$\frac{\text{Δ}{\text{G}}_{\text{m}}}{\text{RT}}={\text{n}}_{\text{A}}\text{ln}{\text{x}}_{\text{A}}+{\text{n}}_{\text{B}}\text{ln}{\text{x}}_{\text{B}}$

(1.1)

where n_A and n_B are the number of moles of components A and B and x_A, x_B are the corresponding mole fractions. As we will see, this equation is as simple as this subject gets and can be obtained by counting the number of arrangements of A and B molecules on a lattice of n_A + n_B sites:

$\text{Ω}=\frac{\left({\text{n}}_{\text{A}}{\text{+ n}}_{\text{B}}\right)\text{!}}{{\text{n}}_{\text{A}}{\text{! n}}_{\text{B}}!}$

(1.2)

[For an athermal system, the natural logarithm of this equation gives the Helmholtz free energy, ΔF_m, but for solids and liquids not too close to the critical point, and at ordinary pressures, terms in PV or VdP are negligible, so that G is practically indistinguishable from F.]

There are a number of critical assumptions that form the basis for this “ideal mixture” treatment, amongst the most crucial of which are:

1. As already mentioned, the forces acting between like molecules and unlike molecules are identical so that mixing is random.

2. The molecules A and B are roughly the same size and shape.

3. “Free volume” is neglected.

These complications conspire to make a precise treatment of the problem of mixing an extremely difficult one, for which tractable solutions have only been obtained through the use of various simplifying assumptions. Our concern here is not a general discussion of these problems, however, but the particular set of assumptions that are necessary for what we believe to be a successful treatment of polymer mixtures, particularly those where there are strong, specific interactions. Accordingly, we will briefly, and more or less qualitatively, consider only those aspects of the subject of mixtures that provide the foundation for the model we will employ – hence the somewhat facetious sub-title of this chapter. The interested reader is referred to other texts for a more complete discussion^2-6. We start by considering the origin and effect of a non-zero energy of mixing.

B. INTERMOLECULAR INTERACTIONS

For “small,” more-or-less spherical molecules (e.g., CCl₄), it is usual to consider interactions between molecules treated as a whole, but for molecules that consist of a number of chemical units linked together in some fashion (e.g., the n-alkanes, any polymer) it is far more useful to consider interactions between segments, sometimes defined in terms of identifiable chemical units, but which can also be defined in terms of some reference volume that may, for example, include part of a polymer chemical repeat unit, or a number of such units.

The interaction energy between molecules or segments can be considered to consist of two components, arising from repulsive and attractive intermolecular forces, respectively. Repulsive forces become significant at short distances and it is convenient to represent the repulsive potential by a term of the form (σ/d)¹², where d is the intermolecular distance. An exponential form [Aexp(-Bd)] has also been used, but there appears to be no fundamental theoretical justification for either choice. Attractive forces are better understood and we will confine our discussion to the energies of these interactions.

We will somewhat arbitrarily categorize the most frequently encountered forces between molecules or polymer segments as illustrated in table 1.1.

Table 1.1
Frequently Encountered Forces

We have employed the usual criterion of “interaction strength,” where interactions between non-polar molecules are considered to be “weak” relative to those that are more polar. We will briefly discuss the nature of these interactions below, but it will prove useful to first place our de...

Table of contents

• Cover
• Half Title
• Title Page
• Copyright Page
• Dedication
• Table of Contents
• Preface, Apologies and Acknowledgments
• Glossary of Common Symbols
• 1. The Thermodynamics of Mixing
• 2. A Practical Guide to Polymer Miscibility
• 3. The Nature of the Hydrogen Bond
• 4. Equilibrium Constants and the Stoichiometry of Hydrogen Bonding
• 5. Vibrational Spectroscopy and the Hydrogen Bond
• 6. Association Models and the Thermodynamics of Mixing Molecules with Strong Specific Interactions
• 7. The Calculation of Phase Diagrams of Strongly Interacting Polymers
• Index