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ORGANIC AND PHYSICAL CHEMISTRY OF POLYMERS ORGANIC AND PHYSICAL CHEMISTRY OF POLYMERS Yves Gnanou Michel Fontanille A JOHN WILEY & SONS, INC., PUBLICATION ORGANIC AND PHYSICAL CHEMISTRY OF POLYMERS by Yves Gnanou and Michel Fontanille Translated by Yves Gnanou and Michel Fontanille Copyright  2008 by John Wiley & Sons, Inc., from the original French translation Chimie et PhysicoChimie des Polym` res by Yves Gnanou and Michel Fontanille  Dunod, Paris 2002 All rights e reserved Published by John Wiley & Sons, Inc., Hoboken, New Jersey Published simultaneously in Canada No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, scanning, or otherwise, except as permitted under Section 107 or 108 of the 1976 United States Copyright Act, without either the prior written permission of the Publisher, or authorization through payment of the appropriate per-copy fee to the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, (978) 750-8400, fax (978) 750-4470, or on the web at www.copyright.com Requests to the Publisher for permission should be addressed to the Permissions Department, John Wiley & Sons, Inc., 111 River Street, Hoboken, NJ 07030, (201) 748-6011, fax (201) 748-6008, or online at http://www.wiley.com/go/permission Limit of Liability/Disclaimer of Warranty: While the publisher and author have used their best efforts in preparing this book, they make no representations or warranties with respect to the accuracy or completeness of the contents of this book and specifically disclaim any implied warranties of merchantability or fitness for a particular purpose No warranty may be created or extended by sales representatives or written sales materials The advice and strategies contained herein may not be suitable for your situation You should consult with a professional where appropriate Neither the publisher nor author shall be liable for any loss of profit or any other commercial damages, including but not limited to special, incidental, consequential, or other damages For general information on our other products and services or for technical support, please contact our Customer Care Department within the United States at (800) 762-2974, outside the United States at (317) 572-3993 or fax (317) 572-4002 Wiley also publishes its books in a variety of electronic formats Some content that appears in print may not be available in electronic formats For more information about Wiley products, visit our web site at www.wiley.com Library of Congress Cataloging-in-Publication Data: Gnanou, Yves Organic and physical chemistry of polymers / by Yves Gnanou and Michel Fontanille p cm Includes index ISBN 978-0-471-72543-5 (cloth) Polymers Polymers—Synthesis Chemistry, Physical and theoretical I Fontanille, M (Michel), 1936– II Title QD381.G55 2008 547 7—dc22 2007029090 Printed in the United States of America 10 CONTENTS Foreword vii Preface ix Introduction Cohesive Energies of Polymeric Systems 13 Molecular Structure of Polymers 19 Thermodynamics of Macromolecular Systems 49 Conformational Structures and Morphologies 89 Determination of Molar Masses and Study of Conformations and Morphologies by Physical Methods 147 Step-Growth Polymerizations 213 Chain Polymerizations 249 Reactivity and Chemical Modification of Polymers 357 10 Macromolecular Synthesis 377 11 Thermomechanical Properties of Polymers 401 12 Mechanical Properties of Polymers 427 v vi CONTENTS 13 Rheology, Formulation, and Polymer Processing Techniques 467 14 Natural and Artificial Polymers 493 15 Linear (monodimensional) Synthetic Polymers 513 16 Three-Dimensional Synthetic Polymers 583 Index 607 FOREWORD Polymers, commonly known as plastics, are perhaps the most important materials for society today They are employed in nearly every device The interior of every automobile is essentially entirely made of polymers; polymers are also used for body parts and for under-the-hood applications Progress in the aerospace industry has been aided by new light, strong nanocomposite polymeric materials Many construction materials (e.g., insulation, pipes) and essentially all adhesives, sealants, and coatings (paints) are made from polymers The computer chips used in our desktops, laptops, cell phones, Ipods, or Iphones are enabled by polymers used as photoresists in microlithographic processes Many biomedical applications require polymers for tissue or bone engineering, drug delivery, and also for needles, tubing, and containers for intravenous delivery of medications Some new applications call for smart or “intelligent” polymers that can respond to external stimuli and change shape and color to be used as artificial muscles or sensors Thus, it is not surprising that the annual production of polymers approaches 200 million tons and 50% of the chemists in USA, Japan or Western Europe work in one way or other with polymeric materials However, polymer awareness has not yet reached the appropriate level, for many of those chemists not fully comprehend nor they take advantage of concepts of free volume, glass transition, and microphase separation; consequently they not know how to precisely control polymer synthesis One may also argue that some polymer scientists not sufficiently appreciate most recent developments in organic and physical chemistry, although polymer science has a very interdisciplinary character and bridges synthetic chemistry with precise characterization techniques offered by the methodologies of physical chemistry Organic and Physical Chemistry of Polymers by Yves Gnanou and Michel Fontanille provides a unique approach to combine fundamentals of organic and physical chemistry and apply them to explain complex phenomena in polymer science The authors employ a very methodical way, straightforward for polymer science novices and at the same time, attractive for more experienced polymer scientists On reading this book, one can easily comprehend not only how to make conventional and new polymeric materials, but also how to characterize them and use them for classic and new advanced applications vii viii FOREWORD I read the book with a great interest, and I am convinced that this book will become an excellent polymer science textbook for senior undergraduate and graduate students Krzysztof Matyjaszewski J.C Warner University Professor of Natural Sciences Carnegie Mellon University Fall 2007, Pittsburgh, USA PREFACE Although the uses of polymers in miscellaneous applications are as old as humanity, polymer science began only in the 1920s, after Staudinger conclusively proved to sceptics the concept of long chain molecules consisting of atoms covalently linked one to another Then came the contributions of physicists: Kuhn first accounted for the flexibility of certain polymers and understood the role of entropy in the elasticity of rubber Flory subsequently explained most of the physical properties of polymers using very simple ideas, and Edwards found a striking analogy between the conformation of a polymer chain and the trajectory of a quantum mechanical particle The aim of this textbook is to justice to the interdisciplinary nature of polymer science and to break the traditional barriers between polymer chemistry and the physical chemistry and physics of polymers Through the description of the structures found in polymers and the reactions used to synthesize them, through the account of their dynamics and their energetics, are conveyed the basic concepts and the fundamental principles that lay the foundations of polymer science We tried to keep in view this primary emphasis throughout most of the book, and chose not to elaborate on applicative and functional aspects of polymers At the core of this book lie three main ideas: —the synthesis of polymer chains requires reactions exhibiting high selectivity, including regio-, chemo- and sometimes stereoselectivity Mother Nature also produces macromolecules that are useful for life (proteins, DNA, RNA) but with a much higher selectivity; —polymers represent a class of materials that are by essence ambivalent, exhibiting at the same time viscous and elastic behaviors Indeed, a polymer chain never behaves as a purely elastic material or as an ideal viscous liquid Depending upon the temperature and the polymer considered, the time scale of the stress applied, either the viscous or the elastic component dominates in its response; —an assembly of polymer chains can adopt a variety of structures and morphologies and self-organize in highly crystalline lamellae or exist as a totally disordered amorphous phase and intermediately as mesomorphic structures ix x PREFACE Polymers are thus materials with peculiar physical properties which are controlled by their methods of synthesis and their internal structure The first chapters (I to III) introduce the notions of configuration and conformation of polymers, their dimensionality, and how their multiple interactions contribute to their overall cohesion The three next chapters are concerned with physical chemistry, namely the thermodynamics of polymer solutions (IV), the structures typical of polymer assemblies (V), and the experimental methods used to characterize the size, the shape and the structures of polymers (VI) Four chapters (VII to IX) then follow that elaborate on the methods of synthesis and modification of polymers, and the engineering of complex architectures (X) Chapters XI to XIII subsequently describe the thermal transitions and relaxations of polymers, their mechanical properties and their rheology These thirteen chapters are rounded off by monographs (chapters XIV to XVI) of natural polymers and of some common monodimensional and tridimensional polymers Since the 1920s, polymer science has moved on at a dramatic rate Significant advances have been made in the synthesis and the applications of polymeric materials, paving the way for the award of the Nobel Prize in five instances to polymer scientists Staudinger in 1953, Ziegler and Natta in 1963, Flory in 1974, de Gennes in 1991, and more recently McDiarmid, Shirakawa and Heeger in 2000 indeed received this distinction Their contributions and the many developments witnessed in the area of specialty polymers have made necessary to write a book that provides the basics of polymer science and a bridge to an understanding of the huge primary literature now available This book is intended for students with no prior knowledge or special background in mathematics and physics; it can serve as a text for a senior-level undergraduate or a graduate-level course In spite of our efforts, some mistakes certainly remain; we would appreciate reports about these from readers Last but not least, we wish to mention our debt and express our gratitude to Professors Robert Pecora (Standford University), Marcel van Beylen (Leuven University) and colleagues from our University who read and checked most of the chapters We are also indebted to Professor K Matyjaszewski for accepting to write the foreword of this book Yves Gnanou Michel Fontanille Summer 2007, Bordeaux, France DISPERSITY OF MOLAR MASSES— AVERAGE MOLAR MASSES 39 Σ Ni i Mp log Mi Figure 3.8 Diagram representing the integral (cumulative) chain-length distribution of a polymer sample After defining the average molar masses in the next section (Section 3.4.2), the various methods of quantifying the molar-mass dispersity will be presented Remark Although it is not recommended by the competent committees, the terms polydispersity and monodispersity (as well as the corresponding adjectives: polydisperse and monodisperse, respectively) are commonly used instead of high dispersity and low dispersity For reasons of etymology, the terms mono- and polydispersity are to be rejected 3.4.2 Average Molar Masses They are defined using Ni and Mi , which are a number of moles of species (i ) and their molar mass, respectively The number-average molar mass (Mn ) is defined as the sum ( i ) of the molar masses (Mi ) of all the i families of species present in the system times their number fraction, that is, Ni / i Ni : Mn = Mi i Ni = i Ni i Ni Mi i Mi The bar over M symbolizes the averaging process, but the IUPAC Nomenclature Subcommittee recently suggested n as symbol of an average number, and it is easier to type Because Ni Mi denotes the total mass of each family of species (i), their sum i Ni Mi represents the total mass of the sample Thus Mn is equal to the total mass of the sample divided by the total number of moles of polymeric species present For the determination of M n , methods that count the number of molecules forming the sample are appropriate; these methods will be described in detail in Section 4.3 40 MOLECULAR STRUCTURE OF POLYMERS The mass-average molar mass (M w )∗ is defined as the sum i of the molar masses (Mi ) of the i families of species present in the sample times their mass fraction, that is, Ni Mi / i Ni Mi : Mw = Ni = i Ni Mi Mi i Ni Mi2 i Ni Mi i M w of a polymer is then the ratio of the second moment of the number distribution of molar masses to the corresponding first moment According to the latter definition, longer chains have a higher statistical weight than that of shorter chains The measurement of M w is based on phenomena whose intensity is proportional to the size of the particles to be measured (see Section 4.3) The Z-average molar mass (M z ) is seldom used to characterize synthetic polymers; it can be defined by a similar logic as above: Mz = Ni Mi3 i Ni Mi i The three preceding average molar masses result from logical definitions and are measured in g·mol−1 Viscometric-average molar masses (M v ) cannot be defined with the same rigor since they result from an empirical expression, the Mark–Houwink relation correlating the intrinsic viscosity (or limit index of viscosity) [η] to the molar mass M of a nondispersed (isomolecular) fraction of a polymer sample The way to establish the relation [η] = KM α is discussed in Section 4.3 In the case of a dispersed system, every family (i ) is characterized by its own intrinsic viscosity, [ηi ] = KMiα and for an assembly of macromolecules of a sample, [η] is equal to the sum of [ηi ] multiplied by their mass statistical weight: [η] = i [ηi ] · Ni Mi i Ni Mi ∗ This quantity, often miscalled weight-average molecular weight, reflects the chemist’s legendary carelessness to distinguish between mass and weight DISPERSITY OF MOLAR MASSES— AVERAGE MOLAR MASSES 41 If one sets [η] is equal to α [η] = KM v = K i Ni Mi(1+α) i Ni Mi then the definition of M v can be deduced: Mv = i Ni Mi(1+α) i Ni Mi 1/α For α = 1, which is generally the case, M v cannot be expressed in g·mol−1 because it is a relative average molar mass Remarks (a) Generally, 0.50 < α < 0.90 and thus M n < M v < M w < M z (b) The number-average, the mass-average, and so on, degrees of polymerization (X) can be deduced from the corresponding average molar masses For example, Xn = M n /m0 , in which m0 is the molar mass of the monomeric unit There are two ways of characterizing the dispersity The first one consists in calculating the deviation ε from the number-average molar mass for every family of polymer chains exhibiting a molar mass Mi , that is ε = |Mi − M n | The number-standard deviation σ is the average of the square of deviations i Ni (Mi − M n )2 i Ni i Ni (Mi2 − M n − 2Mi M n ) i Ni σ2 = σ2 = σ2 = i σ2 = i Ni Mi2 − 2M n i Ni i Ni Mi + Mn i Ni Ni Mi2 2 − 2M n = M w M n − M n i Ni which is generally used in the form σ = [M n (M w − M n )]1/2 42 MOLECULAR STRUCTURE OF POLYMERS Although very logical by its definition, the standard deviation σ is rarely used to measure the dispersity of polymer samples The dispersity index is generally used: DM = M w /M n (often called Ip ) The dispersity index can vary from unity for perfectly nondispersed samples (“isometric” systems) up to several tens for samples characterized by a strong dispersion in the size or molar mass of the constituting macromolecules (highly dispersed systems) 3.5 POLYMER NETWORKS 3.5.1 Description of the Networks Polymer chains can form a three-dimensional network when each one of them is connected to more than two of its homologs through monomeric units whose valence is equal to or higher than These units are also called junction points In contrast to crystal lattices, the term polymer network does not imply the idea of long-range order; it is used to indicate that the macromolecular structure created in this manner ranges over a three-dimensional space until it occupies the volume of its container—that is, the dimension of a macroscopic object One can then speak of an infinite network whose ends are connected one to another through a single macromolecule of macroscopic dimensions Such networks are insoluble and nonfusible; they can, at most, swell in a good solvent The mass of a mole of such a macroscopic molecule is obviously “infinitely large,” but by no means infinite, which is often stated as a figure of speech If n grams of monomer form one single macromolecule after cross-linking, the molar mass of the latter would be M = n × 6.02 × 1023 g·mol−1 (according to Avogadro), which is an extremely large value of molar mass but certainly not an infinite one Just before cross-linking, an abrupt transition occurs during which the reaction medium passes from the state of a solution to that of an elastic gel, undergoing a sol–gel transition (which is also called gel point): at this stage, which corresponds to a specific conversion of the reactive functional groups (pgel ), the mass-average molar mass of the polymer as well as its viscosity diverge abruptly toward infinity Before the gel point the chains present in the reaction medium have finite size; that is, they are still soluble and fusible Hence, the sol–gel transition is a critical phenomenon, which corresponds to a transition of connectivity that can be analyzed within the framework of the theory of percolation The first condition to obtain an infinite three-dimensional macromolecule and consequently a network is to introduce in the polymerizing system a branching agent whose valence is at least equal to three Its role is to allow a nonlinear growth of the chains However, this condition is not sufficient to generate an infinite macromolecule We note that the probability for two branching units to be connected by the sequence of units POLYMER NETWORKS A + A A + B 43 A [B-BA-A]xB-BA B A is pa [pb (1 − ρ)pa ]x · pb ρ where ρ is the proportion of a function carried by the branching unit, and pa and pb are the probabilities that A and B, respectively, have reacted Then the general case of a branching unit being linked to another one by an elastic chain of any size can be expressed by the following probability (α): ∞ α= [pb (1 − ρ)pa ]x · pa pb x=0 which can also be written α= pa pb ρ − [pa pb (1 − ρ)] If all A functions belong to the branching units and for stoichiometric conditions (ρ = 1, pa = pb = p), the previous equation reduces to α = p2 The critical value (αc ) at which the system grows toward an infinite network can be deduced as follows: for a system with trivalent branching units (v = 3), each elastic chain reacting with one of these units has the possibility to be succeeded by two more chains For the branching of successive chains to continue indefinitely, α should be higher than 1/2; under these conditions, only the expected number of elastic chains in succeeding generations can be greater than the number of chains in the preceding ones In other words, n chains can give rise to 2n α chains with 2n α > n only for α > 1/2 For a system whose branching units is of valence v, gelation will occur for αc = v−1 1/2 and at the critical conversion pc = v−1 Polymer networks can be obtained in various ways: • By step-growth polymerizations between monomers (or precursors) carrying antagonistic functional groups and having an average valence higher than two • By chain copolymerization of a divalent vinyl or related monomer with a multi-unsaturated comonomer • By random cross-linking of chains carrying reactive functional groups (vulcanization, etc.) 44 MOLECULAR STRUCTURE OF POLYMERS In each of the cases, cross-linking occurs only under certain conditions that depend on the proportion and valence of the agent responsible for branching For a threedimensional step-growth polymerization involving antagonistic molecules A and B with the valence distribution Av1 + Av2 + · · · + Avk + Bg1 + Bg2 + · · · + Bgj the gel point (pgel ) occurs at the following critical conversion: pgel = (ve − 1)(ge − 1) where v e and ge denote the average valence of monomers A and B, respectively fe = fi af i i and ge = gj bgj j afi and bgj being the molar fractions of the various species present in the reaction medium and v i and gj their valence If the ratio 1/(v e − 1)(ge − 1) is higher than 1, the system cannot generate an infinite macromolecule For chain copolymerization reactions involving a tetravalent monomer (e.g., a bis-unsaturated monomer [CH2 = CH(A) + CH2 = CH(A) − (A)HC = CH2]), the conversion at the gel point can be written as pgel = 1−q (v − 2)aq where a is the molar fraction of the tetravalent monomer and q is the probability of chain growth [q = Rp /(Rp + Rt ), Rp is the rate of propagation, and Rt is the rate of termination] In the case of chain vulcanization, where the latter react through side groups carried by the monomeric units or through unsaturations present in the backbone, the gel point (pgel ) occurs for pgel = Xn−1 ≈ Xn POLYMER NETWORKS A 45 A A A A A A A A A A A A A A A Beyond the gel point the cross-linked fraction increases at the expense of the sol fraction as the reaction proceeds (see Figure 3.9) In addition to the chemical nature of its constituting monomeric units, a polymer network can be defined through two essential parameters, namely, the number (ν) of its elastic chains and the number (µ) of its junction points or cross-links According to their definition, elastic chains are connected by their two ends to junction points from which emanate at least two other elastic chains However, a network not only is a collection of ν chains connecting µ junction points, but also contains also a fraction of loose ends, or dangling chains, loops, and entanglements A model network is defined as a defect-free one where all cross-links have the same functionality and all elastic chains are of the same size (identical number of monomeric units) M(g.mol−1) WS W We 0.8 50000 Mw WP 40000 0.6 30000 0.4 20000 0.2 Mn 10000 0 0.2 0.4 0.8 0.6 p pgel Figure 3.9 Variation of average molar masses and of mass fractions related to extractable chains (W s ), elastic chains (W e ), and dangling chains (W p ), versus conversion (p) for a cross-linking step-growth reaction 46 MOLECULAR STRUCTURE OF POLYMERS trivalent cross-links o tetravalent cross-link o elastic chain o o o o chain-end o o loop dangling chain o Figure 3.10 Elastic chains, cross-links, and defects in a polymeric network From a practical and experimental point of view, one resorts to the notions of cross-links, degree of cross-linking, or molar mass (Mc ) of the elastic chains (Figure 3.10) The density of cross-linking (γ) corresponds to the number of moles of cross-links per unit of volume (or per unit of mass), whereas µ represents their molar fraction with respect to the monomeric units of the chains µ, γ, and Mc are interrelated to one another by ρ = γ/N0 Mc = M0 N0 /γ where N0 is the number of moles of the monomeric units per unit of volume and M0 is their molar mass Practically, the chemical properties of polymer networks depend on the chemical nature of elastic chains and on the type of cross-links The mechanical properties are, in contrast, essentially governed by the cross-linking density and the mobility of the elastic chains Polymer networks may thus be soft or hard and exhibit rubbery or brittle behavior 3.5.2 Characterization of the Networks Due to their infinitely large size, which confers insolubility and nonfusibility to them, polymer networks cannot be characterized by the traditional methods used for linear and soluble polymers It is mainly through the study of their mechanical properties (experiments of traction and compression) that one can attain a better knowledge of the structure of networks As for other solid materials, polymer networks behave like Hookean bodies (within the limit of moderate deformations); that is the deformation is directly proportional to the applied stress At this stage, it is necessary to make a distinction between rigid networks, made up of crystallized POLYMER NETWORKS 47 Table 3.1 Comparison between respective characteristics of flexible and rigid polymeric networks Characteristic Elastic modulus Reversible strain Variation of the temperature while stretching Variation of the length while heating Rigid Networks (Elasticity from Enthalpic Origin) Elastomeric Networks (Elasticity from Entropic Origin) High (2–3 GPa) Low (0.1%) Decreasing low (1 × 10−3 GPa) High (100% and more) Increasing Stretching Shrinking chains or whose glass transition temperature is higher than the service temperature (Ts ), and elastomeric networks whose chains are characterized by a glass transition temperature lower than Ts The marked differences that characterize these two types of networks are due to the nature of their elasticity; the latter has (a) an enthalpic origin in networks made up of rigid chains and (b) an entropic one for those that consist of flexible chains (Table 3.1) The mechanical properties of elastomeric networks are described within the theory of rubber elasticity, which accounts for the behavior of a network—in fact, its elastic modulus—as a function of its molecular parameters (number of elastic chains and cross-links) Valuable information regarding the structure of networks can also be obtained from swelling measurements The swelling ratio is directly related to the number of elastic chains by the Flory–Rehner equation (see paragraph on rubber elasticity) Networks that swell in a solvent are called gels; the elasticity of gels—even if they result from rigid networks in a dry state—can be analyzed using the theory of rubber elasticity From a simple experiment such as the extraction of the soluble fraction (ωs ), one can obtain useful information regarding the conversion ratio (p) or the fraction of dangling chains thanks to relations that connect ωs to these two structural parameters In addition, more sophisticated techniques have been developed for a more precise analysis of the structure and the behavior of networks: dynamic light scattering proved to be useful to determine the fluctuation of density in networks, and neutron scattering allows one to understand the mechanism of deformation of chains when submitting the network to a macroscopic deformation 3.5.3 Physical Networks In addition to networks comprising covalent cross-links (irreversible by nature), there are also transient networks known as “physical networks.” The mechanism of formation of these physical networks is somehow similar to the chain vulcanization previously described, but in this case the cross-links formed have a reversible character Chain “bridging” can occur through the establishment of either weak 48 MOLECULAR STRUCTURE OF POLYMERS molecular interactions (van der Waals bonds) or moderate interactions (hydrogen bonds) or by self-assembly of rigid polymer blocks in ABA-type block copolymers Such physical networks are schematized in Figure 3.11 Nature also provides a number of examples of physical networks The most common example is that of aqueous gelatin solutions whose cross-linking corresponds to a transition: from coil to helix of polypeptide chains and to their partial renaturation into native collagen In such networks, some polypeptide chains self-assemble and adopt triple-helix organization while others remain as statistical coils Figure 3.11 Schematic representation of a physical network LITERATURE H G Barth and J W Mays, Modern Methods of Polymer Characterization, Wiley, New York, 1991 F A Bovey and L W Jelinski, Chain Structure and Conformation of Macromolecules, Academic Press, New York, 1982 T R Crompton, Analysis of Polymers: An Introduction, Pergamon Press, Oxford, 1989 R N Ibbnett (Ed.), NMR Spectroscopy of Polymers, Chapman & Hall, London, 1993 K Matsuzaki, U Toshiyuki, and T Asakura, NMR Spectroscopy and Stereoregularity of Polymers, Karger, Basel, 1996 THERMODYNAMICS OF MACROMOLECULAR SYSTEMS 4.1 GENERAL CHARACTERISTICS OF POLYMER SOLUTIONS By definition, a solution contains more than one component A solution can be gaseous, liquid, or solid The term macromolecular (or polymer) solution will be used to indicate a mixture of a polymer with a small-molecule solvent and polymer blend when solvent and solute are both polymers In this chapter the thermodynamics of polymer solutions and of solid polymer blends will thus be discussed separately The study of the behavior of polymer solutions is important for several reasons One of the most important is that most methods used to characterize polymers are usually applicable to liquid solutions An exception to this is neutron scattering, which is often applied to polymers that are condensed as liquids or solids Polymers are best characterized in solutions with small-molecule solvents Their molar mass and “statistical” molecular dimensions can be determined using osmometry, size exclusion chromatography, viscometry, and/or light scattering—that is, all of which methods require dilute polymer solutions From these methods, information about the polymer radius of gyration, the distribution of the molar masses, their average value, and so on, can be obtained In addition to their importance for polymer characterization, polymer solutions have wide applications (paints, varnishes, oils for engine lubrication, etc.) that take advantage of their particular properties (high viscosity, etc.) The true nature of polymers was initially revealed from investigations of their solution properties These studies established that macromolecular chains are made of repetitive units connected by covalent bonds Organic and Physical Chemistry of Polymers, by Yves Gnanou and Michel Fontanille Copyright  2008 John Wiley & Sons, Inc 49 50 THERMODYNAMICS OF MACROMOLECULAR SYSTEMS Like simple organic molecules, polymers can be dispersed or dissolved in a solvent but their properties differ completely from those exhibited by common solutions of simple molecules For the same range of concentration, the viscosity of a solution of macromolecules is much higher (due to the chain entanglement) than that of a solution of simple homologous molecules In addition, marked deviations in their colligative properties (i.e., properties related to the number of species present in the system) are observed compared to the ideal behavior or even to the behavior of real systems consisting of small molecules This experimental fact—the unusual deviations from Raoult’s law—and also the stability of macromolecular solutions led to the final demise of the theory of molecular aggregates glued together by weak interactions (the micellar theory as opposed to the macromolecular theory) Mixing two chemical compounds, whether simple molecules or macromolecular chains, leads to a change in the entropy (S ), the enthalpy (H ) and even the volume of the solution For an isothermic mixing, these changes cause a variation of the free energy, which can be expressed as Gmix = Hmix − T Smix where T is the absolute temperature The miscibility of two compounds is thermodynamically favored when Gmix < Solutions can be classified into five main categories—ideal, athermic, regular, irregular, and “theta”: • • • • • An ideal solution is characterized by an enthalpy of mixing equal to zero and an entropy of mixing equal to the conformational entropy (or combinatorial entropy); An athermal solution is also characterized by an enthalpy of mixing equal to zero but its entropy of mixing is higher than the conformational entropy It thus exhibits an excess entropy; A regular solution is characterized by a nonzero enthalpy of mixing and an entropy of mixing equal to the conformational entropy; An irregular (or real) solution corresponds to a solution whose enthalpy of mixing does not equal zero and whose entropy of mixing comprises an excess entropy in addition to the conformational term; A polymer solution is in “theta” conditions when the enthalpy of mixing compensates the excess entropy of mixing at a given temperature Thus, at this temperature, the solution can be regarded as ideal Remark The combinatorial entropy related to the multiplicity of conformational arrangements, which can be adopted by a macromolecule, is sometimes erroneously called configurational , because the configuration of a polymer refers to different levels of structure FLORY–HUGGINS THEORY 51 4.2 FLORY–HUGGINS THEORY Statistical thermodynamics can be used to calculate both the enthalpic and the entropic contributions to the free energy of mixing by means of statistical calculations Attempts were first made to apply a theory appropriate to simple regular solutions, the Hildebrand model, to the case of macromolecular solutions This model does not adequately account for the specific behavior of polymer solutions primarily because the entropy of mixing in a polymer solution is strongly affected by the connectivity of the polymer—that is, by the existence of covalent bonds between the repetitive units This prompted Flory and Huggins to propose a model specific to polymers, which is an extension of the lattice fluid theory but better considers the specificity of polymers; it compares the free energy of polymer–solvent systems before and after mixing 4.2.1 Entropy of Mixing In the Flory–Huggins theory, also referred to as the Flory–Huggins mean-field theory, solutions are depicted as three-dimensional crystalline lattices comprising nt cells (Figure 4.1) Each solvent molecule occupies one cell of this lattice while the macromolecules occupy neighboring cells in a number equal to that of the degree of polymerization (X ) It is assumed that each repetitive unit occupies one cell of the lattice, and thus a volume identical to that of a solvent molecule Each molecule of solvent or solute possesses a number of nearest neighbors (z )—called the coordination number —which is also the number of possible interactions for the resident of the cell Moreover, the volume of such cells is considered to remain unchanged after mixing of the two components Figure 4.1 Representation of a polymer solution by a lattice Flory– Huggins model 52 THERMODYNAMICS OF MACROMOLECULAR SYSTEMS Using this model, it is possible to calculate the conformational entropy ( Sco ) resulting from the mixing of n1 molecules of solvent with n2 macromolecules having the same degree of polymerization X , considering that the total number of elements (nt ) is equal to (n1 + n2 X ) As the solution is relatively dilute and as the solvent molecules and the macromolecules are randomly distributed, it is assumed that both the conformational entropy (which reflects all possible positional combinations of solvent molecules and of the macromolecules) and the mixing entropy are identical The calculation of the entropy of mixing of a macromolecular solution begins with the determination of the number of ways of placing n1 solvent molecules and n2 macromolecules in a total of nt spaces In a system where j macromolecules have already been placed, let us introduce the (j + 1)th macromolecule At this stage, the number of vacant cells is equal to nt − jX , a number that also corresponds to the number of ways (number of combinations) of placing the first monomer unit of the (j + 1)th macromolecule The number of ways of placing the next monomer unit is equal to the product of the coordination number (z ) of the lattice times the fraction (1 − fj ) of the vacant cells—that is, z (1 − fj ) Remark Equations are numbered in the following part only when necessary Each chapter has its own independent numbering The third monomer unit and all subsequent units will find one of the adjacent cells occupied by the preceding unit so that only (z − 1)(1 − fj ) vacant cells will remain for them Hence, the number of ways of placing this (j + 1)th macromolecule can be expressed as νj +1 = (nt − j X)z(z − 1)X−2 (1 − fj )X−1 (4.1) The number of ways of arranging all the macromolecules of the solution which correspond to the number of possible combinations can therefore be written as P2 = ν1 ν2 νn = n2 ! n2 ! n2 −1 νj +1 = j =0 n2 ! n2 νj (4.2) j =1 The presence of n2 ! in this expression as denominator is essential for the probability P2 to be independent of the order of introduction of the n2 macromolecules Now it remains to place the n1 molecules of solvent: the solvent molecules being indistinguishable, their number of combinations is equal to Thus, expression (4.2) represents the number of combinations of the solution The fraction of vacant cells (1 − fi ) is expressed as (1 − fj ) = nt − j X nt (4.3) FLORY–HUGGINS THEORY 53 and the number of combinations is given by P2 = n2 ! n2 z(z − 1)X−2 [nt − X(j − 1)]X (4.4) nX−1 t j =1 considering (4.1), (4.2) and (4.3) After regrouping the terms independent of j , this leads to P2 = n2 zn2 (z − 1)(X−2)n2 n2 !n(X−1)n2 t [nt − X(j − 1)]X (4.5) j =1 Such a product can be written in terms of factorials: n2 n2 [nt − X(j − 1)]X = Xn2 X j =1 j =1 nt −j +1 X X ∼ X n2 X (nt /X)! (n1 /X)! X Now equation (4.5) becomes P2 = zn2 (z − 1)(X−2)n2 Xn2 X n2 !n(X−1)n2 t (nt /X)! (n1 /X)! X (4.6) Using Stirling’s approximation (ln a! ∼ a ln a − a) to remove factorials, (4.6) can be formulated as follows: ln P2 = −n2 ln Xn2 n1 − n1 ln nt nt + n2 [ln X − X + + ln z + (X − 2) ln(z − 1)] (4.7) The conformational entropy of the solution can then be derived from the Boltzmann equation (S = k ln P2 ): Sco = −k n2 ln Xn2 n1 + n1 ln nt nt + kn2 [ln X − X + + ln z + (X − 2) ln(z − 1)] (4.8) where k is the Boltzmann constant The conformational entropies of pure solvent and solute can be easily calculated from (4.8), by setting n2 = leading to (Sco )1 = (4.9) .. .ORGANIC AND PHYSICAL CHEMISTRY OF POLYMERS ORGANIC AND PHYSICAL CHEMISTRY OF POLYMERS Yves Gnanou Michel Fontanille A JOHN WILEY & SONS, INC., PUBLICATION ORGANIC AND PHYSICAL CHEMISTRY OF POLYMERS. .. methodologies of physical chemistry Organic and Physical Chemistry of Polymers by Yves Gnanou and Michel Fontanille provides a unique approach to combine fundamentals of organic and physical chemistry and. .. from F ) (cm3 ·mol? ?1 ) (J1/2 ·cm3/2 ) from δ) Formula Polyethylene 33.0 Poly(vinyl chloride) 17 ,800 18 ,200 18 .2 32,000 36,000 21. 3 19 ,300 17 ,800 60.7 n 16 .3 17 .1 17,500 17 ,600 14 3.2 20.5 62,000