Advanced Polymer Chemistry

Advanced Polymer Chemistry

Advanced Polymer Chemistry Problem Solving Guide

Manas Chanda Indian Institute of Science Bangalore, India

MARCEL

MARCEL DEKKER, INC. DEKKER

NEW YORK ¯ BASEL

ISBN: 0-8247-0257-3 This bookis printed on acid-free paper. Headquarters Marcel Dekker,Inc. 270 Madison Avenue, NewYork, NY10016 tel: 212-696-9000;fax: 212-685-4540 Eastern HemisphereDistribution Marcel Dekker AG Hutgasse 4, Postfach 812, CH-4001Basel, Switzerland tel: 41-61-261-8482;fax: 41-61-261-8896 World Wide Web http://www.dekker.com Thepublisher oilers discounts on this bookwhenordered in bulk quantities. For moreinformation, write to Special Sales/Professional Marketingat the headquartersaddress above. Copyright © 2000 by Marcel Dekker,Inc. All Rights Reserved. Neither this book nor any part may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying,microfilming, and recording, or by any information storage and retrieval system, without permission in writing from the publisher. Currentprinting (last digit): l0 9 8 7 6 5 4 3 2 l PRINTED IN THE UNITED STATES OF AMERICA

Preface

Polymers are among the most commonof all materials. A proper understanding and appreciation of the principles governing the nature of polymers, their behavior in solid and liquid states, and the methods of building polymeric structures from simple molecules are often required for a study of chemical, physical, and biological sciences. It is my opinion that these principles taught in a course of polymer chemistry can be learnt properly only through their applications to solving problems and that the existing texts which are highly deficient in this respect do not prepare a student sufficiently in the diverse applications of the fundamentalprinciples of polymer chemistry. This book has been written to fill this gap. It may be described as a combination of textbook and study guide harmoniously blended into one. Polymerchemistry is not just a branch of organic chemistry. It has its own methods of synthesis and its owntechniques of characterization. However, all fundamental and applied efforts of polymer synthesis and manufacturing processes can be eventually condensed in a few guiding principles which represent, so to speak, the essence of our present understanding and knowledge. Such principles are, of course, mainly qualitative generalizations and must be supplemented for proper appreciation by quantitative considerations and enumerations at every step. The aim of this book is to present these general principles of polymer synthes.is, with emphasis on the mechanisms and kinetic aspects of these reactions, elaborated with the help of numerous worked out problems alongside the text and addressed repeatedly through many unsolved problems appended as exercises. The problems both

iv

Pteface

in the text and the exercises have been formulated to require, in most cases, numerical rather than essay-type answers, since numerical calculations call for more intimate involvement with theory and demandan awareness of the dimensions and units to be used. The instructor also mayfind an incidental advantage in that answers to such problems can be graded relatively rapidly with the aid of the SolutionsManualthat is available. Before discussing the detailed chemistry, kinetics, and mechanismsof the various pathways of polymer synthesis, it is necessary to introduce some of the fundamental concepts of polymer science in order to provide essential background to such a development. Weneed to know what a polymer is and howit is namedand classified. It is also necessary to obtain an appreci~ition of the molecular size and shape of polymer molecules, the molar mass characteristics, the important transition temperatures of polymers, and their distinctive behavior both in solid state and in solution. These concerns are addressed in the first four chapters of the book while the remaining six chapters deal with the important categories of polymerization processes and their mechanismsand kinetic aspects. Throughout this journey the narrative in the text is illuminated with thoughtfully worked out examples which not only complement but also supplement, where necessary, the theoretical development in the text. The initial four chapters of the book concern several important aspects of polymer science which are relevant to a course in polymer chemistry. Following Chapter 1, which is a general introduction aimed at giving the reader an appreciation for the language, applications, and versatility of synthetic polymers, Chapter 2 is devoted to polymer characterization dealing with the size and shape of a polymer chain, polymer isomerism, polymer conformation, and thermal transitions in polymers. Since most methods of molecular characterization involve analysis of polymers in dilute solution, it is advantageous to introduce the relevant theories for polymers in solution before considering the individual methods of determination of molecular weight. Chapter 3 therefore deals with thermodynamics of polymer solutions and also considers the swelling and solubility behavior of polymers. Polymer molecular weight distributions and averages, a centrally important topic of polymer science, seem to be less comprehended, though there is nothing particularly difficult in the topic. This may be because many textbooks only present basic equations for molecular weight averages with. no explanation of their origin and significance. ’Chapter 4 therefore focuses on the fundamentals of molecular weight statistics and the measurementof molecular weight averages. The subsequent six chapters are devoted entirely to five types of polymerization reactions--step, chain, ionic, coordination, and ring-opening--and to copolymerization which may involve any of these mechanisms. Polymeriza-

Pre[ace

v

tion reactions are characterized as to their mechanismsand kinetic features, with emphasis placed on understanding the reaction parameters which are important in controlling polymerization rates, polymer molecular weight, and structural features, such as branching and cross-linking. Numerous worked out problems involving numerical calculations are given in each chapter and these are intended to illustrate and expand the text material. The versatility of polymerizationresides not only in the different types of polymerization reactions and types of reactants that can be polymerized, but also in variations allowed by step-growth synthesis, copolymerization, and stereospecific polymerization. Chain polymerization is the most important kind of copolymerization process and is considered separately in Chapter 7, while Chapter 9 describes the stereochemistry of polymerization with emphasis on the synthesis of polymers with stereoregular structures by the appropriate choice of polymerization conditions, including the more recent metallocene-based Ziegler-Natta systems. Synthetic approaches to starburst and hyperbranched polymers which promise to open up new applications in the future are considered in an earlier chapter dealing with step-growth polymerization. WhileSI units are being increasingly used in all branches of science, nonSI units like the older cgs systemare still in common use. This is particularly true of polymer science. Manyactive practitioners have developed intuitive understanding of the meanings and magnitudes of certain quantities in nonSI units, and it could be a needless annoyance to change them abruptly. In this book, both SI and non-SI units have therefore been used. However, in most places where non-SI units have been used, ,equivalent values are given in SI terms. A suitable conversion table is also provided as an appendix. Anactive, developing technology flourishes in synthetic polymersbecause of the great commercial importance of these materials. Synthesized polymers are utilized increasingly in our daily life, and manifoldindustrial applications have contributed to their phenomenal growth and expansion. Following this gigantic industrial development, which requires polymer chemists and specialists in polymers, manyuniversities throughout the world have set up teaching programs in polymer chemistry, science, and engineering. Their students are drawn from various disciplines in science and engineering. The present book is designed primarily for both undergraduate and graduate students and it will serve specially as a classroom text for a one-year course in polymer chemistry. Polymerindustry is the single largest field of employmentfor students ’of both science and engineering. However,most workers entering the field have little backgroundin polymer chemistry and are forced to educate themselves in its basic principles. This book with its easy style and many illustrative, worked-out problems will be useful to them as a self-contained

vi

Preface

text that guides a beginner in the subject to a fairly advanced level of proficiency. The book originated from a course in polymers that I taught during my sabbatical year (1985-1986) at the University of Waterloo, Canada, where I have also been a summer-termvisiting faculty memberover the last two decades. Students in the polymer course came from both chemistry and chemical engineering disciplines. The materiai has been tested since then and improved year after year to its present state as the course has been offered every year to a mixedclass of students from various disciplines, including chemistry, chemical engineering, metallurgy, and aerospace engineering at the Indian Institute of Science, Bangalore. A basic knowledge of mathematics, chemistry, and physics is assumed although it has been written to be, as far as possible, self-contained with most equations fully derived and any assumptions stated. In an undertaking of this sort it is impossible to acknowledgeall of one’s debts. Nonetheless, myspecial thanks are due to Professors K. E O’Driscoll, G. L. Rempel, and Alfred Rudin, all of the University of Waterloo, and to the late Professor K. Kishore of the Indian Institute of Science. Thanksalso go to all those students at Waterloo and Bangalore who endured this course before and during the writing of this book. Discussions with the students at these two reowned institutions, both in the classroom and outside, greatly influenced and shaped the course of writing and the contents of the book. I am indeed very much indebted to them. Finally, I wish to thank my wife Mridula and daughter Amrita for their great forbearance, understanding and graceful sacrifice, and words would be inadequate to measure their cooperation in shaping this volume. MANAS CHANDA

Contents

Preface 1

iii

Introductory Concepts and Definitions

Basic Definitions Polymer Monomer Resin and Plastic Molecular Weight and Molar Mass End Groups Degree of Polymerization Copolymers Polymerization and Functionality Polymerization Processes Addition or Chain Polymerization Step Polymerization Molecular Architecture Polymers: Thermoplastics and Thermosets Polymers: Plastics, Fibers, and Elastomers Why are Polymers Useful? Stress-Strain Behavior Polymer Nomenclature Common Nomenclature Bibliography Exercises

1 1 2 5 5 6 6 7 9 11 11 15 18 27 27 -29 35 37 37 40 41

viii 2

Conten~s Chain Dimensions, Structures, and Transitional Phenomena

Introduction Polymer Chains: St~iuctures and Dimensions Conformational Changes Polymer Conformations in the Bulk State Polymer Conformations in Crystals Polymer Size in the AmorphousState Constitutional and Configurational Isomerism Constitutional Isomerism Configurational Isomerism Crystallinity in Polymers Determination of Crystallinity Morphologyof Crystalline Polymers Thermal Transitions in Polymers

43

Secondary Glass Transition First- and Second-Order Transitions Regions of Viscoelastic Behavior Factors Affecting Tg Factors Affecting Tm Relation Between Tm oand T Theoretical Treatment of Glass Transition The Free-Volume Theory The WLFEquation Effect of Molecular Weight on T o Effect of Branching on T9 Effect of Cross-Linking on T o Effect of Diluents on T o Effect of Copolymerization on Tg Relaxation Processes in AmorphousPolymeric State The Reptation Model References Exercises Appendix 2.1 End-to-End Distance of a Freely Jointed Chain

43 44 44 53 53 58 67 67 72 83 83 86 89 89 91 91 93 98 101 102 103 103 107 112 114 115 116 117 121 123 126 128 133

3

139

:to and7’m

Polymers in Solution

Introduction Thermodynamics of Liquid Mixtures Partial Molar Property Low-Molecular-Weight Mixtures: van Laar Model

139 140 140 145

Contents

Lx

Polymer-Solvent Mixtures: Flory-Huggins Model Phase Equilibria in Poor Solvents Upper and Lower Critical Solution Temperatures Solubility Behavior of Polymers Frictional Properties of Polymer Molecules in Dilute Solution Viscosity of Dilute Polymer Solutions References Exercises Appendix 3.1 Regular Solutions: Solubility Parameter

150 188 197 198

4

229

Polymer Molecular

Weights

210 211 219 220 227

Introduction Molecular Weight Averages Arithmetic Mean Molecular Weights in Terms of Moments Arithmetic Mean as a Ratio of Moments Extension to Other Molecular Weight Averages Molecular Weight Determination End-Group Analysis Colligative Property Measurement Light-Scattering Method Dilute Solution Viscometry Gel Permeation Chromatography References Exercises

229 230 230 235 237 237 239 240 242 262 285 295 307 307

5

313

Condensation

(Step-Growth)

Polymerization

Introduction Reactivity of Functional Groups Rates of Polycondensation Reactions Uncatalyzed (Self-Catalyzed) Polyesterification Catalyzed Polyesterification Number-Average Degree of Polymerization Equilibrium Considerations Closed System Open Driven System Reversible Polymerization Kinetics Molecular Weight of Step-G, rowth Polymer Control of Molecular Weight

313 315 318 320 323 330 332 332 334 336 337 338

x

Confenfs

Molecular Weight Distribution (MWD)in Linear Polymerization Breadth of MWD Multichain or Nonlinear Step Polymerization Branching Cross-Linking Recursive Approach for Average Properties Linear Step-Growth Polymerization Nonlinear Step-Growth Polymerization Dendritic Polymers Polycondensation of AzB Monomers Synthetic Approaches Applications Concluding Remarks References Exercises Appendix 5.1 Summations Often Needed in Calculation of Molecular Distribution of Polymers

347 351 366 366 369 394 395 399 416 418 420 421 423 425 427

6

435

Radical Chain Polymerization

Introduction Overall Scheme of Radical Chain Polymerization Rate Expression for Radical Chain Polymerization Integrated Rate of Polymerization Expression A Note on Termination Rate Constant Experimental Determination of _Rqo: Dilatometry Methodsof Initiation Thermal Decomposition of Initiators RedoxInitiation Photochemical Initiation Initiation by Ionizing Radiation Pure Thermal Initiation Dead-End Polymerization Determination of Absolute Rate Constants Nonsteady-State Kinetics Chain Length and Degree of Polymerization Kinetic Chain Length Chain Transfer Degree of Polymerization Determination of Chain Transfer Constants Allylic Transfer

433

435 436 439 443 447 447 452 453 458 462 469 475 477 480 481 488 488 493 496 499 5O9

Con~ents

x]

Deviations from Ideal Kinetics Primary Radical Termination Initiator-Monomer Complex Formation Degradative Chain Transfer Autoacceleration Inhibition and Retardation Kinetics of Retarded Polymerization Energetic Characteristics Rate of Polymerization Degree of Polymerization Polymerization-Depolymerization Equilibrium Molecular Weight Distribution Low-Conversion Polymerization High-Conversion Polymerization Polymerization Processes Bulk Polymerization Solution Polymerization Suspension Polymerization Emulsion Polymerization References Exercises

510 511 512 513 518 522 525 527 527 531 532 542 542 552 553 553 554 554 556 570 573

7

579

Chain Copolymerization

Introduction Binary Copolymer Equation Range of Applicability of Copolymer Equation Types of Copolymerization Alternating Copolymerization Ideal Copolymerization Random-Alternating Copolymerization Block Copolymerization Instantaneous Compositions of Feed and Copolymer Integrated Binary Copolymer Equation Evaluation of MonomerReactivity Ratios Plot of rl Versus Plot of F1 Versus fl Direct Curve Fitting Monomerand Radical Reactivities Resonance Effects Steric Effects Polar Effects The Q -- e Scheme

579 581 586 587 587 587 589 590 590 594 600 601 603 605 607 607 610 610 612

xii

Confen~s

Sequence Length Distribution Rate of Free-Radical Copolymerization Multicomponent Copolymerization Deviations from Terminal Copolymerization Model Penultimate Effect Copolymerization and Cross-Linking Vinyl and Divinyl Monomersof Equal Reactivity Vinyl and Divinyl Monomersof Different Reactivity One Group of Divinyl MonomerHaving Lower Reactivity Block and Graft Copolymerization Block Copolymerization Graft Copolymerization References Exercises

615 620 625 630 631 633 634 638 640 641 641 644 647 648

8

653

Ionic Chain Polymerization

Introduction Ionic Polymerizability of Monomers Anionic Polymerization Anionic Initiation Termination Reactions Polymerization Kinetics Copolymerization Reactivity Gro.ups Block Copolymers Cationic Polymerization Cationic Initiation Propagation of Cationic Chain Chain Transfer and Termination Kinetics Molecular Weight Distribution Cationic Copolymerization References Exercises

653 656 659 659 665 669 695 696 699 704 706 711 713 718 732 735 735 736

9

741

Coordination

Addition Polymerization

Introduction Ziegler-Natta Catalysts Catalyst Composition Nature of the Catalyst Evolution of the Titanium-Aluminum System

741 742 742 744 746

Con~en~s Mechanismof Ziegler-Natta Polymerization Mechanismof Stereospecific Placement Bimetallic and Monometallic Mechanisms Kinetics of Ziegler-Natta Polymerization Typical Shapes of Kinetic Curves Effect of Catalyst Particle Size Chain Termination Kinetic Models for Ziegler-Natta Polymerization Early Kinetic Models Adsorption Models Models Involving Monometallic Mechanism Av, erage Degree of Polymerization Concluding Remarks Supported Metal Oxide Catalysts Polymerization Mechanism Ziegler-Natta Copolymerization Metallocene-Based Ziegler-Natta Catalysts Catalyst Composition The Active Center Polymerization Mechanism Kinetic Models Concluding Remarks References Exercises

747 748 749 755 755 758 759 760 761 764 773 778 779 779 781 789 791 792 795 796 797 803 803 805

10

Ring-Opening Polymerization

809

Introduction Polymerization Mechanism and Kinetics Anionic Polymerization of Epoxides Cationic Polymerization Of Cyclic Ethers Degree of Polymerization Polymerization of Lactams References Exercises

809 812 815 820 827 830 837 837

Appendixes 1: Conversion of Units 2: Fundamental Constants

839 841

Index

843

Advanced Polymer Chemistry

Chapter 1 Introductory Definitions BASIC

Concepts and

DEFINITIONS

Several important terms and concepts must be understood in order to fully discuss the synthesis, characterization, structure, and properties of polymers. Most of these will be defined and discussed in detail in subsequent chapters. However, some are of such fundamental importance that they must be defined at the beginning. Polymer The term polymer stems from the Greek roots poly (many) and meros (part). The word thus means "many parts" and designates a molecule made up by the repetition of some simpler unit called a mer. Polymers contain thousands to millions of atoms in a molecule that is large; they are also called macromolecules. (Although the words polymer and macromolecule are used interchangeably, the latter strictly defines the molecules of which the former is composed.) Polymers are prepared by joining a large number of small molecules called monomers. The structure of polystyrene, for example, can be written as

"-"-- CH 2 - CH- CH 2- CH- CH 2 - CH- CH 2 - C H-"---"

(I)

2

C1~apter 1

or, more conveniently, as (II), which depicts the mer or repeating unit of the molecule within parentheses with a subscript, such as n, to represent the number of repeating units in the polymer molecule.

(~) The difference in behavior between ordinary organic compounds and polymeric materials is due mainly to the large size and shape of polymer molecules. Common organic materials such as alcohol, ether, chloroform, sugar, and so on, consist of small molecules having molecular weights usually less than 1000. The molecular weights of polymers, on the other hand, may extend, on the higher side, to several millions. The macromolecule is a synonymfor a high-molecular-weight polymer. Often the term high polymer is also used to emphasize that the polymer under consideration is of very high molecular weight. For low-molecular-weight polymers, no firm lower limit of molecular weight can be defined. Strictly, a relatively small molecule composedof only, say, 3 mers might also be called a polymer. However, the term polymer is generally accepted to imply a molecule of large size (macromolecule). Accordingly, the lower-molecular-weight products with relatively small numberof repeating units in the molecular structure should preferably be called oligomers (Greek: oligo = few) to distinguish them from polymers. Generally speaking, a species will be called polymeric if articles made from it have significant mechanical strength and oligomeric if such articles are not strong enough to be practically useful. The distinction between the sizes of oligomers and the corresponding polymers is left vague, however, since there is no sharp transition between most properties of interest.

Monomer Monomers are generally simple organic molecules containing a double bond or a minimumof two active functional groups. The presence of the double bond or active functional groups acts as the driving force to add one monomermolecule upon the other (of the same or different type) to make a polymer molecule by a process known as polymerization. As a consequence, the structure of the repeating unit of a polymer is essentially that or closely related to that of the monomermolecule(s) from which the

IntroductoryConceptsandDefinition, s

3

polymer molecule is made. The formula of the polystyrene repeating unit (II) is thus seen to be essentially same as that of the monomerstyrene CH2 =CH-C6Hs. The repeating unit of a linear polymer (defined below) is a portion the macromolecule such that the whole polymer molecule (except for the end units) might be produced by linking together a required number of these units through bonds between specified atoms. A repeating unit may comprise a single identifiable precursor, such as (II) for the polymer(I). repeating unit may also be composed of the residues of several smaller molecules, as in poly(ethylene terephthalate), which has the structure:

"-’~OCH2CH2OC~COCH?CH2OC~COCH2CH20C 0

0

0

0

0

0

(III) The repeating unit in (III) may be written

(iv) Thus, the whole molecule of (III) can be built by linking the left-hand atom shown in (IV) to the right-hand atom, and so on. Though (III) also be built by similarly linking a sequence of the following structure:

-- CH2OC.@-’)-.-COCH 2_

II ’--’

0

(v)

I1 0

the structure (V) would not normally be written for the repeating unit, because the nominal break is in the middle of -O~H2CH20- unit derived from the monomer HOCH2CH2OH. It has been stated previously that structures of repeating units are essentially those of the monomersfrom which the polymers are made. It is, however, not necessary for the definition of the term repeating unit that

Chapter1 such a synthesis be possible. For example, -CH2-CH(OH)the repeating unit of poly(vinyl alcohol):

is evidently

~ CH2 - CH-CH2- CH-CH2- CH-CH2- CH .......OH

OH

OH

OH

(VI) The ostensible precursor monomer for this polymer is vinyl alcohol, CH2=CH-OH, which, being the unstable tautomer of acetaldehyde, does not exist. Poly(vinyl alcohol) is instead made by alcoholysis of poly(vinyl acetate),

.~.CH2-CH-

CH2-CH -CH2-CH-

I

OCCH 3

II

O

CH2- CH..-~

I

I

OCCH 3

OCCH3

II

I

II

O

O

OCCH 3

II

O

which, in turn, is synthesized by polymerization of the monomervinyl acetate, CH2=CHOOCCH3. Similarly, protein fibers like silk, which are polyamidesand are degradable to mixtures of amino acids, cannot be directly synthesized from the latter. Another exampleis cellulose, which is a carbohydrate with molecular ’formula (C6H100~)~,where n is a few thousand. The structure

CH2OH

H OH

H~ N OH

C.H2OH H O 0

ON H H

CH2OH

H OH H

OH

CH2OH

(VIII) Complete hydrolysis of cellulose by boiling with concentrated hydrochloric acid yields D-glucose, C6H1206,in 95%yield. Cellulose can thus be con-

Introductory Concepts and Definitions

~

sidered chemically as a polyanhydroglucose, though it cannot be synthesized from glucose. The concept of an identifiable simple repeating unit loses some of its utility with branched polymers, with network polymers, or with those macromolecules that are synthesized from more than a few different smaller precursor monomers.Such deficiency is, however, not serious, since the concept of a repeating unit is in fact only employed where such groupings of atoms are readily apparent. Resin and Plastic Both the terms ’resin’ and ’plastic’ are in commonuse but there are no explicit definitions for them. The term ’resin’ originally referred to vegetablederived organic products of relatively high molecular weight, the best known of which are rosin and balsam obtained from coniferous trees. But, the term is now used much more broadly, and rather loosely, to include the manmade polymeric substances used in a variety of applications, such as in plastics, textiles, and paints. It is often used interchangeably with the term ’plastic’. The term ’plastic’ is used for a material that is capable of being molded or shaped, with or without the application of heat. Strictly speaking, a plastic is therefore a permanently deformable and moldable material (plastikos fit for molding). While polymers of thermoplastic type can be repeatedly softened by heat and molded, those of the thermosetting type cannot be remolded after molding once. Thus, according to the above definition, the former is a plastic and the latter is not. Commonly,however, both are referred to as plastics; the term ’plastic’ thus denotes a material which has been shaped by plastic deformation at some stage during the process of the fabrication. The term ’plastic’ is also used in classifying polymers and includes all polymers which are not considered to be elastomers (rubbers) and fibers, i.e., which show neither the elastic properties of elastomers nor the high crystallinity and strength of fibers, but rather fall in betweenthem in these respects. Molecular

Weight

and Molar Mass

The term ’molecular weight’ is still widely used instead of ’molar mass’, though it can be somewhatmisleading. Molecular weight is really a dimensionless quantity. It is the sum of the atomic weights in the formula of the molecule and the atomic weights, in turn, are dimensionless ratios of the masses of the particular atoms to 1/12 of the mass of an atom of the most abundant carbon isotope 19C6 to which a mass of 12 atomic mass units

6

Chapter1

(AMU)is assigned. (AMU,a unit used for expressing the atomic masses of individual isotopes of elements, is approximately 1.6604x10-24 g.) The molar mass of a substance, on the other hand, is the mass of 1 mol of the substance and usually is quoted in units of g/mol or kg/mol. By multiplying the numerical value of molecular weight by the specific units g/mol it can be converted into an equivalent value of molar mass. For example, a molecular weight of 100,000 is equivalent to a molar mass of 100,000 g/mol which in turn is equivalent to a molar mass of 100 kg/mol. In this book, v/e shall retain the term ’molecular weight’ because of its widespread use in the polymer literature. However, when using in numerical calculations we shall substitute the molecular weight by its numerically equivalent molar mass (g/mol) to facilitate dimensional balancing. End Groups In none of the above examples of the structural representation of polymers the end groups have been shown. This is partly because the exact nature of the end groups is often not knownand partly because end groups constitute an insignificant fraction of the mass of high molecular weight polymer and so usually have negligible effect on polymer properties of major interest. Problem1.1 Calculate the end group content (weight fraction) of polystyrene of molecular weight 150,000, assumingthat phenyl (C6H5-)groups constitute both the end groups of an average polymer molecule. Answer : Molar mass of phenyl group = 6x12 ÷ 5xl = 77 g mo1-1 2(77 gmo1-1) Wt. fraction of end groups = (1.5 × 10~ -~) g mol = 0.001

Degree of Polymerization This term refers to the numberof repeating units in the polymer molecule. We shall use the abbreviation D_Pfor the degree of polymerization defined in this way. The subscript n used on the parentheses in the foregoing structural formulas for polymers, represent this DP. The relation between degree of polymerization and molecular weight Mof the same macromolecule is given by (1.1) M = (DP)Mo where Mois the formula weight of the repeating unit.

Introductory Conceptsand Definitions

7

Copolymers Whenonly one species of monomeris used to build a macromolecule, the product is a homopolymer, normally referred to simply as a polymer. The word homopolymeroften is used more broadly to describe polymers whose structure can be represented by repetitiorr of a single type of repeating unit which may contain one or more species of monomer unit. Thus, a hypothetical polymer -[-AB-]n- is also a homopolymer,e.g., poly(ethylene terephthalate) (III). The formal definition of a copolymer is a polymer derived from more than one species of monomer.However, in accordance with use of the word homopolymer, the word copolymer is commonlyused to describe polymers whose molecules contain two or more different types of mer or repeating unit. There are several categories of copolymer, each being characterized by a particular form of arrangement of the repeating units along the polymer chain. For simplicity, these different categories will be illustrated here by copolymers containing only two different types of repeating unit (A and B). The copolymer with a relatively random distribution of the different mers or repeating units in its structure is commonlyreferred to as a random copolymer. Representing two different mers by A and B, a random copolymer can be depicted as -ABBABBBAABBAABAAABBA(Strictly speaking, random copolymers are copolymers in which the distribution of repeating units is truly random. They should be considered only as a special type of statistical copolymer, because the term statistical copolymer includes all copolymers in which the sequential distribution of the repeating units may follow different statistical laws such as zero-, first- or second-order Markov, depending on the specific reactants and the methodof synthesis. However,most literature references use the term ’randomcopolymer’ independent of the type of statistical distribution, which is seldom known.) Three other copolymer structures are known: alternating, block, and graft copolymerstructures (Fig. 1.1). In the alternating copolymer the two mers alternate in a regular fashion along the polymer chain: -ABABABABABABABABABABA block copo!ymer is a linear polymer with one or more long uninterrupted sequences of each mer in the chain: -AAAAAAAAAABBBBBBBBBBDifferent block copolymers are distinguished by the number of blocks per molecule, for example, A~B, A~B~A~ A~B~A~B~ (A~B~)~ Diblock Triblock Tetrablock Mult~lock that are referred to as AB diblock, ABA triblock, ABABtetrablock,

8

Chapter1

(b) ~ (c)

(d)

Figure 1.1 Copolymerarrangements: (a) Twodifferent types of mers (denoted by open and filled circles) are randomlyplaced. (b) The mers are alternately arranged. (c) A block copolymer. (d) A graft copolymer. and ABmultiblock copolymers, respectively. The values of m and p as well as n are average values; thus, there is a distribution of block lengths and number of blocks along the copolymer chain. A graft copolymer, on the other hand, is a branched copolymer with a backbone of one type of mer to which are attached one or more side chains of another mer:

B I Copolymerization, which, in its objective, maybe compared to alloying in metallurgy, is very useful for synthesizing polymgr with the required combination of properties. For example, polystyrene is brittle, and polybutadiene is flexible; therefore copolymers of styrene and butadiene should be more flexible than polystyrene but tougher than polybutadiene. The general purpose rubber SBR(styrene-butadiene rubber), the first practical synthetic rubber, is a copolymer of styrene and butadiene.

Introductory Conceptsand Definitions

~

POLYMERIZATION AND FUNCTIONALITY The process of transformation of monomermolecules to a polymer molecule is knownas polymerization. It is a chemical reaction in which the product molecules are able to grow indefinitely in size so long as reactants are supplied and suitable conditions exist. Polymerization may occur if the monomersinvolved in the reaction ha~,e the proper functionalities. Functionality is a very useful concept in polymer science. The functionality of a molecule is the number of sites available for bonding to other molecules under the specific conditions of polymerization reaction. A bifunctional monomer,i.e., a monomerwith functionality 2, can link to two other molecules under appropriate conditions. Styrene, CrHsCH=CH2, for example, has functionality 2 because of the presence of a carbon-carbon double bond. Since every new linkage consumes two bonding sites, the product formed when a styrene molecule bonds to another styrene molecule will still have 2+2-2 or 2 bonding sites and so can continue to add in the same way other styrene molecules in an addition or chain polymerization reaction (described later) leading to the formation of a long-chain macromolecule (I). The minimumfunctionality required for polymerization to take place is 2. Note that the usage of the term "functionality" here is not the same as in organic chemistry where a carbon-carbon double bond, for example, is classified as a single functional group. A polyfunctional monomeris one that can react with more than two other molecules under the conditions of the polymerization reaction. Examples are divinyl benzene:

CH=CH 2

CH=CH 2

Ox) which is tetrafunctional in reactions involving additions across carbon-carbon double bonds, and glycerol

HOCH2- CH-CH20H

I

OH

(x)

I0

Chapter 1

which is trifunctional or pentaerythritol, C(CH2OH)4,which is tetrafunctional in esterifications or other reactions of alcohols. Functionality in polymerization is, however, defined only for a given reaction. Thus, a glycol, HOROH,has a functionality of 2 in esterification or ether-forming reactions, but its functionality is zero in amide forming reactions.

Problem1.2 What is the fl, nctionality of the following monomersin reactions with (i)

(a)

styrene,

C6HsCH=CH2 and (ii)

adipic

acid,

HOOC(CH~)4COOH

CH3 I CH2 =C O--C-OCH3

(b)

HOCH2CH2OH

(c)

H~.C=CH-CH2OH

(d)

CH2=C-- C-O-CH2-CH2-O-C-C = CH2 I II II I CH~ 0 0 CH -- CO CH--

CO

Answer: (i) In reaction with styrene, the functionalities of the monomersare: (a) 2 (one reactive carbon-carbon double bond) (b) 0 (-OH groups do not take part in addition reactions) (c) 2 (one reactive carbon-carbon double bond; -OH group nonreactive) (d) 4 (two reactive carbon-carbon double bonds) (e) 2 (one reactive carbon-carbon double bond) (ii)

In (a) (b) (c)

reaction with adipic acid, the functionalities of the monomersare: 0 (carbon-carbon double bond nonreactive) 2 (two reactive-OH groups) 1 (one reactive-OH group; carbon-carbon double bond nonreactive) (d) 0 (carbon-carbon double bond nonreactive) (e) 2 (one reactive anhydride group, which is equivalent to two carboxylic acid groups, each of functionality 1)

11

Introductory Conceptsand Definitions POLYMERIZATION

PROCESSES

There are two fundamental polymerization mechanisms. Classically, they have been differentiated as addition polymerization and condensation polymerization. In the addition process, no by-product is evolved, as in the polymerization of vinyl chloride (see below); whereas in the condensation process, just as in various condensationreactions (e.g., esterification, etherification, amidat!on, etc.) of organic chemistry, a low-molecular-weight by-product (e.g., H20, HCI, etc.) is evolved. Polymers formed by addition polymerization do so by the successive addition of unsaturated monomer units in a chain reaction promoted by the active center. Therefore, addition polymerization is called chain polymerization. Similarly, condensation polymerization is referred to as step polymerization since the polymers in this case are formed by stepwise, intermolecutar condensation of reactive groups. (The terms condensation and step are commonlyused ~nonymously, as we shall do in this book, and so are the terms addition and chain. However, as it will be shownlater in this section, these terms cannot always be used synonymously.In fact, the condensation-addition classification is primarily applicable to the composition or structure of polymers, whereas the step-chain classification applies to the mechanismof polymerization reactions.) Addition

or Chain Polymerization

In chain polymerization a simple, low-molecular-weight molecule possessing a double bond, referred to in this context as a monomer,is treated so that the double bond opens up and the resulting free valences join with those of other molecules to form a polymer chain. For example, vinyl chloride polymerizes to poly(vinyl chloride): nCH2==CH C1 Vinylchloride

Polymerization ,

-(CH2-CH ~-

!CI

(1.2)

Poly(vhylchloride)

It is evident that no side products are formed; consequently the composition of the met or repeating unit of the polymer-(-CH2-CHC1-)-is identical to that of the monomer(CH2=CHCI).The identical composition of the repeating unit of a polymer and its monomer(s) is, in most cases, an indication that the polymer is an addition polymer formed by chain polymerization process. The commonaddition polymers and the monomers from which they are produced are shownin Table 1.1. Chain polymerization involves three processes: chain initiation, chain propagation, and chain termination. (A fourth process, chain transfer, may also be involved, but it may be regarded as a combination of chain termination and chain initiation.) Chain initiation occurs by an attack on

12

Chapter 1

Table1.1 Typical Addition Polymers (Homopolymers)

Ethylene

Po~yeth~’lene(PE)

2. Propylene

PolypropyleneIPP/

3, S~rene CH~ = CH

Polystyrene(PSI *-~CH~-CH-~.

"High density polyethylene (HOPE)and low density ~olyethylene (L~PE):moldedobjects, tubing, film. dectric~i ~sulation. us~ householdproduc~,insulators, pipes, toys. ~ttl~. e.K.eAl~thene,

Cowerdensity, stiffer, andhigher temperatureres~tancethan PE: USed for waterpipes,integral hinges,sterilizabte hospital ~luipment. e.g,~Propathene, ~lovolen, ]V~oplen, Hos~Jen, Martex.

CH~ Transoarent andbrittle: usedfor cheapmoldedoi~-cts, e.g. Styeon, Cm’inex, Hostyren, Lustrex. Modified with robbers toim~e toughness,e.g. high-impactpolystyrene (HIPS)and acrylonitriiea blendedblowingagent(e.g. pentane)to produceooly~tyrenefoam. e.g.~ Styrocell, Styrofoam. 4. Acrylonitrile

Polyacrylonitriie (PAN)

CH~ = CH-CN

WideUseis fibers; best alternative to woolfor ~ters. e.g,~ Orlon,

~CH~m~ CN

Poly(vinyl acetate) (PVA)

S. Vinyl acetate CH~ = CH-O-C-CH~

Emulsionpaints, adhesives, sizing, chewinggum,e.g., Flovic, ~vilith, lVtoWieoU..

--~CH~mCH~

Poly(vinyl chloride) (PVC)

6. Vinyl chloride

Waterpipes andgutters, bottles, gramophone records, ~lasti~zed

Cl 7. Tetrafluoroethylene

Polytetrafluoroethylene (PTFE)

CF~ = CF~

Methyl methacryiate

~CF~-CF~

Poly(methyl methacrylate) (PMMA)

~CH,

tCH~

0 OCH~

C=O OCH=

Hightemperature resistance,chemicallyinert, exce~ientelectrical insulator, ve~low coefficient of friction, expensive;moldings,film. coatings;usedfor non-sticksurfaces,insulation, gaSKe¢~: e,g.~Tefion. Fluon. Transparentsheetsand moldlngs:moreexpensivethan PS: usedfor Vedril,

g. Isobutylene

/~olylsobu~tene(PIB)

~H,

(~H~

CH~,

CH~

Isoprene

¢~- 1.4- Polylsoprene

OH= = C,-CH = cx, -(-c~,-c~= C~-CH~-).CHj ~utadiene (H~ = CH-CH = CHz

CH~ ¢~l.4-Polybutadiene ~CHT-CH

= CH-CH~

Lubricatingoils. sealants;copolymerized with 0.~-2.5tool%isoorene to produce Rut~trubberfor tire innertubesandinne~ilners of tuPeie~ tires.

Cartires, mechanical goods,footwear, sealant, caulkingcompounds,

Introductory Conceptsand Definitions

13

the monomermolecule by a free radical, a cation, or an anion; accordingly, the chain polymerization processes are called free-radical polymerization, cationic polymerization, or anionic polymerization. A free radical is a reactive substance having an unpaired electron and is usually formed by the decompositionof a relatively unstable material called an initiator. Benzoyl peroxide is,’.a commonfree-radical initiator and can produce free radicals by thermal decomposition as O O O R-C-O-O--C-R ---+ R--C-O-+ R-+ CO~ (1.3) (R = Phenylgroupfor benzoylperoxideinitiator Free radicals are, in general, very active because of the presence of unpaired electrons (denoted by dot). A free-radical species can thus react to open the double bond of a vinyl monomerand add to one side of the broken bond, with the reactive center (unpaired electron) being transferred to the other side of the broken bond: O

H

--+

O

H

X X (x =- CHa,C~H5,CI, etc.) The new species, which is also a free radical, is able to attack a second monomermolecule in a similar way, transferring its reactive center to the attacked molecule. The process is repeated, and the chain continues to grow as a large number of monomermolecules are successively added to propagate the reactive center: H H [ Successive addition R-C-O-CH2--C’III (1.5) of monomer ) R-C-O-(-CHg--CI)m’u O X O X This process of propagation continues until another process intervenes and destroys the reactive center, resulting in the termination of the polymer growth. There may be several termination reactions depending on the type of the reactive center and the reaction conditions. For example, two growing radicals may combine to annihilate each other’s growth activity and form an inactive polymer molecule; this is called termination by combination or coupling: H H +"(xC~ -CH~ R-0C-O-(CH~-C~ ),~’,, -),~-0-C-Ro HI H R-C-O-( CH,-C)-~--~t- CI -CH2,-)~-~ OII ! X 0 X 0 Inactive polymermolecule

( 1.6)

14

Chap~e~ 1

A second termination following equation : H R-C-O-(CHo-C)~’,, . 0

X

mechanism is disproportionation;

shown by the

H + "(C~-CHz--),,-O-C-R,, X

H ~-o-(-CH2 - L;-)-~_~ II I O X

H I X

+ C = CH( C-CH2--)-,~_ I I X X

~-O-C-R II 0

(1.7) In chain polymerization initiated by free radicals, as in the previous example, the reactive center, located at the growing end of the molecule, is a free radical. As mentioned previously, chain polymerizations may also be initiated by ionic systems. In such cases, the reactive center is ionic, i.e., a carbonium ion (in cationic initiation) or a carbanion (in anionic initiation). Regardless of the chain initiation mechanism-free radical, cationic, or anionic-once a reactive center is produced it adds many more molecules in a chain reaction and grows quite large extremely rapidly, usually within a few seconds or less. (However, the relative slowness of the initiation stage causes the overall rate of conversion of monomer to be slow: The conversion of all monomers to polymers in most polymerizations requires at least 30 minutes, sometimes hours.) Evidently, at any time during a chain polymerization process the reaction mixture will consist only of unreacted monomers, high polymers and unreacted initiator species, but no intermediate sized molecules. The chain polymerization will thus show the presence of high-molecular-weight polymer molecules at all percents of conversion [see Fig. 1.2 (a)].

Problem 1.3 Styrene monomer’ containing 0.02% (by wt.) benzoyl peroxide initiator was reacted until all the initiator was consumed.If at this stage 22%of the monomerremained unreacted, calculate the average degree of polymerization of the polymer formed. Assume100%efficiency of the initiator (i.e., all initiator molecules are actually consumed in polymer formation) and termination of chain radicals by coupling alone. Answer; Basis: 1000 g styrene Molar mass of styrene (CsHs) = 8×12 + 8×1 = 104 g/mol Molar mass of initiator, (C6H5CO)202= 242 g/tool Amountof initiator = 0.20 g -1) = (0.20 g) / (242 g tool or 8.26×10-4 tool

15

Introductory Conceptsand Definitions Initiator fragmentforming polymer end groups = 2 (8.26 x 10-4 mol) 1.652 × 10-3 mol Polymerformed by termination by coupling = (1.652×10-3 mol) / = 8.26x10 -4 mol Styrene reacted = (100 -- 22) (1000 g) / 100 780 g (780 g) / (104 g mo1-1) or 7.50 mol DP = (7.50 tool) = 9080 (8.26 × 10-4 tool)

Step Polymerization Step polymerization occurs by stepwise reaction between functional groups of reactants. The reaction leads successively from monomerto dimer, trimer, tetramer, pentamer, and so on, until finally a polymer molecule with large DP is formed. Note, however, that reactions occur at random between the intermediates (e.g., dimers, trimers, etc.) and the monomeras well as amongthe intermediates themselves. In other words, reactions of both types, namely, n-mer

+ monomer ~ (r~

+ 1)-mer

and r~-mer

+ m-mer

~ (n

+ m)-mer

occur equally. Thus, at any stage the product consists of molecules of varying sizes, giving a range of molecular weights. The average molecular weight builds up slowly in the step polymerization process, and a highmolecular-weight product is formed only after a sufficiently long reaction time when the conversion is more than 98%[see Fig. 1.2(b)]. In contrast, polymerization by chain mechanismproceeds very fast, a full-sized polymer molecule being formedalmost instantaneously after a chain is initiated; the polymer size is thus independent of reaction time [Fig. 1.2(a)]. The above classification of polymers according to polymerization mechanism, as shownby the variation of molecular weight with conversion [Figs. 1.2(a) and 1.2(b)], is not without its ambiguities. Certain polymerizations show a linear increase of molecular weight with conversion [Fig. 1.2(c)] when the polymerization mechanism deviates from the normal chain or step pathway. This is observed in certain ionic chain polymerizations, which involve a fast initiation process coupled with the absence of reactions that terminate the propagating reactive centers. Biological syntheses of proteins also show the behavior described by Fig. 1.2(c) because the various

16

Chapter1

,I ) 20 40 60 80 % Conversion

100

20 40 60 80 100 °/o Conversion

Ca)

0

Cb)

20 Z,O 60 80 100 % Conversion

(c1 Figure 1.2 Variation of molecular weight with conversion. (a) Chain polymerization; (b) step polymerization;(c) nonterminatingionic chain polymerization. monomermolecules are directed to react in a very specific manner in an enzymatically controlled process. Most (but not all) of the step polymerization processes involve polycondensation (repeated condensation) reactions. Consequently, the terms "step polymerization" and "condensation polymerization" are often used synonymously. In a condensation reaction between two molecules, each molecule loses one atom or a group of atoms at the reacting end, which leads to the formation of a covalent bond between the two, while the eliminated atoms bond with each other to form small molecules such as water-hence the term condensation reactions. Consider, for example, the synthesis of - as the characteristic a polyamide, i.e., a polymer with amide xcCONH-)

Introductory Conceptsand Definitions

17

linkage. If we start with, say, hexamethylenediamine and adipic acid as reactants, the first step in the formation of the polymer (nylon) is the reaction (1.8) producing a monoamide: H2N-(CH2)6-NH2

,,

+ HO-~-(CH2)4--C-OH O O

H2N-(CH2)6-NH--C-

(CHz)4--C-OH

+

(1.8)

Monoamide The reaction continues step-by-step to give the polyamide nylon-6,6. The overall reaction may thus be represented as r~H2N-(CH~)6-NH2 + nHO-O~-(CH2)4-~-OH6

H-[NH-(CH2)~--NH~-(CH2) 4-~.].-OH O O Poly(hexamethylene adipamide)

+ (2n-

1)H20

Wesee that the composition of the repeating unit (enclosed in square brackets) equals that of two monomermolecules minus two molecules of water. Thus a condensation polymer may be defined as one whose synthesis involves elimination of small molecules or whoserepeating unit lacks certain atoms present in the monomer(s). Problem 1.4 Poly(hexamethylene adipamide) (Nylon-6,6) was synthesized condensationpolymerization of hexamethylenediamine and adipic acid in 1:1 mole ratio. Calculate the acid equivalent of the polymerwhoseaverage DPis 440. Answer: Polymer

H-[-NH-(CH2)6-NH-C-(CH2) 4-ICI--}n-OH ii o o

n = 440 Molar mass of repeating unit = 226 g/mol Average molar mass = 440×226 + 18 or 99,458 g/tool Wt. of polymer (in grams) containing one acid equivalent =

99,458

With the development of polymer science and the synthesis .of new polymers, the previous definition of condensation polymer is inadequate. For

18

Chapter1

example, in polyurethanes (Table 1.2), which are classified as condensation polymers, the repeating unit has the same net composition as the two monomers(i.e., a diol and a diisocyanate), which react without eliminating any small molecule. To overcome such problems, chemists have introduced a definition which describes condensation polymers as consisting of structural units joined by internal functional groups such as ester (-~n-O-), amide

0

/CO-

0

(-~-NH-), imide (-N\c0),_ urethane (-O-~-NH-),Q

sulfide (-S-), ether

0 (-0-),

carbonate (-O-~-O-),

and sulfone

(-~"-)

linkages.

A polymer

satisfying either or both of these definitions is classified as a condensation polymer. Phenol-formaldehyde,for example, satisfies the first definition but not the second. Somecondensation polymers along with their repeating units and condensation reactions by which they can be synthesized are shown in Table 1.2. Somehigh-performance polymers prepared by polycondensation are listed in Table 1.3. The ring-opening polymerizations of cyclic monomers,such as propyl~ne oxide, nCH~-CH--CH~

~ --[-CH.~CH-O-~.-CH3

¯

(1.10)

or e-caprolactam O ~ --[-NH(CH~) ~CO-].--

(1.11)

proceed either by chain or step mechanisms, depending on the particular monomer, reaction conditions, and initiator employed. However, the polymers produced in Eqs. (1.10) and (1.11) will be structurally classified as condensation polymers, since they contain functional groups (e.g., ether, amide) in the polymer chain. Such polymerizations thus point out very clearly that one must distinguish between the classification based on polymerization mechanism and that based on polymer structure. The two classifications cannot always be used interchangeably. Both structure and mechanismare usually needed in order to clearly classify a polymer.

MOLECULARARCHITECTURE Polymers can be classified, based on structural shape of polymer molecules, as linear, branched, or network (cross-linked). Schematic representations are given in Fig. 1.3.

Introductory Concepts and Definitions

19

o

I~

I

E

C]~ap~er 1

20

f

±

Introductory Conceptsand Definitions

(a) linear

21

(b) branched (short)

(c) branched (tong)

(d) cross-tinked (e) [adder Figure 1.3 Schematic representation of various types of polymermolecules. The branch points and junction points are indicated by heavydots. Linear polymers have repeating units linked together in a continuous length [Fig. 1.3(a)]. In a linear polymer each repeating unit is therefore linked only to two others. Whenbranches protrude from the main polymer chain at irregular intervals [Fig. 1.3(b)], the polymer is termed branched polymer. [Regularly repeating side groups that are a part of the monomer structure are not considered as branches. Thus, polystyrene (I), poly(vinyl acetate) (VII), and poly(methyl methacrylate) (Table 1.1) are called ear polymers although they contain short branches that are part of the monomerstructure.] Branches may be short, forming a comblike structure [Fig. 1.3(b)] or maybe long and divergent [Fig. 1.3(c)]. Branchedpolymers are thus those in which the repeating units are not linked solely in a linear way. Branched polymers may be formed either because at least one of the monomershas functionality greater than 2 or because the polymerization process itself produces branching points in the polymer molecule. An example of the first type is the polymer made, for instance, from styrene and a very small amount of divinyl benzene (IX). A segment of such macromolecule might look like (XI):

22

Chapter1

~ CH2-CH-CH2-CH-CH2-CH-CH2-CH-CH2-CH

~

CH

v "CH

/

~C"

\

v

A major example of the second branched polymer type is the polyethylene that is made by free radical polymerization at temperatures of 100300°C and pressures of 1,000-3,000 atm. The extent of branching varies considerably depending on reaction conditions and may reach as high as 30 branches per 500 monomerunits. Branches in polyethylene are mainly short branches (ethyl and butyl) and are believed to result from intramolecular chain transfer during polymerization (described later in Chapter 5). This branched polyethylene, also called low-density polyethylene (LDPF,), differs from linear polyethylene (high-densitypolyethylene, HDPE)of a low-pressure process so muchso that the two materials are generally not used for the same application. By convention, the term branched implies that the polymer molecules are discrete, whichis to say that they can generally be dissolved in a solvent and their sizes can be measured by at least some of the analytical methods described in Chapter 4. A network polymer [Fig. 1.3(d)], on the other hand, can be described as an interconnected branched polymer. For example, a three-dimensional or space network structure will develop, instead of the branched structure (XI), if styrene is copolymerizedwith higher concentrations of divinyl benzene. In a network structure, all polymer chains are linked to form one giant molecule. Thus, instead of being composed of discrete molecules, a piece of network polymer constitutes, essentially just one molecule. With the formation of network structure polymers acquire greater rigidity, di~ mensional stability, and resistance to heat and chemicals. Because of their network structure such polymers cannot be dissolved in solvents and cannot be melted by heat; strong heating only causes decomposition.

hltroductory Conceptsand Definitions

23

The molecular weight of a network polymer is "infinite," in the sense that it is too high to be measured by standard techniques. In reality, the molecular weight of a network polymer cannot be infinite even if all the atoms in a network polymer sample is part of a single molecule (which is possible in theory), since the size of the sample is finite (see Problem 1.5). However,on the scale applied in polymer measurements, which require the sample to be soluble in a solvent, its molecular weight is infinite. Problem 1.5 For a network polymer sample in the form of a sphere of 1 cm diameter with a density of 1.0 g/cm3, estimate the molecular weight assuming ~3 that the sample constitutes a single molecule. (Avogadronumber= 6.02x10 molecules/mol). Answer : Molar mass = ( Mass "] (Avogadro number) \Molecule./ = =

(Vol. of sphere)(Density)(Avogadro number)

I,(3"14)(~ 0 z] cm)

(1.0 g cm-3) (6.02x102a mo1-1)

-I = 3.15x1023 g mol ~’a Molecularweight = 3.15 xlO If the average functionality of a mixture of monomersis greater than 2, copolymerization reaction to a sufficiently high conversion yields network structures (see Chapter 5). Network polymers can also be made by chemically linking linear or branched polymers. The process whereby such a preformed polymer is converted to a network structure is called cross-linking. Vulcanization is an equivalent term which is used primarily for rubbers. In commercialvulcanization, sulfur cross-links are introduced into a rubber by heating raw rubber with sulfur (1-2% by weight) and accelerating agents. Sulfur reacts with the double-bonded carbon atom to produce a network structure, as shownschematically in Fig. 1.4. The effect of cross-linking is to introduce points of linkage or anchor points between chain molecules, restricting their slippage. The cross-links in elastomers are typically a few hundred carbon atoms apart. Problem 1.6 Assumingthat each cross-link produced by vulcanization [Fig. 1.4(b)] contains an average of two sulfur atoms, calculate the sulfur content vulcanizednatural rubber that is 50%crosg-linked. (Neglect sulfur other than that

24

Chapter 1

is part of the cross-link.) Answer : Molar mass of isoprene [Fig. 1.4(a)], CsHs 5×12 + 8×1 = 58g/mol Accordingto representation in Fig. 1.4(b), each sulfide cross-link joins two isoprene units. With x = 2, one sulfur atom, on the average, is required for cross-linking per isoprene repeating unit. Therefore, Sulfur

content

=

(0.5)(32 g) (0.5)(32 g) + (68

CH3 "-’~CH2-C=CHn I

× 100 =

CH2~

I

cHg[ --CH2-C-CH =CH-(2-

I

CH 3

C~3L

~x

CH2- C = CH- CH2"-~n CH2-CI -CH2- CH

CH3

CH 3

iSx

(b)

(c) Figure1.4 Vulcanization of natural rubber with sulfur. (a) Linear polyisoprene (natural rubber). (b) An idealized structure produced by vulcanization with fi~r. The number(x) of sulfur atoms in sulfide cross-linkages is 1 or 2 in efficient vulcanization systems but may be as high as 8 under conditions where cyclic and other structures are also formedin the reaction. (c) The effect of cross-linking is introduce points of linkage or anchor points between chain molecules, restricting their slippage.

Infroduc~oryConceptsand Definitions

25

Problem 1.7 A rubber contains 60%butadiene, 30%isoprene, 5%sulfur, and 5%carbonblack. If each sulfide cross-link contains an averageof twosulfur atoms, whatfraction of possible cross-links are joined by vulcanization? (Assumethat all the sulfur is presentin cross-links.) Answer : Molar mass of butadiene (C4H6) 4x12 + 6xl = 54g/mol Molar mass of isoprene (C5H8) = 5×12 + 8×1 = 68 g/mol Since one sulfur atom, on the average, is required for crosslinking per repeating unit (cf. Problem1.6), (5 g)/(32 g mo1-1) fraction of cross-links = (60 g)/(54 g mo1-1) + (30 g)/(68 g = 0.101 or 10.1%. The structure of a ladder polymer comprises two parallel strands with regular crosslinks [Fig. 1.3(e)], as in polybenzimidazopyrrolone(XII), which is madeby polycondensation of pyromellitic dianhydride (XIII) and 1,2,4,5tetraminobenzene(XIV). This polymeris practically as resistant as pyrolitic graphite to high temperatures and high energy radiation. It does not burn or melt when heated but forms carbon char without much weight loss. 0

N~C’9~,~~C~ ~"

N

(XII) 0

0

o 0

0 (XIII)

H2N~NH2 H2 N’~ ~ 2

NH

26

ChapterI

The permanence properties of ladder polymers are superior even to those of conventional network polymers. The latter are randomly crosslinked, and their molecular weights can be reduced by random scission of covalent bonds in the network. In contrast, when a covalent bond is broken in a ladder polymer, the second strand maintains the overall integrity of the molecule and the ends of the broken bond are held in such close proximity as to enhance the likelihood of their recombination. A ladder (or two-strand) structure, as shownabove, is one that has uninterrupted sequence of rings joined one to another at two connecting atoms. A semiladder structure, on the other hand, is one in which there are single bonds interconnecting some of the rings. Polyimide (XV) obtained polycondensation of pyromellitic anhydride (XIII) and m-phenylenediamine (XVI) is an example of semiladder polymer. The general approach building rigid polymer chains composedof ring structures, usually aromatic rings, results in polymer structures that can be classified as ladder or semiladder polymers. 0

0

NH 2 0

0

(xv)

(xv

Another approach to obtaining rigid polymer chains is by the synthesis of spiro-polyrners. These are polymers containing ring structures in which there is only one atom commonto two rings. An example is polyspiroketal (XVII) synthesized from 1,4-cyclohexanedione (XVIII) and pentaerythritol

HOH2C~ c/ HOH2 (XVIII)

/CH2OH C~CH20 H

(XlX)

Introductory Conceptsand Definitions POLYMERS:

THERMOPLASTICS

27 AND THERMOSETS

A thermoplastic is a polymer that softens and hardens reversibly on changing the temperature. Both linear and branched polymers are thermoplastic. Thus, they can be softened and made to flow by application of heat. Fabrication processes like injection molding, extrusion molding, and blowing take advantage of this feature to shape thermoplastic resins. The rigidity of thermoplastic resins at low temperatures is attributed to the existence of secondary forces between the polymer chains. These bonds are destroyed at higher temperatures, thereby causing fluidity of the resin. A thermosetting plastic is a polymer that can be caused to undergo crosslinking to produce a network polymer, called a thermoset polymer. Quite commonly,thermosetting resins are prepared, by intent, in only partially polymerized states (prepolymers), so that they can be deformed in a heated mold and then hardened by curing (cross-linking). The most important thermosetting resins in current commercial applications are phenolic resins (Fig. 1.5), amino-resins (Fig. 1.6), epoxy resins (Fig. 1.7), unsaturated polyester resins (Fig. 1.8), urethane foams (Fig. 1.9), and the alkyds (Fig. 1.10). The conversion of an uncross-linked thermosetting resin into a cross-linked network is called curing. For curing, the resin is mixed with an appropriate hardener and heated. However, with some thermosetting systems (e.g., epoxies and polyesters), the crosslinked or network structure is formed even with little or no application of heat. Epoxies are often preferred to polyesters because they have superior corrosion resistance, mechanical properties, and high-temperature properties, but they are more difficult to handle due to higher viscosities. Vinyl esters, which are obtained by reacting epoxies with an unsaturated acid such as acrylic acid or methacrylic acid, combine the benefits of epoxy resins with the lower viscosity and faster curing of unsaturated polyesters. Since vinyl esters contain a double bond (instead of the epoxide group) at each end of the molecule, they can be cured in the same way as polyesters (Fig. 1.8) after being dissolved in styrene. Aging of polymers is often accompanied by cross-linking due to the effect of surroundings. Such cross-linking is undesirable because it greatly reduces the elasticity of the polymer, makingit more brittle and hard. The well-knownphenomenonof aging of polyethylene with loss of flexibility is due. to cross-linking by oxygen under the catalytic action of sunlight [Fig. 1.1 l(a)]. Natural rubber undergoesa similar loss of flexibility with time due to oxidative cross-linking [Fig. 1.11(b)]. This action maybe discouraged add.ing to the polymer an antioxidant, such as a phenolic compound, and an opaque filler, such as carbon black, to prevent entry of light. POLYMERS:

PLASTICS,

FIBERS,

AND ELASTOMERS

Polymers can be classified in manyways, such as by source of raw materials£ methoOof synthesis, fabrication processes, end use, and so on. Someof

28

Chapter 1

OH

H

OH

OH

OH

B- Stageresin (Novo[ac)

OH

A

OH

OH A

--CH2"-~CH2"-~CH2"~CH2-H2C= O H2C=O -- C H2-’~ CH~2 ~j’OH

Fi.gure 1.5

Equations

" -CH2-’~CH2-’~CH2"~CH2--OH ---

CH2 T’~ CH 2 -OH

(idealized)

OH

T

CH2~CH2-~C"2J~CH2 OH ¯ OH C-Stage resin

for the production.

f.

phen01-formaldehyde

reslns.

H--N-CH20H H-NsCH2OH ~’NH2 Alkaline O=C,x ’ = + O=C O C\ + 2 C=O Catalyst NH 2 Urea Formaldehyde " Monomethytol urea /c DimethytOlurea

/

/

id "catalyst ~

¯ ",~

~H2

N -- CH2 ~ 1 C=O I N-- CH2--N -- CH~--y ~

C=O

N~C\N

C=O

I II H2N~C~N~C-Melamine

NH2

I.

I --CH2--N--CH2-Network polymer

Figure 1.6

C=O

C=O N--

CH2--N ~

(UF)

The two important classes of amino-resins are the products of condensation reactions of urea and melamine with formaldehyde. Reactions for’ the formation of urea formaldehyde, amino-resins (UF) are shown. Preparation, melamine-formaldehyde resins is similar.

hltroductory Concepts and Definitions

CH2--CH-CH2C[

29

+

Epichtorohydrin I aq. NaOH~

CH3 Bis pheno[-A

CH 2- CH’- CH2- O Curing of an epoxy resin

~ CH2-

/o\

CH-

CH2 + H2N- F~-NH 2 ~ Diarnine

Epoxy end group

~ ’ CH2-C. H-CH2- NCH2- CH -CH2~ I I OH I~ I

OH

~ CH2-- ~CH-C~2OH OH Network polymer

Figure1.7 Epoxy monomersand polymer and curing of epoxy resins. Polyamines such as diethylenetriamine (H,~NC~H4NHC2H4NH2) are widely used for the production of network polymers by room temperature curing. . these classifications have already been considered in earlier sections. In the present section, polymers ar6 grouped according to end use as plastics, fibers and elastomers. However, before making a formal distinction between these categories, it is imperative to consider-why polymers are useful.

WhyAre Polymers Useful ? The strength properties of polymers, that enable us to convert them-into a myriad of strong products to serye diverse end uses set them distinctly apart, from conventional chemical compounds that are useless for the same applications. For example, n-hexatriacontane [CHa-(-CH~-)~9-CHa] (molecular weight 437), which may be considered as an oligomer of polyethylene, is

30

CH_C//0 II >O CH Ma[eic anhydride

Chapter 1

+ HO-CH2-CIH-OH CH 3 Propy[ene gLycoL

+ r~c~) ~c~ Phtha[ic anhydride

Acid catalyst

O 00 0 II II II II ~ O-C-CH=CH-C-O-CH~-CH-O-C C-O~ ~ J I I Unsaturated I CH3 ~ polyester Styrene ~( [ and peroxide ~ catalyst

I

0 II

0 0 0 I1 II ~-~ C-O-CH2- ICH-O-C C-O ~ u

CH3

~ O-C-CH-CH -0~

(~

C- O-CH2-CH-O-C C-O~ 0 CH3 ~)

Networkstructure

Figure 1.8 Equations for preparation

and curing of an unsaturated polyester resin. The presence of ethylenic unsaturation provides sites for cross-linking by a chain reaction mechanismin the presence of styrene. Phthalic anhydride increases flexibility by increasing spacingof cross-links. a weak and friable waxy solid but the chemically identical material polyethylene (Table 1.1) can be used to make strong films, cable jackets, bottles, pipe, and so on, provided the polymer molecular weight is more than about 20,000. Polymers are thus useful mainly because of their large molecular size. The intermolecular forces of attraction in hexatriacontane cited above and in polyethylene are essentially van der Waals forces. However, in polymers of large molecular size, there are so many intermolecular contacts that the sum of van der Waals forces holding each molecule to its neighbors is appreciable. It thus requires a much. larger stress to break a polymeric material as this involves separating the constituent molecules. Deformation of a polymeric structure for the same reason requires more force as the macromolecules become larger.

Introductory Concepts and Definitions

31

(a) Prepolymer formation: O~C~N-R-N=C=O

+ HO-P-OH--~-~

biisocyanate

O=C=N-~R-NH-C-O-P-O-C-NH-~-R-N=C=O

--

Glycol

~NCO Prepo[ymer (ii)

With

(ill)

~

With

NH

the manufacture of foams): O ,-~- ----~’~--’~-N~H=C=NH,----~ + H20 + OCN Prepolymer - Ur-ea link =

"n

O prepolymer

(in

glycols:~NCO+

(¢) Cross-linking

O

Urethane

(b) Chain extension of prepolymer: (i) With water

II

II

HO-R-OH + OCN ~’----~

amines:~NCO+

H2N-R-NH2+OCN~

+ 2 CO

~ NH-C-O-R-O-C-NH ~ Urethanelink 0 0 -~,~’~NH-C-NH-R-NH-C-NH~ Double urea [ink

of chain-extended polyurethane:

CONH~

-

~

NCONH I

"’NCO ~

NHCONH

H N CO I ~NCONH~

~

Figure1.9 Equations for preparation, chain extension, and curing of polyurethanes. 0 ~ HOCH2-CI H-CH20H 4OH O Gl.ycero!. Phthatic anhydride O O II II HO-C C-O-CH2-1CH--CH2-O-C

0 0 II II C-O-.~--~-

Gtypla!. resin

l

Curing

0 O O O "~’’~ NO-C, C,-O-CH2-C,H-CH2-O-C C-O

Figure 1.10 Equations for preparation of network glyptal resin.

32

Chapter I

- CH2-CH CH 2 -CH2-CH2-- CH2- -2 -

02 , tigh’t

CH2-CH2-CH2-CH 2 -CH2-CH 2-

-CH2-~H-CH2-CH2-CH2-CH 2O I -CH2-CH2-cH --CH2-CH2-CH I

-CH2-C % -CH-CH2-CH2-C--CH-CH 0 CH 23

H3~ I

02 , l.igh~ -CH2~=CH-CH2-CH2-~=cH-CH2CH 3

I

0 I - CH2- ~ -~H--CH 2-CH2-~ = CH-CH2--

CH 3

Figure 1.11 Aging of (a) polyethylene

(b)

H3C O

I

and (b) natural

CH 3

rubber by oxidative

cross-linking. The relationship of such properties as tensile strength, impact strength, and melt viscosity with polymer molecular weight is indicated schematically in Fig. 1.12. Note that the strength properties increase rapidly at first as the chain length increases and then level off, implying that all polymer types reach about the same strength at sufficiently high molecular weights. This would be expected because the sum of intermolecular forces on an individual molecule will equal the strength of its covalent bonds if the molecule is large enough. Most synthetic polymers have carbon-carbon, carbon-oxygen, or carbon-nitrogen bonds in their backbone chains and the strengths of these bonds do not differ very much. The ultimate strengths of polymers with extremely high molecular weights should thus be almost equal. This ideal limiting strength is of more theoretical than practical interest because an optimum balance of different properties is more important than a single outstanding property. Figure 1.12 shows that unlike strength properties, which level off at high molecular weight, the polymer melt viscosity continues to increase rapidly with the size of macromolecules. Thus, samples composed of extremely large molecules will be very strong, but they cannot usually be caused to flow into desired shapes because of very high viscosity. The temperature needed for molding or extrusion of an ultrahigh-molecular-weight polymer can in fact exceed that at which the material degrades chemically. The range of molecular weights chosen for commercial polymers thus represents a compromise between maximum properties and processability. For example, the average molecular weight and the viscosity of natural rubber, which usually has molecular weights between 500,000 and 1,000,000 are reduced to tractable levels by masticating the polymer with chemicals (which promote Scission of carbon-carbon bonds and stabilize the fragmented ends) so that the less viscous rubber can then be mixed easily

Introductory Conceptsand Definitions

33

Tensile/~ strength strength

~

/ ~/VMett /// viscosity

~4o[ecutarweight Figure 1.12 Polymerproperties versus polymer molecular weight (schematic). easily with necessary ingredients, formed, and finally increased in molecular weight and fixed in shape by vulcanization. Since the strength of polymeric materials is attributed to intermolecular attractio.ns, it is not surprising that polymers having functional groups that promote higher intermolecular attractions develop more strength at equivalent molecular weight than polymers in which intermolecular forces are only of weakvan der Waals type. Polyamides (nylons), for example, are characterized by the structure H H -~c- N-R-N- C-R’-C-)~-O. 0 in which hydrogen-bondinginteractions are important and are muchstronger than hydrocarbon polymers like polyethylene having the same degree of polymerization. As the strength of an article made of discrete polymer molecules depends on the sumof intermolecular attractions, it is obvious that any process that increases the numberof intermolecular contacts at the expense of intramolecular contacts of segments buried in the normal coiled conformation

34

Chapter1

of macromolecules,will result in a stronger product. If an article is oriented, polymer molecules tend to become stretched out and mutually aligned; the article thus becomesmuchstronger in the orientation direction. Orientation is therefore an important part of-the process of forming fibers of polymers. If the polymer molecule is rigid, it will have less tendency to coil up on itself, and most segments of a given molecule will therefore contact segments of other molecules rather than those of its own. A prime example is the aromatic polyamide fiber, poly(p-phenylene terephthalamide) (XX) marketed as Kevlar. It is wet-spun from a solution in concentrated sulfuric acid into fibers which can be stretched to two or three times their original length. The products are as strong as steel but have one-fifth the weight, and can be heated without decomposition to temperatures exceeding 500°C.

The requirements that must be satisfied by polymers to be used as textile fibers can thus be summarized as: (1) linear molecules, (2) molecular weight, and (3) a permanent orientation of the molecules parallel to the fiber axis. The molecules must have a high degree of order and/or secondary forces to permit orientation and crystallization. The chain orientation necessary to develop sufficient strength by crystallization is achieved by a process knownas cold drawing, in which the initially formed filaments (unoriented or only slightly oriented) are drawn at a temperature above the glass transition temperature (see Chapter 2), which is the temperature at which sufficient energy is available to the molecular segments to cause them to begin to rotate. Elastomeric materials, like thermoplastic resins and fibers, are essentially linear polymers. But certain distinctive features in their molecular structure give rise to rubberlike elasticity. Conventional elastomers consist of highly flexible nonpolar macromolecules that coil up on themeselves in the unstressed condition, thereby reducing drastically the extent of intermolecular contacts. A large deformation is possible merely by reorienting the coiled molecules. Whenelongated, the macromolecular coils partially open up and becomealigned more or less parallel to the direction of elongation. This results in an increase in the number of intermolecular contacts and in the sum of intermolecular attractions. In the stressed condition, a rubber is thus harder than in the unstressed condition. The aligned configuration, however, represents a less probable state or a state of lower entropy than a random arrangement. The aligned polymer chains therefore have a tendency to return to their randomly coiled state. The large deformability of elastomeric materials is due to the presence of a certain internal mobility

Introductory Conceptsand Definitions

35

that allows rearranging the chain orientation, the absence of which in linear chain plastic materials (at normal temperature) constitutes the essential difference between the two groups. Although the aforesaid requirements are necessary conditions for ensuring a large extent of deformability, the remarkable characteristic of the rubbery state-namely, nearly complete recovery-cannot be obtained without a permanent network structure, since permanent deformation rather than elastic recovery will occur. A small amountof cross-linkage is necessary to provide this essential network structure. However, the amount of cross-linkage must be as small as possible to retain the structure; excessive cross-linkages will make the internal structure too stiff to permit even the required rearrangement of chain orientation during deformation and recovery-in other words, it will destroy the rubbery state. An example of this is best furnished by ebonite, which is a rigid plastic madeby vulcanizing natural rubber with large quantities of sulfur. Stress-Strain

Behavior

The distinction between plastics, fibers, and elastomers is most easily made in terms of the characteristics of tensile stress-strain curves of representative samples. The diagrams shownin Fig. 1.13 are most typical of those obtained in tension for a constant rate of loading. The parameters of each diagram are nominal stress (force on the specimen divided by the original crosssectional area), the corresponding nominal strain (increase in length divided by original length), and the modulus(slope of the stress-strain curve). slope of the curve near zero strain represents the initial modulus. For the present comparison we shall ignore the yield phenomenonand the fact that stress-strain curves are functions of testing temperature, rate of loading, and characteristics of the particular polymer sample (such as molecular weight, molecular orientation, and density). It maybe seen that while fibers have high, initial tensile moduli in the range 3x103 - 14x103 2 (3x104 - 14x104 kgf/cm’~), elastomers have low initial MN/m moduti 2 (71 kgf/cm2), but in tension, typically up to about a range of 7 MN/m they generally stiffen (as shownby higher moduli) on stretching. Plastics 2generally have intermediate tensile moduli, usually 3.5 x 102 - 3 x 103 MN/m (3.6x103 - 3x104 kg/cm2), and their elongation at break varies from a few percent for brittle materials like polystyrene to about 400%for tough, semicrystalline polyethylene. Somepolymers can be used both as fibers and as plastics, depending on the extent of macromolecular alignment in them. Polyamides, polyesters, and polypropylene are prime examples of such polymers used in both areas. In the fiber making process, known as cold drawing, the polymer molecules are aligned in the fiber direction, thereby increasing the intermolecular forces and crystallization. This increases the tensile strength and stiffness and reduces the elongation at break. Thus, typical poly(hexamethylene

36

Chapter 1

~ 8000 ~u 6000 94000 ~ 2000,

--/

0

800 600 ~00 200

r

T 1"0 ’ 20 ’ 30 Strain (%)

I

0

r

I

I

1

I

10 20 30 Strain(°/o) (b)

800

Elongation at break I 150

60O

~100

aoo 2oo

0

~ 5O -

i~ I

I

I

20 40 60 Strain (%)

i

I

I

80

0

200 400 600 800 Strain(°/o)

(d) Figure 1.13 Stress-strain curves. (a) Synthetic fiber, like nylon-6,6. (b) Rigid, brittle plastic, like polystyrene. (c) Tough plastic, like nylon-6,6. (d) Elastomer, like lightly vulcanized natural rubber. (1 kgf/cm2 = 0.098 MN/m2). 2adipami ~.) (nylon-6,6) fibers have tensile strengths around 700 MN/m (7.1x10 kgf/cm 2) and breaking elongations about 25%. Without cold drawing, the same polymer, used as plastics, has tensile strengths only around 2 (7.1 ×102 kgf/cm 2) and breaking elongations 70 MN/m near 100%. The macromolecules in such articles are randomly oriented and much less extended than in the fiber. Not all plastics can be converted into practical fibers, however, because the intermolecular forces or crystallization tendency may be too weak to achieve useful stable fibers by axial orientation. Synthetic fibers are

Introductory Conceptsand Definitions

37

therefore made generally from polymers whose chemical composition and geometry enhance intermolecular attractive forces and crystallization. Such polymers can be converted from plastics to fibers by suitable treatment to cause axial alignment, as explained above. On the contrary, since elastomers are necessarily characterized by weak intermolecular forces and lack of tendency to crystallize, the same polymeric types are not readily interchangeable between rubber applications and uses as fibers or plastics. Thus, polyethylene which has weak intermolecular forces but a high degree of crystallinity due to its highly symmetrical and regular molecular structure, is not capable of elastomeric behavior because the crystallites prevent easy coiling or uncoiling of the macromolecules. In contrast, random copolymers of ethylene and propylene (EPMrubbers) with the two monomers in mole ratios between 1:4 and 4:1, have no long sequences with regular geometry and are therefore noncrystallizing and elastomeric. POLYMER

NOMENCLATURE

The naming of polymers or envisaging the chemical structure of a polymer from its name is often an area of difficulty. This is because the naming of polymers has not proceeded in a systematic manner until relatively recently. Polymer nomenclature has been largely a matter of custom without any one system being universally adopted. Though a systematic IUPAC nomenclature now exists for polymers just as it does for organic and inorganic chemicals, this nomenclature is, however, rarely used for polymers because a commonnaming system is deeply entrenched through the force of usage. The commonnaming system, based either on the source of the polymer, the structure of the polymer, or trade names, usually works without difficulty because the numberof polymers that are of interest to the average worker in the field does not exceed a few dozens, and the burden of memorization is thus not excessive. However, as the newer polymeric species enter the area of commonusage the number will eventually become too large for convenience in the commonnaming system, and the IUPAC nomenclature will probably receive wider acceptance in time. For naming a polymer according to either the commonnomenclature or the IUPACnomenclature one needs to recognize the repeating unit and the parent monomerstructures. The commonnomenclature generally uses trivial names for monomers as well as the corresponding polymers and features quite a few exceptions. In the following, only the commonnaming system will be described. Common Nomenclature For the commonnomenclature the usual practice is to name a polymer according to its source, i.e., the monomer(s)used in its synthesis, and the generic term used is poly"monomer", whether or not the monomeris ¯ real. The prefix ’poly’ is added on to the name of the monomerto form

38

Chapter1

a single word, e.g., polyethylene, polystyrene, and polyacrylonitrile (see Table 1.1). However, when the monomer has a multiworded name, the name of the monomerafter the prefix ’poly’ is enclosed in parentheses, e.g., poly(vinyl chloride), poly(vinyl alcohol) and poly(methyl methacrylate) (Table 1.1). If a same polymer can be synthesized from more than one monomer, the name of the monomerthat is commonlyused to manufacture the polymer is used in the nomenclature. Thus -[-HN(CH2)sCO-],~namedas poly(caprolactam) since it is industrially made from caprolactam by ring-opening polymerization,-in preference to polymerization of the parent amino acid, H2N-(CH2)5-COOH. A few polymers are given names based on the repeating unit without reference to the parent monomer. The primary examples are silicones, which possess the repeating unit R

Thus, if R = CHa, the polymer is named as poly(dimethyl siloxane). The nomenclature of random copolymers includes the names of the monomersseparated by the interfbx -co-. Thus (XXII) is namedas poly(styreneco-methyl methacrylate) or poly(methyl methacrylate-co-styrene), depending on which of the monomersis the major component (if there is one). For alternating copolymers,the interfix -alt- is used, e.g., poly(styrene-alt-maleic anhydride) (XXIII)

C~2- CFt- CH2-C. ~ O-CH 3

~ CFI2- CH-CFI-CH-CFI 2- CH -CH- CFI - CH2 - CFI -CFI -CH ~ ~

CO CO "0"

CO CO "O"

(x~)

CO CO "0"

Introductory Conceptsand Definitions

39

Graft copolymers of A and B monomers are named poly(A-g-B) poly -graft-poly B with the backbone polymer -(-A-)n- mentioned before the branch polymer. Someexamples are poly(ethylene-g-styrene) polyethylene-graft-polystyrene and starch-graft-polystyrene. In the nomenclature of block copolymers, b or block is used in place of g or graft, e.g. poly(A-b-B) or poly A-block-poly B, poly(A-b-B-b-A)or poly A-block-poly Bblock-poly A, poly(A-b-B-b-C) or poly A-block-poly B-block-poly C), and so on. Thus the triblock polymer (XXIV)is called poly(styrene-b-butadiene-bstyrene) or polystyrene-block -polybutadiene-block-polystyrene. Whensuch polymers are articles of commerce they are usually designated by the monomer initials; thus, structure (XXIV) would be named SBS block copolymer.

-"(" CH 2 ~-~ CH 2- CH’-C H - CH 2-’)~ CH 2- C~-)~" z

(XXIV) Condensation polymers are frequently named from the internal linking group between hydrocarbon portions. Thus, (III) is pob’esrer which ca n be written as -(-R-O-CO-R’-)n-- and (XX) is polyarnide written as -(- -R-NH-CO-R’-}n-. Similarly, -(-R-O-CO-NH-R’-)rr- is a polyurethane, -(-R-NH-CO-NH-R’-)n-is a polyurea, and -(--R-SO2-R’-}n- is a polysulfone. In the commonnaming system, condensation polymers are named by analogy with the lower-molecular-weight esters, amides, and so on. Thus, since the namesof all esters end with the suffix ’ate’ attached to the name of the parent acid (e.g., ethyl acetate), polymer(III) is namedpoly(ethylene terephthalate) according to the parent acid terephthalic acid, which is a para diacid. The word ’ethylene’ here implies ’ethylene glycol’ because the alcohol used must be a glycol if the polymer is to be linear. Similar reasoning is also followed in naming polyamides. Thus, the word ’hexamethylene’ in poly(hexamethylene adipamide) obviously implies hexamethylene diamine because the polymeric structure could be made by considering hexamethylene diamine H2N(CH2)6NH~,and adipic acid, HOOC(CH~)4COOH. dibasic acids are named according their trivial names HOOC-COOH oxalic acid, HOOC(CH2)COOH malonic acid, HOOC(CH2)2COOH succinic acid, and so on. There is an alternative naming system for synthetic polyamides from unsubstituted nonbranched aliphatic monomers. A polyamide made from either an amino acid or a lactam is called nylon-z, where z is the number of carbon atoms in the repeating unit. A nylon made from a diamine and a dibasic acid is designated by two numbers, in which the first represents the number of carbons in the diamine chain and the second the number of carbons in the dibasic acid.

40

Chapter 1

Problem 1.8 Namethe polyamides made from the following monomersand drawtheir structural formulas(one repeating unit). (a) Caprolactam;(b) to-aminoundecanoicacid; (c) dodecyl lactam; (d) diamine and sebacic acid, and (e) Ethylene diamine and decanedioic acid.

(a) Nylon-6: -[-NH-(CH2)5-CO-]n(b) Nylon-ll : -[-NH-(CH2)I0-CO-]n(c) Nylon-12: (d) Nylon-6,10: -[-NH-(CH2)6-NH-CO-(CH2)8)-CO-]n(e) Nylon-6,12: -[-NH-(CH2)6-NH-CO-(CH2) 10CO-In-

There are also a few commonpolymers whose accepted names convey relatively little information about the repeating unit structure. The primary examples are polycarbonate, poly(phenylene oxide), polyamide-imides, polysulfones, and polyether ketones (see Table 1.3). A simple repeating unit and name cannot be written for polymer blends and/or graft copolymers such as ABS(acronym for acrylonitrile-butadienestyrene), which are an important class of thermoplastics. Since there are no codified rules for the commonnaming system, this brief review has outlined the general practice which applies to most polymers. It has also highlighted several exceptions where the system breaks down or inconsistencies occur in which case resort is made to generally accepted conventions for assignment of names to particular polymers. The commonnomenclature is thus arbitrary in the final analysis.

BIBLIOGRAPHY BrydsonJ. A. Plastics Materials. 3rd ed. Chap. 2. Butterworths, London(1975). Flory P. J. Principles of PolymerChemistry.Chap.2. Cornell University. Press, Ithaca, N. Y., (1953). HiemenzP. C. PolymerChemistry, The Basic Concepts. Marcel Dekker, NewYork (1984). Odian G. Principles of Polymerization. 3rd ed. Chap. 1. John Wiley, NewYork (1991). Rudin A. The Elements of PolymerScience and Engineering. Chap. 1. Academic Press, NewYork (1982). SeymourR. B. and Carraher C. E. Jr. PolymerChemistry, an Introduction. Marcel Dekker, NewYork (1981).

Introductory Concepts and Definitions

41

EXERCISES 1.1. Represent, by showing a repeating unit, the structure of the polymer which would be obtained by polymerization of the ftllowing monomers: (a) w-aminolauric acid; (b) lauryl lactam; (c) ethylene oxide; oxacyclobutan; (e) ethylene glycol and terephthalic acid; (f) hexamethylene diamine and sebacic acid; (g) ethylene glycol and phenylene diisocyanate; (h) m-phenylene diamine and isophthaloyl chloride 1.2. Drawthe structural formula (one repeating unit) for each of the following polymers: (a) poly(4-methylpent-l-ene); (b) poty(chlorotrifluoroethylene); (c) poly(vinyl ethyl ether); (d) poly(vinylidene chloride); (e) polyethyleneimine; (f) poly(methyl-2-cyanoacrylate); (g) polychloroprene; (h) poly(butylene thalate); (i) poly(1,2-propylene oxalate); (j) poly(dihydroxymethylcyclohexyl terephthalate); (k) polycaprolactam (nylon-6); (1) polyformaldehyde; (m) oxymethylene; (n) poly(propylene oxide); (o) poly(propylene glycol); (p) phenylenesulfone); (q) poly(dimethyl siloxane); (r) Poly(vinyl butyral); (p-phenylene); (t) poly(p-xylylene); (u) polycaprolactone 1.3. What is the degree of polymerization of each of the following polymers with molecular weight 100,000: (a) polyacrylonitrile (b) polycaprolactam (c) poly(trimethylene ethylene-urethane) 1.4. What is the functionality of the following monomersin reaction methyl methacrylate and (b) ethylene glycol (i) Divinyl ,benzene (ii) Maleic anhydride (iii) Phthalic anhydride (iv) Acrylic acid 1.5. What is the functionality

with (a)

of the monomershown

CH2 CH2-COOH I II H~N-CH2-CH~- C- CH2 ---C = CH2 (a) in a free radical or ionic addition reaction through C = C double bonds, (b) in a reaction that produces amide linkages, (c) in a reaction that produces ester linkages 1.6. What is the acid equivalent of polycaprolactam (nylon-6) with average D_P 500 ? 1.7. Howwould you determine experimentally whether the polymerization of an unknown monomer was proceeding by a step or a chain mechanism ?

42

Chapter I

1.8. (a) What is the functionality of the diglycidyl ether of bisphenol A (I) hardening reaction with diethylene triamine (II)?

ICH C~2~,CH2 0

CH2~

O I ~C ~ ~O-CH2-CH-CH NO/ CH 3 (I)

H2N-CH2-CH2-NH-CH2-CH2-NH2 (II) (b) Whatis the functionality of (II) in this reaction 1.9. Each of the following polymers can be synthesized from different

monomers

(a) -{--O(CH2)sCO--}-¢-~ (b) -(-OCH2 CH2CH2--)w(c) --[--NH-( CH2) 7-CO-]~-,~ Show by equations the overall chemical reactions involved in the synthesis of these polymers. 1.10. Classify the polymers in Exercise 1.1 as to whether they are condensation or addition polymers. Classify the polymerizations as to whether they are step, chain, or ring opening polymerizations. 1.11. Which of the following materials is most suitable for the manufacture of thermoplastic pipe ? Briefly tell why. (a) CHa-(’CH2-CH2--) 14-CH3 (b) CH3-(--CH2-CH2-)6,000 --CH3 (c) CHa-(-CH2-CH~-) 100,000-1.12.Namethepolymers obtained in Exercise I.Iaccording to theirsource. 1.13. Writerepeating formulasand namesbasedon commonnomenclature (nonIUPAC)for (a) Nylon-6; (b) Nylon-6,6; (c) Nylon-ll; (d) Nylon-6,10; (e) Nylon-5,7

Chapter

2

Chain Dimensions, and Transitional

Structures, Phenomena

INTRODUCTION As the size and shape of a polymer chain are of considerable interest to the polymer scientist, it is useful to know the factors that govern these properties. Weshall, however, confine ourselves to the models of the random coil for a polymer chain, as this is usually believed to be most appropriate for synthetic polymers; other models-rods, discs, spheres, spheroids--are also postulated, but these need not concern us at this level. The notion that a single polymer molecule assumes random coil shape due to random conformations of the polymer chain leads to the premise that polymer chains in aggregated, unordered (amorphous) state also assume random conformations. The amorphous state, where the molecules are relatively distant and packed in a random manner, represents one extreme in the physical state of the polymer. This is the state that exists in such amorphousstates as solution, melts, or some solids, the randomnessbeing induced by thermal fluctuations. The other extreme is the case where the molecules are able to pack closely in perfect parallel alignment as is found in those polymers that exhibit fibrous behavior-that is, in those possessing a high degree of crystallinity and crystal orientation. A question maybe asked: if polymers are composedof long chains (typically several thousand Angstroms), howdo they fit into unit cells that are knownto be only several Angstroms in size? The answer is that the unit cell contains only a few mers [1], which are repeated in adjacent unit cells. If the adjacent unit cell is considered to repeat in the axial direction, its relationship to the whole chain "is moreeasily visualized. In between these two extremes of amorphous and crystalline polymers there is a wide spectrum of polymeric materials with different degrees of 43

44

Chapter2

crystallinity and amorphouscharacter. These are called semico,stalline. In the cases cited above, two opposing factors appear to be in operation, namely, those that favor strong intermolecular bonding forces and structural regularity contributing to a higher degree of crystallinity and those that favor weak intermolecular bonding forces and structural irregularity contributing to a higher degree of amorphous character. The arrangement of the molecules en masse depends on these two factors which, one might say, involve the molecular state. On the other hand, the morphology and order characteristics of a collection of molecules, in addition to molecular conformation, form the basis of a third factor that influences the physical state of the system. In this chapter we shall be concerned with both these states of the polymer, with a greater focus, however, on the former.

POLYMER CHAINS: DIMENSIONS

STRUCTURES

AND

Somemeasure of polymer size is obtained from molecular weight, but what is the actual length of the molecule and what is its shape? Before trying to answer these questions we should first consider a simple molecule such as butane and examine the behavior when the molecule is rotated about the bond joining two adjacent carbons. This rotation produces different conformational states of the molecule. Conformational

Changes

The term conformation [2] is used to describe the spatial arrangements of various atoms in the molecule that may occur because of rotations about single bonds. (Note that there is no rotation about double bonds, and ~o making a change in the spatial arrangement of atoms in a molecule containing double bonds would need breaking a bond. The term configuration is used to describe structures in such cases where the geometric variations can only be interchanged by breaking a bond. Examples are the cis-trans isomerism in polyisoprene and stereoisomerism in polypropylene, which are discussed in a later section.) The possible coformations of a n-butane molecule produced by rotations about the bond joining carbons 2 and 3 are illustrated in Fig. 2.1. Each carbon atom in the butane molecule is spa hybridized and so is tetrahedral with bond angles of 109.5°. The stabilities of different conformations which arise due to rotation about the C2-C3 bond depend upon the steric interactions between the methyl groups and the hydrogen atoms bonded to C2 and C3. The methyl groups are relatively bulky and so the planar cis conformation (also called eclipsed conformation because the groups would be visually eclipsed if viewed along the bond of rotation), corresponding to their distance of closest approach, is the least stable. Conversely, theplanar

45

ChainDimensions,Structures, and Transitional Phenomena

trans conformation, where the bulky groups are farthest apart, is the most stable. The Newmanand "saw horse" projections of n-butane in these two conformations are shown Fig. 2.2. The potential energy of the molecule is given in Fig. 2.3 as a function of the angle ~b through which the Ca-C4 bond is rotated from the planar trans conformation (q5 = 0) about the plane of the CI-C2-Ca bonds. The minima correspond to the staggered conformations, namely, planar trans = 0) and gauche 4- (q5 = 4- 120°), in which there is maximum separation the substituents on C2 and Ca, i.e., the methyl groups and hydrogen atoms are at 60° to each other. All three can be regarded as discrete rotational states. The staggered conformations are shown as Newmanprojections in Fig. 2.4 along with the eclipsed conformations that give rise to the maxima in Fig. 2.3. The restrictions imposed by short range steric interactions upon the conformations of a polymermolecule occur at a local level in short sequences of chain segments. There is, however, an interdependence of local chain conformations, that is, the conformation about a given chain segment is dependent upon the conformations about the segments to which it is directly connected (see Problem2.1). Such interdependent steric restrictions affect the local chain conformationsall along a polymerchain and have a significant effect upon chain dimensions.

CH3 ~ CH2 --

t I

CH2- CH 3

t 2

3

4 C~

CI

O

/~ Planar

C3 \ ii \I

ci_~s C1-cz, distance = 2.6A

I I I I

Planar trans Cl-Cz, distance =3.9~, Figure 2.1 Effects of bondrotation uponthe conformationof n-butane molecule.

46

Ch~pte~2

1 3 ---~ CH

H

H

(a) H

H

H/~H

4(~H3

4 ~ cH3 H H cH3 (b)

(~..

H3C~k, ~

"~

Figure 2.2 Newmanand "saw horse" projections

for n-butane. (a) A staggered state (planar trans) with angle of bond rotation ~b = 0 and (b) an eclipsed state (planar c/s) with ~b = 180%

2O

~ 15-~

p!.anar

/

710 c"

c 5 13.

0° -180

-120°

Figure 2.3 Potential ~b of bond rotation.

, \/

°-60

0

i ,

°60

°120

°180

energy of a r~-butane molecule as a function of the angle

Chain Dimensions, Structures,

H CH 3

CH3

H3CCH3

~ =-60 o

47

and Transitional Phenomena

H o

H CH3

~ =1800

~ = 60

Eclipsed conformations CH 3 H3C

~

CH3

CH 3

H i

H~CH3

H~ H

CH 3 ° ~:0

° ~=-120

H

° ~=120

Staggered conformations

Figure 2.4 Newmanprojections

of the eclipsed and staggered conformations of

a r~-butane molecule.

Problem23 The interdependence of local chain conformations along a polymer chain is most easily appreciated considering the case of rz-pentane. Represent in skeletal form the conformations of n-pentane in terms of sequences of planar trans (t) and gauche ± (g±) conformations. Answer: Every group of four atoms in a chain has a choice of three possible stable rotational states, namely t and g± (e.g., staggered conformations in Fig. 2.4 for r~-butane). There are thus a total of 9 conformations for n-pentane involving and g± and these comprise 5 distinct conformations. If the three central carbon atoms in rz-pentane are fixed in the plane of the paper, then the five distinct conformations of the molecule can be represented in skeletal form as shown in Fig. 2.5. The g-g+ and g+g- conformations bring the methyl groups into close proximity (with their carbon atoms about 2.5 ~. apart) and so are of high energy and low probability in comparison to the other conformations.

48

Chapter2

tg- (= g+t

g+g+(=g-g-)

g-g+(=g+g-)

Figure2.5 Thefive distinct conformationsof n-pentane. The short-range interactions, as shownabove, are important in determiningthe relative probabilities of existenceof different conformations in a polymerchain. Thusthe ratio of the numberof trans (r~t) to gauche states is given by ng/nt = 2 exp(-Ae/kT) (2.1) where k is the Boltzmannconstant, Ae is the energy difference between the energyminimain the trans and gauchestates, and the factor of 2 arises becauseof the -I- gauchestates available. WhenAeis less than the thermal energy kT provided by collisions of segments, none of the three possible staggered forms will be preferred. If this occurs, the overall conformation of an isolated polymer molecule will be a randomcoil. WhenAe > leT, there will be preferencefor the trans state. Polyethylene The potential energy diagramfor n-butane, shownin Fig. 2.3, is symmetrical. The diagram will vary with the type of molecule and need not be symmetrical,but the butane diagramis very similar to that for the simple polymer polyethylene -(-CH2-CH2--)n-, if the -CH3groups on C2 and Ca (in Fig. 2.1) are replaced by two sections of the chain adjoining the bondof rotation. Thedistribution of trans (t) and gauche(9) states along a chain will be a function of.the temperatureand the relative stability of these states. Consequently,there is an unequaldistribution of each.

Chain Dimensions, Structures,

and Transitional Phenomena

49

Problem2.2 For polyethylene,

the energy difference between the gauche and trans states is about 3.34 kJ/mol. Calculate the ratio of the number of trans and 9auche states along a chain at 100, 200 and 300°K. A1]swer

:

k = 1.38x10 -23 -1 JK -I) -1) = (3.34x 103J tool / (6.02 x 1023 molecules tool .20 -1 = 0.55 x 10 J (molecule)

Ae =3.341dmo1-1,

~,e

From Eq. (2.1):

%/n~-- 2 exp[(-0.55 x 10-2°)/(1.38 x -~3 x T) T : ng/n~ :

100°K 0.035

200°K 0.264

300°K 0.524

At low temperatures, trans states are thus preponderant.

Since the backbone of polyethylene molecule is composed of a chain of tetrahedral carbon atoms, the molecule in all-trans conformation has a linear zigzag structure (like corrugated sheets) shown in (I). (The dotted lines denote bonds below and the wedge signifies bonds above the plane of the page.)

(I) Linear polyethylene is thus capable of close-packing into tight unit cell and is highly crystalline, despite its low cohesive energy, and the alltrans zigzag form is the shape of the molecule in crystalline regions of polyethylene. However, at higher temperature the ng/nt ratio increases (see Problem 2.2) showing that the chain becomes less extended and more coiled as the temperature increases. Since rotation is possible about carbon-carbon bonds, the chain is in a state of perpetual motion, constantly changing shape from one coiled conformation to another conformation, which are equally probable at the given temperature. The speed of this wriggling varies with temperature

Chapter 2

50

and also from one polymer to another. The height of the potential energy barrier AE (cf. Fig. 2.3) determines the rate of bond interchange between the trans and the gauche states. If AE is less than the thermal energy kT, the barrier height is not significant and trans/gauche isomerizations will take place in times of the order of 10-11 second. For polyethylene/~E is about 16.7 kJ/mol, i.e., 2.8x10 -2° J per molecule, which is more than kT at room temperature. The higher the value of /kE, the more difficult is this rotation, but as the temperature is raised the fraction of molecules which possess energy in excess of ZXE increases and rotation from one state to another becomes easier.

CH3 H Chain J -’------"-~"

(a)

J Chain C n-~ C n+l’’~"~

I I

CH3 H Chain

(b) .~Cn+1 H3 c~ ~

"CH3

H3

CH3

H Gauche

H3

H3 H

Trans or anti

Gauche

Figure2.6 (a) Schematic diagram showing two adjacent carbons, C,, and C,~+1, the main chain of isobutylene. (b) Newmanprojections of staggered conformations of adjacent carbons in the main chain.

51

Chain Dimensions,Structures, and Transitional Phenomena Polyisobutylene

To describe the effects of steric restrictions in another polymer, polyisobutylene, consider the Newman projections of the staggered conformations of two adjacent carbons in its repeat unit, as shownin Fig. 2.6. Here the chain substituent on the rear carbon shown is either between a methyl group and polymer chain or between two methyl groups on the front carbon. There is no significant energy difference between the conformers. Since no conformation is favored, polyisobutylene will tend to spiral into a helix (gauche conformers) as well to form into a zigzag (II),

C~3 """ CH3

C.H3 CH3 C.H3 CH3 C~1-t3

~’ "’"

c

/c H~’

""’H

H~’

~’ """

H~’

’"’H

CH3

"" "/"~’C/:""c "’"H

H~’

"’"H

(II) and thus would assume a random coil formation. Rotations between staggered conformations require sufficient energy for the chain to overcome the high barrier represented by the eclipsed forms. At room temperature, however, interconversion between conformers can occur 104 times per second or more, and the chain is thus randomly coiled and highly flexible, exhibiting rubbery properties-but the polymer does not retain its elastic character at low temperatures.

Problem2.3 Accountfor the fact that a stretched polyisobutylene molecule has ¯ a tendencyto revert to the coiled conformation. Answer: Whenstretched, polyisobutylene assumesthe linear conformation(II) exclusively and is thus sufficiently symmetricalto crystallize. Havingonly one conformation, the entropy of the extended state is zero. The entropy changeof stretching (AS) is thus negative and the free energy change (AG) given by AG-TAS is positive. So the extended chain has higher free energy than the randomcoil and tends to revert spontaneouslyto the latter.

52

Chapter 2

H 3

H3~HC~

H

H 3

H

H

~

2

H

~

H

CH3

3

"~H

H

H

CH3

H 3

H H

5

6

Figure 2.7 Newmanprojections of the six possible,conformations of polypropylene. Polypropyle~e Consideringagain two adjacent carbonsin the mainchain of the polymer,six conformationsare nowpossible becauseof the presence of an asymmetrically substituted carbon atom, as shownin Fig. 2.7. Forms 1 and 6 can be neglectedfor steric reasons;so four different conformations are still possible for the polymer.Atactic polypropylene(see Stereoisomerism)has two trans forms(2 and 5) in the fully extendedstate (III) C.H 3 H

¯

:,,.

C.H 3 H

~’c" ""H

H#’

H 3

CH

-/

C’~

C.H 3 H

\’"C

\ "

"’"H

’" ’" C

\

c

H~1’ (III)

H...

~’ ""’H

H

"’"H

H

Chain Dimensions,Structures, and Transitional Phenomena

53

and so, unlike polyisobutylene, is incapable of crystallizing upon being stretched. Isotactic polypropylene, however, having all the methyl groups on one side, crystallizes easily.

POLYMER

CONFORMATIONS

IN

THE

BULK

STATE

Ever since H. Staudinger developed the macromolecular hypothesis in the 1920s [3], polymer scientists have wondered about the spatial arrangement of polymerchains, both in dilute solutions and in the bulk. The development of the random coil by H. E Mark and the many further developments by E J. Flory [4-7] led to a description of the conformationof chains in the bulk amorphousstate. Neutron scattering studies revealed that the conformation in the bulk is close to that found in solution in (0-solvents (see Chapter 3), thus, strengthening the random coil model. On the other hand, some workers suggested that the chains have various degrees of either local or long range order [8-10].

Polymer Conformations

in Crystals

The conformation of a polymer in its crystal will generally be that with the lowest energy consistent with regular placementof structural units in the unit cell. Helical conformationsoccur frequently in pdlymer crystals. Helices are characterized by a number fj where f is the number of monomer units per j number of complete turns of the helix. Thus, polyethylene could be characterized as a 11 helix in its unit cell with an all-trans conformation. The arrangement of the molecules in the polyethylene crystal structure is illustrated in Fig. 2.8. The van der Waals radii of the chain substituents affect the intermolecular space requirements. Thus, since fluorine atoms are significantly larger than hydrogenatoms, an all-trans crystal conformation of polyethylene is too crowdedfor poly(tetrafluoroethylene), which therefore crystallizes instead in a very extended helical conformation that allows the larger F atoms to be accommodated. Below 19°C the molecules are in the form of a 136 helix and at higher temperatures they untwist slightly into a 157 helix. Isotactic polypropylenecrystallizes as a 31 helix, as illustrated in Fig. 2.9, because the bulky methyl substituents on every second carbon atom in the polymer backbone force the molecule from an all-trans conformation into a trans/gaucheltrans/gauche.., sequence with angles of rotation of 0° (trans) followed by a 120° (gauche) twist. The simple helix can be generated by imagining the successive bonds being wrapped around a triangular mandrel [see Fig. 2.9(b)], every other bond lying on a face and the alternating bonds lying on an edge [11]. As seen in an end view [Fig. 2.9(b)], the pendant methyl groups form a second triangular prism that contains the smaller one.

54

Chapfer 2

In syndiotactic polymers(see Stereoisomerism) the substituents are farther apart because the configurations of successive asymmetriccarbons alternate and so the trans/trans/trans.., planar zigzagconformation is generally the lowest energy form. This conformationis observedin syndiotactic poly(vinyl chloride) and 1,2-polybutadiene. Syndiotactic polypropylenecan also crystallize in this conformationbut a trans/trans/gauche/gauche .... sequenceforminga 21 helix is slightly favored energetically. A 21 helix, as shownin Fig. 2.10, can be generated by wrappinga chain around a mandrel with a squarecross section [11]. Unlikethe 31 helix, the 21 helix has bonds only on faces and not on the edges of the mandrel. It is important to note that only single-bond rotations and no bondbendingor stretching are involved in achieving these conformations.

~ ~ ~o

.--o

(a)

(b)

Figure2.8 Crystal structure of orthorhombic polyethylene. (a) Perspective view of the unit cell of polyethylene. (b) View along c (chain axis).

Chain Dimensions, Structures,

and Transitional

Phenomena

55

Figure2.9 (a) All-trans (linear zigzag) conformation o! lsotactic

polypropylene. The large spheres represent the pendant methyl groups and filled circles the hydrogens. (b) The 31 helix of isotactic polypropylene with hydrogens omitted. (c) The pendant methyl groups as a helix. (From Ref. 11)

Cb)

Figure2.10 (a) All-trans (linear zigzag) conformation of syndiotactic pqlypropylene. (b) The 21 helix of syndiotactic

polypropylene. (From Ref.

56

C]3apter 2

Problem2.4 What is the repeat distance between pendant methyl groups that form a row (a) in the 31 helix of isotactic polypropylene and (b) in the 21 helix syndiotactic polypropylene. Assnmeeach carbon-carbon bond length is 1.54 ,~and °each . bond angle is 109.5 Answer: (a) Consider methyl groups 1 and 4 in Fig. 2.9(b). The linear distance between them is made of 3 fidl bond lengths (1.54 ,~) lying on the edges of the triangular mandrel and 3 projections of bond lengths on the edge. Repeat distance = 3 (1.54 .~) + 3 (1.54 ~,) [cos ° - 10 9. 5°)] = 6.16 ~ (b) Consider methyl groups 1 and 3 in Fig. 2.10(b). diagonal ’a’ represents the bond length 1.54 ~.

V"~c/2

= a.

cos

--

In the diagram below

= 0.577a

So, c = 0.815a Also,a2 = b2 + c2, so b2 2 = a2[l- (0.815) b = 0.577a = 0.577 (1.54 .~) = 0.890 Repeat distance

= 4b = 4 (0.890 ~) = 3.55

In polymers having polar groups, intermolecular electrostatic attractions exert strong influence on chain conformation in their crystals. In polyamides, hydrogen bonds form between the carbonyls and NH groups of neighboring chains (Fig. 2.11) and influence the crystallization of the polymer in the form of sheets, with the macromolecules themselves packed in planar zigzag conformations [see Fig. 2.11(b)].

Chain Dimensions, Structures,

57

and Transitional Phenomena

O= C 6+

(a)

I

N~H .....

O~C

~+

N~H ....

O=C

÷ I 6

N~H ....

(b)

Figure 2.11 (a) Hydrogen bonds between neighboring chains of polyamide. (b) Arrangement of chains in hydrogen-bonded sheets in the crystal nylon-6,6.

structure

of

Problem2.5 Explain the fact that conversion of the amide groups -CONHin nylon to methylol groups -CON(CH2OH)-by reaction with formaldehyde, followed by methylation to ethel groups -CON(CH2OCH3)-results in the transformation of the fiber to a rubbery product with low modulus and high elasticity. Answer: Replacing the -NH- hydrogen in polyamides by an ether group curtails

the in-

58

Claap~er2

termolecular hydrogenbonding. Henceat low degrees of substitution the modulus is reducedand a moreelastic fiber is obtained. As the substitution increases, the crystallinity is completelydestroyed and rubbery property appears.

Our discussion above on polymer conformations in single chains and in crystals has assumed regularity of macromolecular structure. However, irregularities such as inversions of monomerplacements (head-to-head instead of head-to-tail), branches, and changes in configuration mayoccur. These irregularities, which are considered in a later section, mayinhibit crystallization and have a profound effect on polymer properties. Polymer

Size

in the Amorphous State

The subject of polymer size or chain dimensions is concerned with relating the sizes and shapes of individual polymer molecules to their chemical structure, chain length, and molecular environment. The shape of the polymer molecule is to a large extent determined by the effects of its chemical structure upon chain stiffness. Polymerswith relatively flexible b~,ckbones tend to be highly coiled and can be represented as randomcoils. But as the backbone becomes stiffer, e.g., in polymers with more aromatic backbone chain, the molecules begin to adopt a more elongated wormlike shape and ultimately becomerodlike. However, the theories which are presented below are concerned only with the chain dimensions of linear flexible polymer molecules. More advanced texts should be consulted for treatments of wormlike and rodlike chains. The main quantitative developments of the randomcoil model of flexible polymers began in 1934 with the work of E. Guth and H. E Mark [12] and W. Kuhn[13]. Using the concept of free rotation of the carbon-carbon bond, Guth and Mark developed the idea of the "random walk" or "random flight" of the polymer chain, which led to the familiar Gaussian statistics of today, and eventually to the famous relationship between the end-to-end distance of the main chain and the square root of the molecular weight, described below. _~eel3~

Jointed

Chains

The simplest measure of chain dimensions is the length of the chain along its backbone and is knownas the contour length. For a chain of n backbone bonds each of length l, the contour length is nl. However, because of the fixed bond angle (109.5 °) of carbon, the maximumend-to-end distance of the polymer chain will be somewhat less than nl (see Fig. 2.12 and Problem 2.6). For linear flexible chains that are more like random coils, the distance separating the chain ends, i.e., the end-to-end distance r (Fig. 2.13) will be even considerably less than

Chain Dimensions, Structures,

and Transitional

Phenomena

59

e-~sin 2 Figure2.12 Skeletal representation of polyethylene chain in planar zigzag form.

Figure 2.13 Schematic representation

of a coiled polymer molecule showing the

end-to-end distance.

Problem2.6 ~For a linear molecule of polyethylene of molecular weight 1.4 × 10 what would be the end-to-end distance of the polymer molecule in the extended (all-trans) state, as compared to the contour length of the molecule ? Answer: The polyethylene molecule may be represented skeletally in a planar zigzag form as shown in Fig. 2.12, where l = 0.154 nm and 0 = 109.5 °. In order ,to perform the calculation, the number r~ of backbone bonds is required. It can be obtained

60

Chapter2

from the molar massof the molecule since there is only one methylenegroup per backbonebond(neglecting chain ends, which makean insignificant contribution to molecularweight for long chains): n = 1.4x10~/14 = 10,000 Thecontour length is the length of the moleculealong its backboneand so is given by Contour length = nl = 10,000 (0.154 nm)= 1540 Since each bond in the fully extended molecule has a projection of I sin(0/2), the end-to-end distance r is given by r = (10,000) (0.154 nm)sin (109.5°/2) = 1258 (Magnifieda million times, the chain could be represented by a piece of wire 1.26 mlong and 0.3 mmin diameter.)

Whenconsidering an isolated polymer molecule it is not possible to assign a unique value of r because the chain conformatioff (and hence r) is continuously changing due to rotation of backbone bonds. Since the single polymer chain can take any of an infinite number of conformations, an average magnitude of r over all possible conformations is computed from the meanof the squares of end-to-end distances and is called the root mean square (RMS)end-to-end distance, represented by (r 2) 1/2, where () meansthat the quantity is averaged over time. It is given by a rather simple equation (see Appendix 2.1):

(r2}}/2 =

(2.2)

where the subscript f indicates that the result is for a freely-jointed chain. Equation (2.2) reveals that (r~)}/~" is a factor of n1/2 smaller than the contour length nl. Since n is large, this highlights the highly coiled nature of flexible polymer chains. In addition to the RMSend-to-end distance, the dimensions of linear chains are often characterized in terms of the RMSdistance of a chain segment from the center of mass of the molecule. It is defined as the square root of the average squared distance of all the repeating units of the molecule from its center of mass and is known simply as the RMSradius of gyration, (S 2)1/2. It is thus given by [14]

(s 2>

E i=1

(2.3)

where si is the vector distance from the center of mass to the ith unit of the chain in one particular conformation of the molecule and n is the total number of units. The angular brackets denote a linear average over all possible conformations of the molecular chain.

Chain Dimensions,Structures, and Transitional Phenomena

61

The radius of gyration has the advantage that it also can be used to characterize the dimensions of branched macromolecules (which have more than two chain ends) and cyclic macromolecules (which have no chain ends). Moreover, properties of dilute polymer solutions that are dependent on chain dimensions are controlled by (S 2)1/2 rather than 1/2. (r2) The radius of gyration is directly measurable by light scattering (see Chapter 4), neutron scattering, and small angle scattering experiments, whereas the end-to-end distance is not directly observable and has no significance for branched species which have more than two ends. However, for high-molecular-weight linear macromolecules that have random coil shapes, (S 2>1/2 is uniquely related to (7~2} 1/2 by

<s2)’l

(2.4)

and so in the theoretical treatment of linear flexible chains it is usual to consider only @2)1/~. The end-to-end distance is also more readily visualized than the radius of gyration and is more directly applicable in the molecular explanation of rubber elasticity discussed later in this chapter. Real Polymer

Chains

While the freely jointed chain is a simple model from which to begin predictions of chain dimensions,it is physically unrealistic. Since each carbon atom in a real polymer chain is tetrahedral with fixed valence bond angles of 109.5°, the links are subject to bond angle restrictions. Moreover, the links do not rotate freely because, as we have seen earlier, there are energy differences between different conformations (cf. Fig. 2.3). Both of these effects cause @~)1/2to be larger than that predicted by the freely jointed chain model. The simplest modification to the freely jointed chain model is the introduction of bond angle restrictions while still allowing free rotation about the bonds. This is known as the valence angle model and for a polymer chain with backbone bond angles all equal to 0~ it leads to Eq. (2.5) for the mean square end-to-end distance (2.5) = nl2 (11- + c°s00) cos where the subscript fa indicates that the result is for chains in which the bonds rotate freely about a fixed bond angle. Since 180° > 0 > 90°, cos 0 is negative and (r2> is greater than n’/2 of the freely jointed chain model [Eq. (2.2)1. For polymers having C-C backbone bonds with 0 ~-’_ 109.5 ° for which cos 0 ~_ --g,1 the equation becomes (~’2>y a = 2nl 2 (2.6) (r2)]~

62

Chapter 2

Thus for polymers such as linear polyethylene, bond angle restrictions cause the RMSend-to-end distance to increase by a factor of ~ from that of the freely-jointed chain. sProblem2.7 For a linear molecule of polyethylene of molecular weight 1.4× l0 what wouldbe the RMSend-to-end distance according to the valence angle model as comparedto that accordingto the freely-jointed chain modeland the end-to-end distance of a fully extendedmolecule. Comment on the values obtained, indicating whichone is a morerealistic estimate of chain dimensions. Answer: Lengthof each bond, l ----- 0.154 nm(see Fig. 2.12) Numberof bonds, r~ = 10,000 (see Problem2.5) End-to-enddistance of a fully extended molecule---- 1258nm(see Problem2.5) The RMS end-to-end distance, according to freely jointed chain model, is obtained from Eq. (2.2): (r2)~/2 = (10,000)V~"(0.154 nm)=15.4 The RMS end-to-end distance according to valence angle model is obtained from Eq. (2.5) (r 2,1/2 ),~a =(10,000)½ (0.154 nm){1°)½ cos(109.5°)} 1 ~ cos(109.5 ---- 21.8 nm Comment:The lowest energy state for individual sequences of three bonds correspondsto the planar trans conformation,and so the molecularconformationof lowest energy for the polymeris the planar zigzag, as shownin Fig. 2.12. However, the chain coils due to the influence of entropy, and it is the contribution from entropy whichresults in there being an extremelylow probability for existence of the planar zigzag conformation.The mostrealistic of the abovethree estimates of chain dimensionsis that afforded by the valence angle model. This is because it takes into accountcoiling of the moleculeand also restriction due to fixed valence bondangle.

The valence angle model, though more realistic than the freely jointed model, still underestimates the true dimensions of polymer molecules, because it ignores restrictions upon bond rotation arising from short-range steric interactions. Such restrictions are, however,moredifficult to quantify theoretically. The usual procedure is to assume that the conformations of each sequence of three backbone bonds are restricted to the rotational

63

Chain Dimensions,Structures, a~d Transitional Phenomena

Table 2.1 Typical Values of cr and C’~ for Common Polymers Polymer Temperature (°C) Polyethylene 140 Polypropylene(isotactic) 140 Poly(vinylchloride) 25 Poly(methylmethacrylate) 25 Poly(vnethylmethacrylate) 72 Polystyrene 25 Polystyrene 70

~r 1.8 1.6 1.8 2.1 1.8 2.3 2.1

C~ 6.8 5.2 6.7 8.6 6.6 10.8 9.2

isomeric sates that correspond to the potential energy minima such as those shownfor n-butane in Fig. 2.3. For the simplest case of polyethylene and for vinylidene-type polymers, the rotational isomeric state theory leads to the following equation

+ where the subscript ha indicates that the result is for a polymer chain with hindered rotation about a fixed bond angle. The quantity cos qb is the average value of cos qS, where q~ is the angle of bondrotation as defined for n-butane (see Figs, 2.3 and 2.4). For unrestricted bond rotation, cos qb = since all values of ~b are equally probable causing the positive and negative cos q~ contributions to cancel each other out. Equation (2.7) then becomes identical to Eq. (2.5). However, due to short-range steric restrictions, values of I~1 < 90° are favored so that cos ~b is positive and (r2)ha is greater than (r2)ya. Moreover, the presence of bulky side groups on the polymer chain, e.g., phenyl group in polystyrene, introduces additional interdependent steric interactions involving side groups and produces further hindrance to bond rotation. Since these effects are very difficult to treat theoretically, Eqn. (2.7) is usually written in a more general form

-+

where cr is a steric parameter and is the factor by which @~)o1/~ exceeds ~/2 measured experr 2~1/2 2Ia" It is usually evaluated from values of @~}o imentally (the subscript o indicating that the result is observed and not calculated) and typically has values between 1.5 and 2.5 (see Table 2.1).

Chapte~ 2

64 An idea

of the stiffness

of a polymer chain can be gained

from the

ratio (r )2 _/( ol/2 )f r2 1/2, which is square the of the root characteristic ratio, 2, Coo ---- (rU)o / and in dicates ho w much gr eater th e RMS en d-to-end distance of a real polymer chain is, compared to that of the freely jointed chain. From the values of Coo given in Table 2.1 it is seen that this ratio is 2-3. Thus, the combination of fixed bond angles and short-range steric interactions causes the end-to-end distances of real polymer chains to be greater than those of freely jointed chains by factors of 2-3.

Problem2.8 A real polymer chain consisting

of n bonds each of length I may be usefully represented by an equivalent freely jointed chain of N links each of length b such that it will have the same end-to-end distance and the same contour length. Obtain N and b in terms of the characteristic ratio Coo of the polymer chain. Answer: If both chains have the same end-to-end distance, 2 o = Nb

then (P2.8.1)

Also, if both have the same contour length, then nl

= Nb

(P2.8.2)

From Eqs. (P2.8.1) and (P2.8.2), nl and

N --

-

2nl


- Cool

(P2.8.3)

(P2.8.4)

Problem 2.9 It has been determined that
Chain Dimensions,Structures, and Transitional Phenomena the chain is n = 4x It is given that (r2)~o/2 = (2.01v/~) -1° m.So, (r2)o = (2.01x10-1°) 2 (4x) 2 z= 16.2×10-20x m

65

(P2.9.1)

Since the length of an isoprene repeating unit is given as 4.60x10-1° m, the contour length (nl) of the chain is nl = 4.6×10-1°:c m FromEq. (P2.8.3) we obtain the length of each link of an equivalent .freely jointed chain as 16.2 × 10-2°x b = - 3.52 x 10-1° m 4.6 x 10-1°x FromEq. (P2.8.4), we get the numberof links of the equivalent freely jointed chain as N = (4.6×10 -1°x)2 = 1.31x 16.2 × 10-~°x By comparison with Eq. (P2.9.1) we obtain N = 0.326n. Therefore, number C-C bonds per equivalent freely jointed link = n/N = 1/0.326 = 3.07 and the numberof isoprene units per equivalent freely jointed link = 3.07/4 = 0.77. Whenan average end-to-end distance of a macromolecularcoil is t 2x 1/2 as given by Eq. (2.7) or more generally (r2)lo/2 as given by Eq. (2.8), the polymer is said to be in its "unperturbed" state (see below). Henceforth the subscript ’o’ will be used to indicate the unperturbed state of a polymer molecule. The mathematical model for evaluating (r2)~o/2 is that of a series connected vectors (representing backbone bonds) which are restricted local level to certain allowed conformations on consideration of steric interactions. The bond vectors are volumeless lines in space and the model does not impose any restriction upon the relative positions of two bond vectors widely separated in the chain. In other words, the model does not prevent remotely connected bond vectors occupying the same volume in space, and thus allows self-intersections of the chain. In a real isolated polymer molecule, however, each part of the molecule excludes other more remotely connected parts from the volume. These long-range steric interactions cause the true RMSend-to-end distance, (r2) 1/2, to be greater than the unperturbed dimension (r2)o 1/2 and usually are considered in terms of an excluded volume (see Chapter 3). The extent to which unperturbed dimensions are perturbed in real chains is defined by an expansion factor

Clmpter2

66 o~ such that

=

(2.

Perturbations in chain dimensions are also caused by interactions of the chain with its molecular environment, e.g., with solvent molecules or other polymer molecules. The effects of these interactions are also embodied by the expansion parameter o~. In a theta solvent (see Chapter 3), o~ is reduced to unity and the end-to-end distance of a polymer chain is then the same as it would be in bulk polymer at the same temperature.

Problem 2.10 In a bulk amorphouspolymer containing no diluent the polymer molecules adopt their unperturbeddimensions. Explain this with reasoning. Answer: In the bulk state each polymermoleculeis surroundedby other polymermolecules of the sametype. Expansionof a given chain to relieve long-range intramolecular steric interactions only serves to create an equal numberof intermolecular steric interactions with neighboring chains. These opposing volumeexclusion effects exactly counteract each other and so in a bulk amorphouspolymer the polymer molecules adopt their unperturbed dimensions(i.e., a = 1). Problem 2.11 Assumingthat the RMSend-to-end distance is an approximation to the diameterof the spherical, coiled polymerin dilute solution, calculate the volumeoccupiedby one moleculeof polystyrene (molecular weight 106) in theta solvent at 25°C. (carbon-carbon bond length = 1.54x10-s cm; tetrahedral bond angle "-" °) 109.5 Answer: Molarmassof styrene ----- 104g mo1-1 Degree of polymerization = (106 g mo1-1)/(104 g mo1-1) = 9615 Numberof backbone C-C bonds = 2x9615 = 19230 FromEq. (2.8):

-s ° cr = 2.3 (fromTable2.1),n = 19230,I = 1.54xi0 cm, , O = 109.5 cos 0 "~ -1/3. Therefore, (r~)o = (2.3) 2 (19230) (1.54 x10-8) ~ (2) = 2.4x10-ll 2cm Volumein a theta solvent (c~ = 1) = (Tr/6)(2.4x10-1~)3/2 = 4.4x10-16 a. cm

Chain Dimensions,Structures, and Transitional Phenomena CONSTITUTIONAL ISOMERISM

67

AND CONFIGURATIONAL

The discussion in previous sections have focused on the occurrence of conformational changes, that is, changes in spatial arrangement of a polymer molecule due to bond rotation during ordinary thermal movementand its effect on size and shape of the polymer chain. When, on the other hand, the different spatial arrangements cannot change from one to the other or when such a change requires a more drastic molecular event than a thermal movement,namely, breaking of chemical bonds, then we are talking about isomers. Different isomers have different configurations, that is, spatial arrangements that cannot be interchanged (without at least momentary breaking of bonds). This is referred to as configurational isomerism. Another type of isomerism, which involves constitutional variations of a molecule, is referred to as constitutional isomerism. By constitutional variation we mean the differences in ways in which atoms in the molecule are linked together and the types of bonds involved therein. In polymers, the major types of constitutional differences involve positional isomerism and branching. Constitutional Positional

Isomerism Isomerism

Except in monomerslike ethylene and tetrafluoroethylene that have identical substituents, the two ends of the double bond in a vinyl type monomer are distinguishable. One C of the double bond can be arbitrarily labeled the head and the other the tail of the monomer,as shown below for vinyl chloride (IV)

tail

l CH2=C head I Ct (IV)

During polymerization, the monomerin principle can be enchained by head-to-tail linkages or head-to-head/tail-to-tail enchainments(V). However, head-to-tail enchainment is the predominant constitution of most vinyl monomersbecause of the influence of resonance and steric effects, as described below.

H I ~ CH2-C-

Head

H H H I I I CH:2-C-CH2-CCI I

to tail. (v)

H i CH2-CH2-C I

~

Head to head Tail to tail

Vinyl monomerspolymerize by attack of an active center (V’I) on the double bond. Equation (2.10) shows the propagation step in head-to-tail enchainment and Eq. (2.11) that in head-to-head/tail-to-tail enchainment.

X X X I I I ~-’-CH2-C e ~ + CH2=C ~--,,CH2-C-CH2-C Y

(2.10)

Y

Y

(VI)

Y (VII)

X X I I ~-CH 2-C ~+cH2=c Y (VI)

X I

Y

X X I I ~ C-C-cH~ ~-,CH 2Y Y (VIII)

The active center involved in the propagation reaction may be a fleeradical, ion, or metal-carbon bond (see Chapters 6-10). A propagating species will be more stable if the unpaired electron or ionic charge can

ChainDimensions,Structures, and Transitional Phenomena

69

be delocalized across either or both substituents X and Y. Such resonance stabilization is possible in (VII) but not in (VIII). MoreoverwhenX and/or Y is bulky there will be more steric hindrance to approach of the two substituted C atoms as in reaction Eq. (2.11) than in attack of the active center on the methylene carbon as in reaction Eq. (2.10). So, in. general, head-to-tail addition as in Eq. (2.10) is considered to be the predominant modeof propagation in all polymerizations.

Problem 2.12 A chemical method of determining head-to-head structures poly(vinyl alcohol) is by meansof the followingdifference in diol reactions:

H H I I ~ C~C "~’’~ + HIO 4 ~ "--"-~C I I HO OH

in

H H I I + C~’- + HIO3+ H20 II II 0 0

(1,2 Diot)

H J ~CCH2-C’ I OH

H I I OH

HI04 ~- No reaction sotn.

(1,3 Diot) Poly(vinyl acetate) of number-averagemolecular weight 250,000is hydrolyzed base-catalyzedtransesterification with methanolto yield poly(vinylalcohol). Oxidation of the latter with periodic acid yields a poly(vinyl alcohol) with number-average degree of polymerization485. Calculate the percentages of head-to-tail and headto-headlinkages in poly(vinyl acetate). Answer: Repeat unit of poly(vinyl acetate): -[-CH2-CH(OCOCH3)-]Molar mass of repeat unit = 85 g mo1-1 X~ = (250,000 g mo1-1) (86 g mo1-1) = 2907 Assume that poly(vinylacetate) is completelyhydrolyzedso that ~,, of the resulting poly(vinyl alcohol) is also 2907. The fact that ~,, is reduced from 2907 to 485 upon treatment with periodic acid meansthat each poly(vinyl alcohol) molecule

70

Chapter2

is, on the average, cleaved to yield 2907/485or 6 smaller molecules each of X,~ = 485 and so the number of cleavages = 6 - 1 = 5. Therefore, the average polymermolecule has 5 head-to-head linkages out of a total of 2907 - 1 = 2906 linkages in the polymerchain. 6 × 100 % head-to-head - 0.20 %. 2906 %head-to-tail = 100 - 0.2 = 99.8 %.

Polymers of dienes (hydrocarbons containing two carbon-carbon double bonds), such as butadiene and isoprene, have the potential for head-totail and head-to-head isomerism and variations in double-bond position as well. The conjugated diene butadiene, for example, can polymerize to produce 1,4 and 1,2 products:

Headto head

HH I I CH2=C-C =cH2 ~ 1 2 3 t, Butadiene

I I H CH II CH 2 1,2- poIybutadiene

Headlo tail (2.12)

H H I I

4cH2-c= c-c.z 1,4-po[ybutadiene

Thus, there are three possible constitutional isomers and, in addition, the possibility of mixed structures. The carbon atoms in butadiene are numbered in reaction Eq. (2.12) and the polymers are named according to the particular atoms which are involved in the enchainment. There is no 3,4-polybutadiene because it is identical with 1,2-polybutadiene. This is not the case with 2-substituted conjugated butadienes like isoprene and chloroprene. Thus, as can be seen from Eq. (2.13), there are six possible constitutional isomers of isoprene or chloroprene, to say nothing of the potential for mixedstructures. The properties of polymers of conjugated diolefins tend to be like those of thermoplastics if the monomerenchainment is 1,2 or 3,4. Elastomeric behavior is shown by 1,4-polymer, particularly if the polymer structure is cis about the residual double bond. Natural rubber is head-to-taft c/spolyisoprene.

71

ChainDimensions,Structures, and Transitional Phenomena

R I -~CH2CI -’~ n

~ head-to-

head-to-tail

H-C=CH 2 1,2-polyrner H I 2 1 CH2=C-C=CH 2 I 3 4 Monomer Isoprene:R=CH 3 Chloroprene: R

head

H

(2.13)

-(-CH2-i-~ n C-R II CH 2 3,4- potymer R H I

r-~

head-to-head

--(-CH2C= C-CH2~n 1 ~ 4- pol y mer

head-to-tai [

Branching Linear and branched polymer structures were defined earlier in Chapter 1. A polymer molecule having branches forms a more compactcoil than a linear polymer molecule of the same molecular weight with the result that the flow properties of the two types can differ significantly in the melt as well as in solution. Branchingoccurs in some .free-radical polymerization of monomers like ethylene, vinyl chloride, and vinyl acetate in whichthe propagatingpolymer radical is very reactive and can lead to branchingby chain transfer to the backbone chain of another polymer molecule or onto its own chain (see Chapter 6). Since branchedpolymermolecules possess a multiplicity of ends, it is more convenient to speak in terms of the radius of gyration (~j~2)1/2. A branched molecule occupies a smaller volum6 than a linear one with the same numberof segments, that is, the same molecular weight. It is convenient to express this diminutionof size as a factor 9 = (S2)branched/(S2)linear, which can be determined statistically for degrees and types of branching [15]. For randombranching, five trifunctional branch points per molecule reduce g to about 0.70. The change in size is the basis of a methodto measurebranching [16].

72

Chapter 2

ComSgurationalIsomerism The configuration of a molecule specifies the relative spatial arrangement of bonds in the molecule (of given constitution) irrespective of the changes molecular shape which can arise because of rotations about single bonds. A given configuration can thus be changed only by breaking and reforming of chemical bonds. There are two types of configurational isomerism in polymers and these are analogous to geome~cal and optical ~ome~smin micromolecular chemistry. Geometrical

Isomerism

Since rotation cannot take place about a double bond between two carbon atoms, two nonsuperimposable configurations (geometrical isomers) are possible if the substituents on each carbon differ from each other. For example, the two monomers maleic acid (IX) and fumaric acid (X) geometrical isomers, designated cis and trans, respectively.

H- C - COOH II H-- C-- COOH

H- C- COOH II HOOC- C --H

qx)

(x)

The molecules in solid trans isomers pack more tightly and crystallize more readily than those in cL~ isomers. (The melting point of fumaric acid is thus 160°C higher than that of maleic acid.) The corresponding differences in polymersare also significant. Natural rubber is 1,4-polyisoprene and the polymer configuration is c/s at each double bond in the chain, as shownin (XI). Consequently, the polymer molecule has a bent and less symmetrical structure. Natural rubber does not crystallize at room temperature and is amorphousand elastomeric.

H/

\CH3~ n (XI)

H/

\Cbl2 (XII)

73

ChainDimensions,St’ructures, and Transitional Phenomena

Balata (guttapercha) is also 1,4-polyisoprene, but the polymer configuration is trans at the double bond (XII). The molecule is more extended and has symmetrical structure. The trans isomer is thus a nonelastic, hard and crystalline polymer. It is used as a thermoplastic.

Stereoisomerisrn OpticM Activity

in Polymers

Stereoisomerism in polymers is formally similar to the optical isomerism of organic chemistry. In a vinyl polymer with the general structure shownin (XIII) every other carbon atom in the chain, labeled C*, is a site of steric isomerism, because it has four different substituents, namely, X, Y, and two sections of the main chain that differ in length. However,optical activity is influenced only by the first few atoms about such a site and, for the two sections of the main chain, these will be identical regardless of the length of the whole polymer chain. The carbons marked C* in (XIII) are thus not truly asymmetric and are termed pseudoasymmetric or pseudo chiral carbons. Only those C* centers near the ends of a polymer molecule will be truly asymmetric, and there are too few chain ends in a high molecularweight polymerto confer any significant optical activity on the molecule as a whole. Hence, most polymers of this type are not optically active.

X -,--~--CH 2¥

X -CH 2-

X ~

--CH 2¥

Y

(XIII) It is easy to see that there are two distinct configurational arrangements of the repeat unit of (XIII), viz., (XIV) and (XV), where the wedge the dotted lines denote bonds which are extending above and below the plane of the paper, respectively. These two stereoisomers of the repeat unit cannot be interchanged by bond rotation and exist because the substituted carbon atom, labeled C*, is attached to four different groups. Thus, every C* may have one or other configuration. This configuration is fixed when the polymer molecule is formed and is independent of any rotations of the main chain carbons about the single bonds connecting them.

Chapter 2

X

C

Y

¥

%,

and

\

C H

CXIV)

(xv)

It is important to note that even when optically different forms of a polymer can exist, they are obtained in practice only if an asymmetric influence is operative during propagation. This is necessary to ensure that throughout polymerization there is a preference for the formation of one particular absolute configuration of the asymmetric chain atoms. (The order of arrangement of four groups around a chiral carbon atom is called the absolute configuration.) In the absence of such an influence individual chains would usually be inherently compensated by having equal probabilities for D and L sequences of monomer placements. Tactlcity

in Polymers

As explained above, the C* atoms in (XIII) are pseudochiral and hence do not give rise to optical activity; the two mirror image configurations remain distinguishable, however. The different possible configurations or spatial arrangements that occur as a consequenceare called the tacticity of the polymer. The usual wayto picture the configurational nature of a vinyl polymeris to consider the polymer backbone stretched out so that the bonds between the main chain carbons form a planar zigzag pattern in the plane of the paper and the X and Y substituents on successive pseudochirat carbons lie either above or below the plane of the backbone, as shown in Fig. 2.14. If the substituents, say X, are then all aboveor all below the plane [Fig. 2.14(a)], the configurations of successive pseudoasymmetriccarbons are the same and the polymer is termed isotactic. If, however, a given substituent appears alternately above and below the reference plane in the planar zigzag conformation [Fig. 2.14(b)], the polymer is termed syndiotactic. In both cases, configurations of successive pseudochiral carbons are regular, and the polymer is said to be stereoregular or tactic. Whenthe configurations of pseudoasymmetric carbons are more or less random [Fig. 2.14(c)], the polymeris not stereoregular and is said to be atactic. Those polymerizations that yield tactic polymers are called stereospecific. Someof the more

75

ChainDimensions,Structures, and Transitional Phenomena

X H X H X H X H Ca) I I

~ I

I I

I I

I I

’/ I I

I

I

Y H Y H Y H Y H

IIIIIIII Y

H

Y

H

Y

H

Y

H

X H Y H X H Y H

- 7177 I IIIIII --C--C--C--C--C--C-C--C-C C

(b)

C ~ C ¥

H × H ¥ H × H

¥ H X ~ ~ N X N

Y H X H X H Y H (c)

--I-ix/c,\ ~ C

c,\CI--]1C~17 /I C

--]7 C ~ I I , , I X H Y H Y

I I I H X H

¥ H X H X H Y H -c-c-c-c-c-c-c-cX H Y H Y H X H

Figure 2.14 Diagramsof (a) isotactic, (b) syndiotactic, and (c) atactic configurations of -~CH~-CXY-)~ polymer. The corresponding Fisher projections are shownon the right.

important stereospecific polymerizations of vinyl monomersare described in Chapter 9. It is useful at this point to reiterate that stereoisomerism does not exist if the substituents X and Y are the same. In specifying the tacticity of the polymer,the prefixes it and st are placed before the name or structure to indicate isotactic and syndiotactic structures, respectively. For example, /t-polystyrene means that the polystyrene is isotactic. The absence of these terms denotes that the polymer has an atactic structure. The importance of stereoregularity in vinyl polymers lies in its effects on crystallizability of the material. Polymerchains with stereoregularity are better able to pack together in a regular array and crystallize. The increase in crystallinity is reflected in higher melting point, greater rigidity, and less solubility compared to amorphous species with the same constitution. A striking example is provided by polypropylene. Isotactic polypropylene has a crystal melting point of 176°C and is widely used in fiber, cordage,

76

Chapter 2

and automotive, and appliance applications, while the atactic polymer is a rubbery amorphous material that per se has no important use. Syndiotactic polymers, as we have seen above, are stereoregular and so are crystallizable. They, however, do not have the same mechanical properties as isotactic polymers, because the different configurations affect the crystal structures of the polymers. Most highly stereoregular polymersof commercialimportance are isotactic, and relatively few syndiotactic polymers are made. Atactic polymers, on the other hand, are usually completely amorphous unless the side group is so small or so polar as to permit some crystallinity. Thus, while atactic poly(vinyl acetate) has never been crystallized, poly(vinyl alcohol), which is derived from it and is also atactic, has been found to crystallize. It is important to note at this point that completely tactic and completely atactic polymers represent extremes of stekeoisomerism that are rarely encountered in practice. Manypolymers exhibit intermediate degrees of tacticity and their characterization requires measurementof the extent of stereoregularity as well as the lengths of the tactic chain sections. The most powerful tool for analyzing the stereochemical nature of polymers is nuclear magnetic resonance (NMR)spectroscopy. Meso- and Racemic Placements The Fisher projections in Fig. 2.14 show that placement of the groups in isotactic structures corresponds to a meso (same) or m placement of pair of consecutive pseudochiral centers, whereas the syndiotactic structure corresponds to a racemic (opposite) or r placement of the corresponding pair of pseudochiral centers. It must be emphasized that the m or r notation refers to the configuration of one pseudochiral center relative to its neighbor. Thus, the meso dyad is designated m, and the racemic dyad r. The system of nomenclature can be extended to sequences of any length. Thus, an isotactic triad is mm,a heterotactic triad is mr, and a syndiotactic triad rr. Several such configurational sequences are illustrated in Table 2.2. Each of these, and even more complicated combinations, can be designated through NMRstudies. Let us assume that the probability of generating a meso sequence when a new monomer unit is added at the end of a growing chain can be denoted by a single parameter, which we shall call Pm(and similarly Pr for a racemic sequence). This implies an assumption that the polymer obeys Bernoullian statistics, that is, the probability of forming an m or r sequence is independent of the stereochemical configuration of the chain already formed. It follows of course that Pmq- Pr = 1, because addition can be only m or r type. A triad sequence involves two monomeradditions; the Bernoullian probabilities of ram, mr, and rr are therefore P~, 2Pro(1 - Prn) and (1 -- -Pro)2, respectively. A plot of these relations is shownin Fig. 2.15. It will be noted that the proportion of mr, that is, heterotactic units, "rises

Chain Dimensions, Structures,

and Transitional Phenomena

77

.g

E E E

I.--

"

E

"-

78

Chapter2

to a maximumat Pm=0.5, corresponding to random propagation. For a random polymer (Pro = 0.5) the proportion mm: mr : rr will be 1:2:1. The Pmvalues of 0 and 1 correspond to completely syndiotactic and isotactic polymers, respectively.

Problem2.13 Whatwill be the shape of a plot of dyad frequencies against P,~ ? Answer: Since the Bernoullian probability of the formation of an m-dyadsequence is P,~ and that of an r-dyad sequenceis (1 - P,~), the plot of dyad frequencies against P, wouldobviously be two straight lines with slopes of +1 and -1 for m and r, respectively.

For any given polymer, if Bernoullian, the ram, mr, rr sequence frequencies, as estimated from the relative areas of the appropriate peaks in NMRspectra (see below), should lie on a single vertical line in Fig 2.15, corresponding to a single value of /gin (see example below). If this is not the case, then the polymer’s configurational sequence deviates from the simple Bernoullian statistics. Plots such as Fig. 2.15 can be made [18] for tetrad and other higher order probabilities (or fractions) as a function of Pmusing the relationships given in Table 2.2. These are useful for peak assignments in NMRspectra (in which finer structures are observable), aided by certain necessary relationships (Table 2.3) amongthe frequencies of occurrences of sequences, which must hold regardless of the configurational statistics (Bernoullian or not). NMRMeasurement of Tacticity The application of high resolution NMRspectroscopy made the quantitative measurement of sequence distribution feasible and allowed one to compare actual polymer chains with theoretical predictions. A simple illustration is given here considering only triad sequences. Three consecutive monomerunits in a chain considered to define a configuration is called a triad. The term heterotactic is now used to define a triad which is neither isotactic nor syndiotactic. The three types of triads are shown in Fig. 2.16 for poly(methyl methacrylate). The three protons an o~-methyl group being equivalent absorb radiation at a single frequency, but this frequency will be different for each of the three kinds of triad, because the environment of o~-methyl groups in each is different. Thus the NMRspectra of solutions of poly(methyl methacrylate) show marked differences depending upon the conditions of polymerization that influence the tacticity of the polymer. Examples of two spectra are shown in Fig. 2.17 [18]. The o~-methyl groups give the set of three peaks at 8.78"r, 8.95r,

Chain Dimensions, Structures,

and Transitional

Phenomena

79

1.0

~

0.8

(rr)

= (1-Prn) 2 (ram)=

/ p2m

o.~ o.a ~0.2

0.2

O.Z,

0.6

0.8

1.0

Pm

Figure2.15 The probabilities (or fractions) of isotactic (ram), heterotactic (mr), and syndiotactic (rr) triads as a function of P,~, the probability of isotactic monomer placement during propagation. (From Ref. 18.)

H

H

\ .-. c

/\c

H

/ 1".

H

\.... c

H

/\c/\c

CH3 R

H

\ .. c

1".

CH3 R

l".. CH3 R

Isotoctic

Figure2.16 Isotactic,

H

H

H H

\.... /c.

\c / l"..

H

\....H

\ ... c

CH3 R

/ \c

c\

1"..

R

H H

/c

/"..

CH3 CH 3R

Syndiotactic

syndiotactic, poly(methyl methacrylate).

and heterotactic

\:.. /c.

H H

\/ c\

/\c

/",

CH 3R

H

H

\,-" c

c / \c

1"..

CH3 R

/

/"-.

R

CH 3

Heterotactic

triad configurations

for

80

Chapter 2

aTable 2.3 SomeNecessary Relations AmongSequence Frequencies Dyad: (m) + (r) Triad: (mm) + (mr) + (rr) Dyad-Triad:

(,~) = (m.~) (~-) = (Tr)

Triad-Tetrad: (mm) = (mmm) + ½(mmr) (mr) = (mmr) + 2(rmr) + (mrr)

(~) = (~) Tetrad-Tetrad:

sum

--

1

(,~m~)+ 2(~,~) = 9.(.~.~) (. ~) Pentad-Pentad: sum = 1 (mmmr) + 2(rmmr) = (mmrm) + (mmrr) (mrrr) 2(mrrm) = (rrmr) + (rrmrn) Tetrad-Pentad: (mmm) = (mmmrn) + ½(mmmr) (mmr) = (mmmr) + 2(rmmr) = (mmrm) + (rmr) = ½(mrmr) + ½(rmrr) (mrm) = ½(tartar) + ½(mmrm) (rrm) = 2(mrrm) + (mrrr) = (mmrr)

(~) = (~) ½(.~)

aFromRef. 17. and 9.09~-. For a mainly isotactic sample the peak at 8.78~- is muchlarger than the other two. It is therefore assigned to the a-methyl groups which are flanked on each side by a monomerunit with similar configuration (see Fig. 2.16). Such a group of three units constitutes an isotactic (ram) triad. ’A mainly syndiotactic sample showsonly a small peak in this position and the peak at 9.09T is the most prominent of the three. It is assigned to o~-methyl protons in the middle of syndiotactic (rr) triads. The remaining peak at 8.95"r is therefore attributed to the oe-methyl group in the middle of heterotactic (mr) triads. The numbers of isotactic, heterotactic and syndiotactic triads in any particular sample, and hence ram, mr and rr, are proportional to the areas under the appropriate peaks. The NMRmethod of identifying stereoregular structures can be applied also .to polymers that are not predominantly stereoregular, yielding information about the probability Pm that a monomer adding on to the

Chain Dimensions,Structures, and Transitional Phenomena

81

(a}

r

(b)

Figure 2.17 NMR spectra of poly(methyl methacrylate) (15% solution in chloroform, r = tetramethyl silane reference peak). (a) Mainly syndiotactic; mainly isotactic. The methyl ester group appears at 6.40r in both spectra and is unchangedby chain configuration. (FromRef. 17.)

end of a growing chain will have the same configuration as the ufiit it is joining. The fraction of each configuration, ram, rr, and mr, is measured, respectively, from the isotactic, syndiotactic and heterotactic peak areas in the NMRspectra. The results are then fitted to the probability curves by placing one of them, say, the fraction rr triad on the curve and letting the other fractions, viz., rr~m and mr, fall where they may. If the latter fractions also fall satisfactorily close to the calculated curves, then the configurational statistics is Bernoullian and can be described by a single value of Pra. Problem2.14 Determinationof the individual peak areas in the a-methyl region of the NMR spectra of poly(methyl methacrylate) shownin Fig. 2.17 yielded the followingvalues [18]:

82

Chapter 2

Polymer No. 1 2

Area % Polymerization conditions Iso Hetero Syndio Benzoyl peroxide in bulk, 100°C 8.6 37.5 53.9 n-BuLi in toluene,-62°C 63 19 18

Determine whether the polymerization in each case can be determined by a single value of P,~ in agreement with Bernoullian statistics. Answer: Wemight test for Bernoullian fitting by placing the (ram), (mr), and (rr) triad intensities (that is, the fractional areas representing polymer fractions in iso, hetero and syndio configurations, respectively) on the probability curves of Fig. 2.15. This is done in Fig. 2.18 by placing the (rr) point on the curve to obtain the corresponding P,~ value and then letting the other two points fall where they may on the vertical line at this P,~. It is observed that the fractions of mr and mmunits fall satisfactorily close to the calculated curves. This maybe interpreted as indicating that the free radical propagation in the given case is Bernoullian and can be described by a single value Of -P,~ (= 0.27). For polymer2, the results are fitted similarly to the probability curves in Fig. 2.18 by placing the (ram) point on the curve and letting the other points fall where they may. It is observed that the (mr) and (rr) points are significantly awayfrom the calculated curves, an indication that the polymer is non-Bernoullian. Thus the anionic polymerization mechanismin the given case is not describable by a single value of

1.0 2 (rr) = (1-Pm)

¯"= 0.8

~

/

(mml=P~--~

,o 0.6 "~_

~ 0.2

0.2

0.4

0.6

0.8

1.0

Pm

Figure 2.18 Test of Bernoullian model. Data for polymer 1 of Problem 2.14.

ChainDimensions,S~ructures,andTransitionalPhenomena CRYSTALLINITY

83

IN POLYMERS

The crystallization of polymers is of great technological and practical importance. Many thermoplastics will crystallize to some extent when the molten polymer is cooled below the melting point of the crystalline phase. This is a procedure that is done during polymer processing and the presence of crystals thus formed has an important effect on polymer properties. There are many factors that can affect the rate and extent to which crystallization occurs in a particular polymer. They can be processing variables such as the rate of cooling and the melt temperature. Other factors include the tacticity and molecular weight of the polymer, the amount of chain branching and the presence of any additives such as nucleating agents.

Determination of Crystallinity Polymerscrystallized from melt are never completelycrystalline. This is because there are enormousnumberof chain entanglements in the melt that makeit impossiblefor the polymerchains to be fully aligned to form a 100% crystalline polymerduringcrystallization. Thedegreeof crystallinity is of great technologicalandpractical importanceas it influencesthe properties and hence applications of polymers. Several methodshave been devised to measurethe degree of crystallinity, two widely used amongthembeing the density methodand the wide-anglex-ray scattering (WAXS) method. The crystalline regions in the polymerhave a higher density than the non-crystalline amorphousregions and this difference in densities, which is relatively large (up to 20%), provides the basis of the density method for the determinationof the degree of crystallinity. This methodcan yield both the volumefraction and massfraction of the crystalline regions from measurementof the sampledensity and knowledgeof the densities of the fully crystalline and amorphousstates of the samepolymer. Problem2.15 Derive an equation relating the degree of crystallinity of a semicrystallinepolymerto the sampledensityanddensities of the crystalline and amorphouscomponents. Answer: If V~is the total volume of the crystallinecomponents andV~that of the amorphous components, the total specimenvolumeVis given by V=~+V~ (P2.15.1) Similarly, the weightof the specimenWis given by W=Wc+Wo (P2.15.2) Expressing in termsof densities, pv = pov~ + (P2.15.3) Substitutingfor VafromEq. (P2.15.1)and rearrangingleads

84

~hap~er 2

=

V

=

(P2.15.4)

"~a

where ~c is the volume fraction of crystalline components. The weight fxaetion wc of the crystalline componentsis similarly defined as wc = Wc/W = pcV~/pV

(P2.15.5)

and combining Eqs. (P2.15.4) and (p2.15.5) gives wc = -P

(P2.15.6)

In terms of specific volumes vc and v~ of the crystalline respectively, Eq. (P2.15.6) becomes

and amorphouscomponents,

wc =

(P2.15.7) ~a -- ~c

Thedcgrccof crystaIlinity of a scmicrystaIIinc polymer canthusbc determined fromthe specimen densityp and thedensities #c and p~ of the crystalline and amorphous components, respectively, usingEqs.(P2.15.4) and(P2.15.6).

Problem2.16 Estimate the fraction z. polyethylene of density 0.983 g/cm

of crystalline

material

in a sample of

Data: 3Density of amorphous polyethylene = 0.866 g/cm Unit cell (Fig. 2.8) dimensions of polyethylene containing 4 CH2groups are: =7.41~i, b=4.94.~1., c(fiber axis) =2.55 ~, a =/3 = ’)’= ° Answer: Volumeof unit cell

= 7.41×4.94×2.55×10 -24 3cm 3 = 93.34×10 -24 cm

4×(14 g mo1-1) -~) Cell contains 4 CH2groups or (6.02 × 1023 mol

Density of crystalline

(9.3 x 10-z~ g) material = (93.34 x 10-24 cm3) -3 = 0.996 g cm

Using Eq. (P2.15.6), % Clystallinity

= 9.3x10_2ag

[(0.996 g = 100 [(0-0-~ = 91%.

cm-Z r(0. ss-0.8°°) 1

cm-3) 1 g cm-3)]

[~ 0.866)

g -3]

Chain Dimensions,Structures, and Transitional Phenomena

26

2/-,

22 "’---

20

18

16

85

14

20

Figure 2.19 A typical WAXS curve for semicrystalline polyethylene where the intensity of scattering is plotted against diffraction angle 20. Theamorphous hump is shownshaded.

The density of a polymer sample can be readily determined by allowing it to float in a density-gradient column, which is a vertical columncontaining a mixture of liquids with different (known)densities. The density of a small piece Of polymer is determined from the position it adopts when it is dropped into the column. The density of the crystalline regions Pc can be calculated from a knowledge of the crystal structure [19]. The amorphous density Pa can sometimes be measured directly if the polymer can be obtained in a completely amorphous form, for example by rapid cooling of a polymer melt. Otherwise it can be determined by extrapolating either the density of the melt to the temperature of interest or that of a series of semicrystalline samples to zero crystallinity. WAXS is a powerful methodof determining the degree of crystallinity. Since the scattering from the crystalline regions gives relatively sharp peaks compared to a broad ’hump’ due to scattering from non-crystalline areas (see Fig. 2.19), the degree of crystallinity can, in principle, be determined from the relative areas under the crystalline peaks and the amorphous hump. The mass fraction of crystalline regions zc is given to a first approximation by

xc = Ac/(Ac + A,~)

(2.14)

where Aa is the area under the amorphous hump and Ac is the remaining area under the crystalline peaks.

Chapter2

86 Morphology of Crystalline

Polymers

Most polymers are partially crystalline. The degree of crystallinity of polymers may, however, range from 0 percent for noncrystallizable polymers through intermediate crystallinities, such as 20 percent for unoriented poly(vinyl chloride),. 50 percent for branched polyethylene, 70 percent for isotactic polypropylene, and up to nearly 100 percent for polytetrafluoroethylene and linear polyethylene. In this context, crystalline polymers are more correctly referred to as semicrystalline, but the prefix "semi" may be dropped with the same understanding that we display in dropping the prefix "high" from the term "high polymers." Lamellae The most direct evidence of the crystallinity in polymers is provided by x-ray diffraction studies. The x-ray patterns of many crystalline polymers show both sharp features associated with regions of three-dimensional order, and more diffuse features characteristic of molecularly disordered substances like liquids. The occurrence of both types of feature is evidence that ordered regions (called crystallites) and disordered regions coexist in most crystalline polymers. X-ray scattering and electron microscopy have shown that the crystallites are made "up of lamellae which are built-up of folded polymer chains as explained below. In their simplest form, polymer single crystals appear in the electron microscopeas thin, flat platelets (called lamellae) on the order of 100 to 200 .~(0.01-0.02 micron) thick and several microns in lateral dimensions. While the simplest crystals are monolayer, more complicated crystals contain 10 to several hundred lamellae all originating from the same nucleus. The electron diffraction patterns of individual lamellae indicate that polymer molecular axes are perpendicular to the plane of the lamellae in the crystal. Since the lamellae are only 100 ~ or so thick and polymer molecules are generally on the order of 1000 to 10,000 ~ long, the molecules must be folded back and forth within the crystal. This arrangement has been shownto be sterically possible. In polyethylene, for example, the molecules can fold in such a way that only about five chain carbon atoms are required for the fold, that is, for the chain to reverse its direction. Each molecule folds up and down in a regular fashion to establish afoldplane [Fig. 2.20(a)]. As illustrated Fig. 2.20(b), a single fold plane may contain manypolymer. Also shown this figure are someof the imperfections that mayoccur in polymer crystal structure, viz., chain ends, branches, loops, or irregular fold heights. The height of the fold plane is knownas the fold period. It corresponds to the thickness of the lamellae. It is evident from the ductility and strength of polymers that the ties between lamellae must be stronger than the van der Waal’s forces holding neighboring, parallel fold planes together. Evidently some molecules (tie molecules) must participate in the growth of two or more adjacent lamellae, thereby providing relatively short molecular links between the lam~llae.

Chain Dimensions, Structures,

F

87

m~d Transitional Phenomena

Fotd surface

(a)

plane (side view) Irregular fold heights Chain ends

Loops

~

(b)

Branch point Loose chain end

Figure2.20 (a) Model of a lamellar crystal showing regular, adjacent re-entry folds. (b) Modelof fold plane illustrating chain folding with imperfections which may occur in the structure. (From Ref. 21.)

The formation of such fibrilar demonstrated [20].

links

between lamellae

has actually

been

Spherulites The most prominent structural organization in a material crystallized from a polymer melt is the spherulite. Electron microscopic evidence indicates that spherulites are lamellar and the lamellar structure persists throughout the body of spherulites. Spherulites are spherical aggregates, the sizes ranging from microscopic to millimeters in diameter. They are recognized by their characteristic appearance in the polarizing microscope [21], where they are seen as circularly birefringent areas possessing typical Maltese cross patterns (Fig. 2.21).

88

C.hapter2

Figure 2.21 Spherulites in a sillconelike polymer, observed in the optical microscope between crossed polarizer. The large and small spherulites were grown by crystallization at different temperatures. (Adaptedfrom Ref. 21.)

The Avrami equation [Eq. (2.15)], which was originally proposed the general context of phase changes, has provided the starting point for manystudies of polymer crystallization and spherulitic growth. It relates the fraction of a sample still molten, 0, to the time, ~, which has elapsed since crystallization began. The temperature must be held constant.

0 = exp(-Zt’*)

(2.15)

For a given system under specified conditions, Z and n are constants and, in theory, they provide information about the nature of the crystallization process. Taldng logarithm twice in succession gives

ln(-ln0)

= lnZ + ~ln~

(2.16)

A plot of In(--In0) against ln/~ should be a straight line of slope makingan intercept of in Z with the vertical axis. In practice this method of evaluating r~ and Z is very prone to error and curve fitting methods using Eq. (2.16) and the raw data are usually preferred. The exponent of the Avrami equation, r~, has a theoretical value of 3 when crystallization takes the form of spherulitic growth of nuclei which came into being at the same instant in time. Integral values of n rahging from 1 to 4 can be attributed to other forms of nucleation and growth [22].

Chain Dimensions,Structures, and Transitional Phenomena

89

Spherulite growth may start at a nucleus, which may be a foreign particle (heterogeneous nucleation) or may arise spontaneously (homogeneous nucleation), and proceeds (at the expense of the melt) in a radial fashion from each nucleus until the growth fronts from the neighboring structures impinge. The lamellae are parallel to the nucleus, but as they begin to grow outward, they diverge, twist, and branch to form an overall structure that is the radially symmetric spherulite. The spherulites formed from a melt have different sizes and degrees of perfection, and they completely fill the volumeof a well-crystallized material. THERMAL

TRANSITIONS

IN

POLYMERS

The term "transition" refers to a change of state induced by changing the temperatures or pressure. Twomajor thermal transitions are the glass transition and the melting, the respective temperatures being called Tg and

Tg and All polymers are hard rigid solids at sufficiently low temperatures but as the temperature rises a thermoplastic polymer eventually acquires sufficient thermal energy to enable its chains to movefreely enough for it to behave like a viscous liquid (assuming there is no degradation). There are several ways in which a polymer can pass from the solid to the liquid phase, depending on the structure of the polymer chains and their arrangement in the sample. The different types of thermal response in this transition from a rigid solid to an eventually liquid state can be illustrated in several ways. One of the simplest and most satisfactory is to trace the change in specific volume, as shownschematically in Fig. 2.22. A thermoplastic polymer may be completely amorphous in the solid sts.te, which means that the polymer molecular chains in the specimen are arranged in a totally random fashion. The volume change in amorphous polymers follows the curve ABC.In the region C-B, the polymer is a glassy solid and has the characteristics of glasses, including hardness, stiffness, and brittleness. In the glassy region, the available thermal energy (/~T energy units/mol) is insufficient to allow rotation about single bonds in the polymer backbone overcoming intramolecular energy barriers, and movements of large (some 10-50 consecutive chain atoms) segments of the polymer chain cannot take place. But as the sample is heated, it passes through a temperature T~, called the glass transition temperature, above which it softens and becomes rubberlike. This is an important temperature and marks the onset of extensive molecular motion, which is reflected in marked changes in properties, such as specific volume, refractive index, stiffness, and hardness. Above T~, the material may be more easily deformed. A continuing increase in temperature along B-A leads to a change of the rubbery polymers to a viscous liquid without any sharp transition:

90

C~ap~er2

A

$~r~tite Temperature,T Figure 2.22 Schematic representation of the changes ot specific volumeof a polymer with temperature for (a) a completely amorphous sample (A-B-C), (b) a semicrystalline sample (A-D-E), and (c) a perfectly crystalline material (A-F-G). In a perfectly crystalline polymer, all the chains would be contained in regions of three dimensionalorder, called crystallites, and no glass transition would be observed, because of the absence of disordered chains in the sample. A perfectly crystalline polymer, on heating, would thus follow curve G-F-A, melting at T~° to becomea viscous liquid. Perfectly crystalline polymers are, however, not encountered in practice and real polymers may instead contain varying proportions of ordered and disordered regions in the sample. These semicrystalline polymers usually exhibit both Tg and T,~ (not Tm°) corresponding to the disordered and ordered regions, respectively, and follow curves similar to E-H-D-A. As Tr~ is the melting temperature of a perfectly crystalline polymer of high ° and more often represents a melting molecular weight, Tmis lower than Tm range, because the semicrystalline polymer contains crystallites of various sizes with manydefects which "act to depress the melting temperature. Both Tg and Tm are important parameters that serve to characterize a given polymer. While Tg sets an upper temperature limit for the use of amorphousthermoplastics like poly(methyl methacrylate) or polystyrene and a lower temperature limit for rubbery behavior of an elastomer-like SBR rubber or 1,4-cis-polybutadiene, Tm or the onset of the melting

Chain Dimensions,Structures, and Transitional Phenomena

91

range determines the upper service temperature. Between Tm and semicrystalline polymers tend to behave as a tough and leathery material. As a general rule, however, semicrystalline polymers are used at temperatures between Tg and a practical softening temperature, which lies above Ta and below Tin. The onset of softening is usually measured as the temperature required for a particular polymer to deform a given amount under a specified load. These values are knownas heat deflection temperatures. Such data do not have any direct relation with Tin, but they are widely used in designing with plastics.

Problem 2.17 Polycaprolactone has Tg = - 60°C and T,~ = + 60°C, while poly(ethylene terephthalate) has Tg = + 60°Cand 2r’,~ = 250°C. Whichpolymer wouldbe more suitable in a study of biodegradability at 25°C for which it is desirable to vary the crystallinity holdingall other variables constant? Answer: Crystallization can take place wheneverthe polymer is at a T above Tg. At 25°C, polycaprolactoneis aboveTa and so it maycrystallize. Onthe other hand, poly(ethyleneterephthalate) can be quenchedto 25°Cat various rates resulting various degrees of crystallinity whichwill not changeas long as the temperature does not exceedTg. So this polymerwill be moresuitable for the biodegradability, study.

Secondary Glass Transitions In addition to the main or primary glass transition discussed above, many polymers also exhibit secondary glass transitions. The primary glass transition occurs in the temperature region where long segments of the polymer chain backbone becomefree to move. The secondary glass transitions occur in the temperature ranges where short sections of the main chain or side chains becomefree to move-or oscillate. Since these groups are small compared to the polymer chain, they require less volume and energy than the longer chain segments for movement.Secondary glass transitions therefore occur at lower temperatures than primary glass transitions. The transitions are sometimes denoted 0~, /3, % etc., in order of descending temperature, the transition of highest temperature o~ being called Ta, which thus refers to the primary glass transition. First-

and Second-Order

Transitions

In first order transitions, such as melting, there is a discontinuity in the volume-temperature Not (see Fig. 2.22) or enthalpy-temperature plot

92

Chapter 2

Cb)

H

Cp

Figure 2.23 Idealized variations in volume(V) and enthalpy (H). Also shown are c~, the volumecoefficient of expansionand C’p, the heat capacity, whichare, respectively, the first derivatives of Vand Hwith respect to temperature(2r’).

the transition temperature. In second-order transitions, only a change in slope occurs and there is thus a marked change in the first derivative or temperature coefficients, as illustrated in Fig. 2.23. The glass transition is not a first-order transition, as nodiscontinuities art observed at Tg when the specific volume or entropy of the polymer is measured as a function of temperature. However, the first derivative of the property-temperature curve, i.e., the temperature coefficient of the property (e.g., heat capacity and volumetric.coefficient of expansion), exhibits a marked change in the vicinity of Tg; for this reason it is sometimescalled a second-ordertranxition. The volumetric coefficient of expansion, o~, is defined as

where V is the volume of the material, and c~ has the units °I£-1. While this quantity increases rather sharply at T~, the increase actually occurs over

Chain Dimensions,Structures, and Transitional Phenomena

93

a range (10-30°C). Similar changes occur in the heat capacity at constant pressure (Cp), which is the first derivative of enthalpy H with respect temperature. REGIONS

OF VISCOELASTIC

BEHAVIOR

As we have seen above, the transition that separates the glassy state from the viscous state is knownas the glass-rubber transition. This transition attains the properties of a second-order transition at very slow rates of heating or cooling. In order to clearly locate the region of this transition and to provide a broader picture of the temperature dependence of polymer properties the principal regions of viscoelastic behavior of polymerswill be briefly discussed. Broadly, there are five regions of viscoelastic behavior for linear amorphous polymers [23-24] as shown in Fig. 2.24. In region I (a to b), the polymer is glassy and frequently brittle. Typical examples are polystyrene and poly(methyl methacrylate) at room temperature. ¥oung’s modulus this region just below the glass transition temperature is approximately 3 x 101° dyne/cm2 (3 x 109 Pa) and it is nearly the same for a wide range polymers. RegionII (b to c) is the glass transition region. Typically, the modulus drops a factor of about a thousand in a 20-30°C range. The behavior of polymers in this region is best described as leathery. For static or quasistatic measurements,such as illustrated in Fig. 2.24, the glass transition temperature, Tg, is often taken at the maximumrate of turndown of the modulus at the elbow, i.e., where d21~/dT2 is at a maximum. Region III (c to d) in Fig. 2.24 is the rubbery plateau region. After a sharp drop that the modulus exhibits in the glass transition region, it again becomes almost constant in the rubbery plateau region, with typical values of 2x107 dyne/cm2 (2x10 ~ Pa). In this region, polymers exhibit high rubber elasticity so muchso that an elastomer in this region can be stretched perhaps several hundred percent and it snaps back to substantially its original length on being released. In region III (c to d) of Fig. 2.24, three cases need be distinguished: (1) If the polymeris linear, the solid line is followed on whichthe modulusdrops off slowly with increasing temperature. The width of the plateau is governed primarily by the molecular weight of the polymer; the higher the molecular weight, the longer the plateau. (2) If the polymer is semicrystalline, the dotted line in Fig. 2.24 is followed. Since the crystalline regions in the polymer matrix tend to behave as a filler phase and also as a type of physical cross-link between the chains, the height of the plateau (i.e., the modulus)will be governed by the degree of crystallinity. (3) If the polymer is cross-linked, the dashed line in Fig. 2.24 is followed, and improved rubber elasticity is observed, with the creep portion suppressed. A common exampleof a cross-linked polymer above its glass transition temperature is the ordinary rubber band. For crosslinked polymers region (III) remains

94

Chapter2

11

10

I0

9 8 -7 -6 5

o 6

3 Temperature Figure 2.24 Five regions of viscoelastic behavior for a linear, amorphouspolymer: I (a to b), II (b to c), III (c to d), IV (d to e), and V ( e to illustrated are effects of c~stallinity (dotted line) and cross-linking(dashedline).

in effect at higher temperatures up to the decomposition temperature and regions IV and V do not occur. In the case of linear amorphouspolymers, raising the temperature past the rubbery plateau region brings them to the rubbery flow region-region IV. In this region, the polymer exhibits both rubber elasticity and flow properties depending on the time scale of the experiment. For experiments performed in a short time, the physical entanglements of polymer chains are not able to relax and the material still behaves rubbery. For longer duration experiments, the increased molecular motion imparted by the increased temperature causes chains to move, resulting in a visible flow. An example of a material in the rubbery flow region at ambient temperature is Silly Putty, which bounces like a ball when thrown (short time experiment) but deforms like a taffy when pulled (a much slower experiment). RegionV (e to f) in Fig. 2.24 is the liquid flow region, which is reached at still higher temperatures where the increased kinetic energy of the chains permits them wriggle out through entanglements rapidly and move as individual molecules, often producing highly viscous flow. This is the melting temperature and it is always above the glass transition temperature.

Chain Dimensions, Structures,

and Transitional

Phenomena

95

Problem2.18 A new polymer was reported to soften at 60°C but it is not known for sure whether the softening was a glass transition or a melting point. Describe a simple experiment to distinguish between the two possibilities. It is known, however, that the new polymer is essentially a linear polymer. Aas~er: If 60°C is a glass transition, then heating the polymer slowly past 60°C would take it to the rubbery plateau region (region III in Fig. 2.24), where the modulusE, and hence hardness, would remain fairly constant with increase of temperature. For a melting transition, however, the modulus would drop rapidly and the polymer would becomeincreasingly softer in a similar experiment. Pressing one’s thumb in an object is a simple way to gauze the object’s hardness. Its scientific analogue is the measurementof hardness by indentation. In practice, the point of a weighted needle is allowed to rest on the polymersurface as the temperature is raised. The movementof the needle as it penetrates the surface can be monitored by means of an amplification gauge. Though less accurate than other more sophisticated methods, it is useful for the preliminary engineeringoriented examination of systems.

Problem2.19 Show schematically

the results that would be expected from the following experiments with the new polymer of Problem 2.18 if the reported softening temperature (60°C) is indeed a melting transition: (a) Specific volume (v) as a function of temperature (T). (b) Differential scanning calorimetry. (c) Young’s modulus (E) as a function of temperature (T). (d) X-ray diffraction. The experiments are carried out in the temperature range 50°C-70°C. Answer: (a) The specific volume would show a first order transition. However,for reasons explained in the text (see p. 48), the melting transition for polymers usually occurs over a range of temperature,

I

I

I

50 60 70

T(°C)

96

Chapter 2

(b) DSCwould show an endothermic peak.

50 60 70 T (°C) (c) A plot of logE versus T would show a sharp downturn at about 60°C.

Io 0 E I

50 60 70 T(°C) (d) If the polymer has a high degree of crystallinity, x-ray diffraction would show sharp lines and only diffuse rings or halo at 70°C.

50°C

at 50°C

70°C

Problem2.20 What type of results would be obtained if the reported softening temperature (60°C) for the polymerin Problem 2.18 were indeed a glass transition

Chain Dimensions, Structures,

and Transitional

97

Phenomen~

Answer: (a) The specific volume would show a second order transition

(see p. 92).

! 50 60 70 T (°C) (b) There occurs a change in heat capacity at the glass transition. Tg is taken as the temperature at w~ich one-half of the change in heat capacity-AC’p has occurred.

"~~.~

Cp

50 60 70 T (%)

(c) For quasistatic is at Ta.

measurements, maximumrate of the turndown of the modulus

I I

I

I

50 60 70

T(°C) (d) A glass is a material which has lost most of its ductility

but which has not

98

Chapter 2

otherwise changedon cooling. Thus, on the basis of x-ray diffraction a glass is indistinguishable from the corresponding rubber on the other side of the glass transition.

50°C

70oc

FACTORSAFFECTING Tg As Tg marks the onset of molecular motion, a number of factors that affect rotation about links (necessary for movementof polymer chains) will also influence the ~’g of a polymer. These include (a) chain flexibility, (b) molecular structure (steric effects), (c) molecular weight, and (d) branching and crosslinking. The flexibility of the chain is undoubtedly the most important factor influencing the ~9 of a polymer. The chain flexibility depends more on the rotation or torsion of skeletal bonds than on changes in bond angles. When a randomly coiled chain is pulled out into an elongated conformation, the skeletal bonds "unwind" rather than undergo angular distortion (see Fig. 2.25). Thus, flexibility on a macroscopic scale depends on torsional mobility at the molecularlevel. If a highly flexible chain is present, ~’9 will generally be low and if the chain is rigid, the ~’9 value will be high. For symmetrical polymers, the chemical nature of the backbone chain is the important factor determining the chain flexibility and hence Tg. Chains made up of bond sequences which are able to rotate easily are flexible, and hence polymers containing -(-CH2-CH2-)-, -(-CH2-O-CH2-)-, or Si-O-Si-)- links will have correspondingly low Values of T~. For example, poly(dimethyl siloxane) has one of the lowest ~9 values known(-123°C) presumably because the Si-O bonds have considerable torsional mobility. The value of Tg is raised markedly by the insertion of groups which stiffen the chain by impeding rotation, so that more thermal energy is required to set the chain in motion. Particularly effective in this respect is the p-phenylene ring. Thus, a chain consisting entirely of p-phenylene rings, namely, poly(p-phenylene) (XVI)

(xv

ChainDimensions,Structures, and T~ansitional Phenomena

99

Figure 2.25 Elasticity of a polymer such as silicone rubber depends on the ease with which a randomcoil chain can be stretched out. The stretching is a consequence of the unwinding of bonds rather than a markedwidening of bond angles.

has a highly intractable, rigid structure with no softening point. This structure can be modified by introducing flexible groups in the chain to produce tractable polymers with high values of Tg. Some examples are poly(phenylene oxide) (X’VII), T~ = 83°C and poly(p-xylylene) (XVIII), Tg ~- 280°C as compared to polyethylene, Tg = -93°C and poly(ethylene oxide), T~ = -67°C.

~~-CH2--CH2~n (xvn)

(xviii)

Whenthe polymer chains are unsymmetrical, with repeat units of the type -(-CH~-CHX-)-,an additional restriction is imposed by steric effects depending on the size of the pendant group X. Bulky pendant groups hinder the rotation about the backbone and cause Ta to increase.

Problem2.21 Presence of flexible pendant groups reduces the glass transition of the polymer,whereasbulkyor stiff side group,s increase it. Why? Answer: In general, factors that increase the energy required for the onset of molecular motion increase Ta and those that decrease the energy requirement lower Ta. Flexible pendant groups act as "inherent diluents" and lower the frictional

Chapter 2

1 O0

interaction between chains reducing the Tg thereby. Bulky or stiff side groups, on the other hand, increase interchain friction and energy requirement for molecular motion, thereby increasing Tg.

Problem2.22 Account for the differences in glass transition

temperatures for the following pairs of isomeric polymers having similar chemical structures: (a) (b) (c) (d)

Poly(but-l-ene) (-24°C) and poly(but-2-ene) (-73°C). Poly(ethylene oxide) (-41°C) and poly(vinyl alcohol) (85°C). Poly(methyl acrylate) (10°C) and po[y(vinyl acetate) Poly(ethyl acrylate) (-24°C) and poly(methyl methacrylate) (105°C)

AI:ISWeF :

._(_ _ CH-)-fin CH

(a) .-~ CH2_CIH _.-~n

I I CH3 CH3

CH2 I

CH3

Tg = -73°C

T~ =-24~C The side group is long and bulky; therefore, rotation is difficult.

(b) "-4’ CH2-CH2-O ~

The side groups are shorter; fore rotation is easier.

there-

"-(CH:zCIH’~n OH

Tg =-41°C

Tg =85°C

The backbone chain is flexible; rotation is easy.

so

(c)

The steric effect of the -OH group restricts rotation. Superimposed on this is the effect of polarity, whichincreases the lateral forces in the bulk state and leads to higher

-(-CH2-C~ H-~n 0 o~CXocH3 0~c 3 ~ CH

T~ = lO°C The bulk~ group hinders causing T~ to increase.

Tg =.32°C rotation

The bulky part of the side group is farther from the chain, making rotation moredifficult.

101

Chain Dimensions,Structures, and TransRional Phenomena (d)

CIH3 O#C 5~0 C2H

#C’OCH~t 0 Tg :105°C

Tg =-24~C The side group is long but flexible, so rotation is easier.

The 1,1-disubstituted side groups makerotation moredifficult leading to higher Ta.

FACTORS AFFECTING Tm The application of macroscopic thermodynamics leads to some useful generalizations of factors affecting Tin. At Tin, the free energy change is zero, i.e., AGm = AHm -- TmASm = 0 whence Tm = AHm/ASm. This expression for Tm predicts that a high melting point can be the result of a high value of the enthalpy change At-~m and/or a small value of the entropy change ASm in melting. The former corresponds to stronger binding of adjacent but unbondedunits in the polymer lattice and thus to higher degree of crystallinity. The factors that affect crystallinity and Tmcan be classified as symmetry, intermolecular bonding, tacticity, branching, and molecular weight. These are discussed below. If /kSm is small, melting does not result in a large gain of conformational entropy and, to some degree, the structure of the solid must persist in the melt. For example, molecules of isotactic polypropylene crystallize in the form of helices and these are thought to occur in the melt also, thus making ASm small and Tm high. Linear symmetrical molecules such as polyethylene, polytetrafluoroethylene, and other linear molecules with more complex backbones containing -(--O--)-, --(-COO--)-, and --(---CONH--)- groups, such as polyethers, polyesters and polyamides, all possess a suitable symmetryfor crystallite formation and usually assume extended zig-zag conformations when aligned in the lattice. On the other hand, chains containing irregular units, which impair the symmetry,reduce the ability of the polyfner to crystallize. Thus c/s-double bonds (XIX), o-phenylene groups (XX) and m-phenylene groups (XXI) all encourage bending and twisting in the chains and make closepacking very difficult. H\

/

/H

C=C

(XlX)

(xx)

(xxD

Chapter2

102

If, however, the phenylene rings are para-oriented, the chains retain their axial symmetryand can crystallize more readily. Similarly, double bonds in trans configuration maintain the chain symmetrythus allowing for crystallite formation. This is highlighted by a comparison of the amorphouselastomeric c/s-polyisoprene (Tin = 28°C) with highly crystalline trans-polyisoprene (Tin = 74°C) which is a non-elastomeric rigid polymer, or c/s-l,4-polybutadiene (Tin = - ll°C) with trans-l,4-polybutadiene (Tin = 148°C). Any interaction between polymer" chains in the crystal lattice serves to hold the structure more ~irmly and raise the melting temperature. In polyethylene, the close packing in crystallites achieved due to high chain symmetry and lack of substituents on the chains allows the van der Waals forces to act cooperatively and provide additional stability to the lattice. In polymers containing polar substituents, e.g., C1, CN, or OH,the chains can be aligned and held rigidly by the strong dipole-dipole interactions between the substituents. This effect is more obvious in the symmetrical polyamides. Chain flexibility has a direct bearing on the melting point. Insertion of groups that stiffen the chain increases Tin, while introducing flexible groups into the chain lowers the value of Tm (cf. Factors Affecting Tg). Branching in the side group tends to stiffen the chain and raiseTm, as shownin the series poly(but-l-ene) (Tin = 126°C), poly(3-methyl but-l-erie) (Tin = 145°C), poly(3,3t-dimethyl but-l-erie) (Tin > 320°C). However, if the side group is flexible and nonpolar, Tmis lowered. Also, if the chain is substantially branched to reduce the packing efficiency, the crystalline content is lowered and hence the melting point. A good example is lowdensity polyethylene where extensive branching lowers the density and Tm of the polymer. :

RELATION

BETWEEN T,,,

AND T~

While Trn is a first order transition, Tg is a second order transition and this precludes the possibility of a simple relation between them. There is, however, a crude relation between Tm and Tg. Boyer [25] and Beamen [265] inspected data for a large numberof semicrystalline polymers, some of which are shownin Table 2.4. They found that the ratio Tg/Trn ranged from 0.5 to 0.75 when the temperatures are expressed in degrees Kelvin. The ratio is closer to 0.5 for symmetrical polymers such as polyethylene and polybutadiene, but closer to 0.75 for unsymmterical polymers, such as polystyrene and polyisoprene. The difference in these values maybe related to the fact that in unsymmterical chains with repeat units of the type -(CH2-CHX-)-an additional restriction to rotation is imposed by steric effects causing 2Pa to increase, and conversely, an increase in symmtery lowers T~.

103

ChainDimensions,Structures, and Transitional Phenomena Table 2.4 Tin, Tg, and Tg/Tm afor SomeSelected Polymers

Volymer Silicone rubber Polyethylene Polypropylene Polystyrene Poly(vinylchloride) Poly(vinylidenefluoride) Polyisoprene Nylon-6,6 aData fromRef. 27.

T,~,°c - 58 135 176 230 180 210 28 265

T~,°c - 123 - 68 - 18 100 82 - 39 - 70 50

T~/T,~,°K 0.70 0.50 0.57 0.75 0.78 0.48 0.67 0.60

Problem 2.23 A new atactic polymer of the type -(-CH2-CHX-),,- has of 80°C.Whatis its Tg likely to be ? Answer: The polymer being unsymmetrical, Tg/Tmmaybe assumedto be about 0.75. T/T,~ = 0.75, T,~ = 80°C = 353 °K T~ =265 °K=- 8°C

THEORETICAL TREATMENT OF GLASS TRANSITION Experiments to measure T9 often show that the measured value of Tg is dependent on the time allotted to theexperiment and that T9 decreases as the time allotted is increased. One may therefore ask: Is there an end to the decrease in Tg as the experiment is slowed? Howcan the transition be explained on a molecular level? These are the questions to which the theories of the glass transition are addressed. There are three main groups of theories of the glass transition [28-30] : (a) the free volumetheory, (b) the kinetic theory, and (c) the thermodynamic theory. Although these three theories may at first appear to be different, they really examine three aspects of the same phenomenon and can be unified successfully, though only in a qualitative way.

The Free-Volume Theory According to the hole theory of liquids, first developed by Eyring [31], molecular motion in liquids depends on the presence of holes or voids, i.e., places where there are vacancies. When.a molecule moves into a hole. a

104

Chapter 2

000000 000-O0 000000 O0 000 000000 000000 Figure 2.26 A quasicrystalline lattice exflibiting vacanciesor holes. Circles represent molecules; arrow indicates molecularmotion. new hole is created in its place, as illustrated by the motion indicated in Fig. 2.26. For real materials, however, Fig. 2.26 has to be visualized in three dimensions. A similar model can also be constructed for the motion of polymer chains, the main difference being that more than one "hole" will now be required to be in the same locality, as cooperative motions of a host of consecutive chain atoms are required for the movementof polymer chain segments. Thus, for a polymer chain segment to movefrom one position to an adjacent site, a critical void volumemust first exist before the segment can jump. That empty spaces exist can be inferred from the fact that when a sample of polystyrene is dissolved in benzene there is a contraction in the total volume. This indicates that the polymer can occupy less volume when surrounded by benzene molecules and there must have been unused space in the polymer matrix to allow this increase in packing efficiency to occur. On this basis, the observed specific volume of a sample, v, can be described as a sum of the volume actually occupied by the polymer molecules, v0, and the free volume, v1, in the system [see Fig. 2.27(a)], i.e., v = v0 + vy (2.18) The free volume is a measure of the space available for the polymer to undergo rotation and translation and the magnitude of this space will increase with temperature as the molecular motion increases. The glass transition can thus be visualized as the onset of coordinated segmental motion madepossible by an increase of the free space in the polymer matrix to a size sufficient to allow this type of motion to occur. Conversely, if the temperature is decreased, the free volumewill contract and eventually reach a critical value when it becomesinsufficient to allow large scale segmental motion to take place. The temperature at which this critical value is reached is the glass transition temperature (Tg). As the temperature decreases further below Tg, the free volume, i.e., v.f in Eq. (2.18), will remain essentially constant at v}, since the chains have now been immobilized, and frozen in position. In contrast, the occupied volumeVo will alter because

ChainDimensions,S~ruc~ures, and Transitional Phenomena

V

105

Ca)

Temperature(T)

(b)

o>Vo,G~

.- ... -"

..=.

¯0 -- V Q.

¯

Temperature (T) Figure 2.27 (a) Schematicillustration of the variation of specific volumewith temperature. The free volume(represented by shaded area) is assumed to constant at v~ belowTg and to increase as the temperatureis raised aboveTg. (b) Schematicillustration of free volumeas calculated by Simhaand Boyer[34[. of the changingamplitude of thermal vibrations in the chains and, to the first approximation, will be a linear function of temperature. The temperature coefficient of the specific volumev will therefore change markedly at According to the iso-free-volume state theory of Fox and Flory [32,33], same free volumeexists at the respective glass temperature (Tg) independent of molecular weight and this same free volumeis retained at all temperatures below T9. This can be explained by considering that at temperatures below T~ the segmental rotations become frozen and the holes that are present at the glass temperature become immobilized. Because the holes are no longer able to diffuse out of the structure, the free volume, which is the sum of the holes, remains nearly unchanged at all temperatures below Te. However,the internuclear separation can still adjust itself belov~ T~ as

106

Chapter2

the thermal vibrations of the atoms becomereduced. At temperatures above the glass transition, both the internuclear separation between segments of neighboring chains and the number and size of holes adjust themselves continuously with changing temperature (provided the changes occur sufficiently slowly to allow time for segmental diffusion). The coefficient of expansion above 7"g is therefore higher than that below 7"g. Simha and Boyer [34] thereafter postulated that the free volume at T = Tg should be defined as v.f

= v - vo,~(1

(2.19)

+ (~GT)

where v is the specific volume (i.e., volume per unit mass), vy is the specific free volume, and v0,R is the extrapolated specific volume. Figure 2.27 illustrates these quantities. It is seen that vo,R and Vo,a are the hypothetical specific volumes extrapolated from melt or rubbery state to 0°K using o~R and aG as the coefficients of expansion. Thus at T = Tg v = v0,R(1

+ a2{T)

(2.20)

v : vo,(::;(1

O~GT)

(2.21)

and Substitution of v from Eq. (2.20) in Eq. (2.19) at Tgle adsto the following expression for free-volume fraction [34]:

(-R - c)Tg = w/v0,R w/v

(2.22)

[since v] << VO,Rand c~GT<< 1 in Eq. (2.19)] Simha and Boyer [34] postulated that the free-volume fraction defined by Eq. (2.22) is the same for all polymers, that is, (at{ - o~G)Tg = constant = /(1 (2.23) From the values of (GR -- cec)Tg based on data of o~R and eta determined for various polymers, Simha and Boyer [34] concluded that K~ in Eq. (2.23) is 0.113, i.e. (oe~ - o~a)Tg : 0.113 (2.24) This implies that the free volumefraction at the glass transition temperature is the same for all polymers and constitutes 11.3%of the total volumein the glassy state. (Many simple organic compoundshave a 10%volume increase on melting, it maybe pointed out.) This is the largest of the theoretical values derived, but the first. Other estimates placed the free volume at about 2%. The use of czG in Eq. (2.19) results from the conclusion that expansion in the glassy state occurs at nearly constant free volume; hence o~GTis proportional to the occupied volume. A less exact but simpler relationship 1S

o~RTg = constant

= K2

(2.25)

Chain Dimensions, Structures,

It follows from the assumption that the fractional [cf. Eq. (2.20)]: (v

- VO,_Ft)/V

107

and Transitionad Phenomena

= aRT/(1

+ aRT)

free volume defined

= aRT + O[(aRT)

21

as

(2.26)

is constant at T Equation (2.26) neglects the contribution to total expansion supplied by aGT. From the values of aR determined for various polymers [34] Eq. (2.25) is given aRTg = K2 = 0.164 (2.27) The quantities K1 and K2 provide a criterion for the glass temperature, especially for new polymers or when the ~,alue is in doubt, e.g., in systems with multiple transitions and in semicrystalline polymers, where Tg may be obscured. The relation expressed in Eq. (2.24) has been a subject of more recent research. Simha and coworkers [35, 36] found Eq. (2.24) still acceptable, -4 while Sharma et al [37] found a_a -- &c roughly constant at 3.2×10 -1. deg

Problem2.24 A new polymer is found to have glass transition

at 90°C and its cubic coefficient of thermal expansion is 5.6×10-4 °K-1 at 120°C. Can this polymer be used in an application which requires the cubic coefficient of thermal expansion to be less than 4×10-4 °K-1 at 60°C? Answer: Above Tv, the expansion coefficient

is ~R. Hence ~R = 5.6x10-4 °K-1.

T~ = 90°C -- 353 °K at~Tg ---- (5.6x 10-4 °K-~)(363 °K) = 0.203 From Eq. (2.52): OzGT ~ = 0.203 -- 0.113 = 0.09 ac = 0.09/(363 °K) = 2.5 × -4 °K-a Also from the relationship [37] -4 C~R -- aG = 3.2x10 ac = (5.6 - 3.2) I0-4 = 2.4 x I0-4 °K-I Hencethepolymer is likelyto satisfy thethermal expansivity requirement.

The

WLF

Equation

Polymers soften and flow at temperatures of molecular motion, requires a critical

near and above Tg. Flow, a form amount of free volume. The free

108

Chapter2

volume at Ta arises as a fundamental constant in the WLF(WilliamsLendel-Ferry) equation which provides a quantitative relationship between polymer melt viscosity and free volume. The WLFequation has applications to viscosity and other polymer problems. The derivation of the equation is considered here. Early work of Doolittle [38] on the viscosity (r/) of nonassociated pure liquids such as n-alkanes led to an equation of the form lnr/

= lnZ

+ B(V-~l

)

(2.28)

where A and B are constants, and v0 and v,f are, as before, the occupied volume and specific free volume, respectively, related by Eq. (2.18). The Doolittle equation can be derived by considering the molecular transport of a liquid consisting of hard spheres [39, 40]. The relevance of the Doolittle equation in the present context is that it provides a theoretical basis for the WLFequation [41]. On the molecular level, the ratio (vo/vf) is a measure of the average volume of the polymer relative to that of the holes. Thus, when v0 > vf, i.e., the polymerchain is larger than the average hole size, the viscosity will be correspondingly high, and when vo < v f, the viscosity will be low. Wecan nowuse a symbol f for the free volume fraction defined earlier by Eq. (2.22): vf vy f -- vy -~ -(2.29) v Vo + v.f Vo since vI << v0. Substituting lnr/

Eq. (2.29) in Eq. (2.28), = lnA

+ B/f

(2.30)

Let r/T be the viscosity of a polymer melt at temperature T and r/g that at a reference temperature Ta. So, from Eq. (2.30),

Here fT and fa are the fractional free volumes at T and Ta, respectively. Referring to Fig. 2.26(a) we note that at and below Tg, the fractional free volume f, defined as f = vy/v [cf. Eq. (2.22)], is given fa = v~f/v and can be considered as being effectively constant (see p. 65 for explanation). Above a t here will b e a n i mportant c ontribution t o vI from the expansion of the polymer in the rubbery or molten state. The free volume above Ta is then given by )vf

= v~ + (T

- Ta)(Ov/OT

(2.32)

Chai~Dimensions,Struc~ares, and Transitional Phenomena

109

Dividing through by v gives

f = f~ + (T- Tg)~

(2.33)

where o~$ is the thermal expansion coefficient of the free volume. (In a region above Tg but sufficiently close to it, af will be given by the difference between the thermal expansion coefficients of the rubbery and glassy polymer.) Using a subscript T for f to indicate temperature, Eq. (2.33) can substituted in Eq. (2.31) to give

in UT "=

B f~ _

+ af(T-

T~)

Ba.f(TTg) :~{f~ + af(TTg)}

(2.35)

Dividing both numerator and denominator by oq, Eq. (2.35) is rewritten as

(f~/o~:)

+ (T-

Equation (2.36) is one form of the WLFequation. Since the amount of flow is proportional to the flow time t and density p and inversely proportional to viscosity, one maywrite

and hence

neglecting small differences in density. Substituting Eq. (2.37) in Eq. (2.36) then gives

ogl0 = -(B/2.303fg)(TT : (IST) This can be compared with the WLFequation loglo a/r =

-CI(TC2 + (T-

Tg) T~)

(2.39)

I 10

Chapter2

where aT is called the reduced variables shift factor [42], (71 and C2 are constants that can be evaluated from experimental data. For many linear polymers, independent of chemical structure, C1 = 17.44 and C~ = 51.6 when Ta is the reference temperature, and the WLFequation is then log10 aT "-

-17.44(T

- T,)

1.6 + (T-

(2.40)

Problem 2.25 A new linear amorphous polymer has a Ta of +10°C. At 27°C it has a melt viscosity of 4x10s poises. Estimateits viscosity at 50°C. Answer: Eqs. (2.37)-(2.39): T = 27°C, r/= 4x10s poise, log[4Xl0S] I_ r/~ j

_

-17.44x17 51.6 + 17

Tg = 10°C,

T - Ta= 17°C

- -4.32

r/g = 8.3 x 1012poise At T = 50°C, T - Ta = 40°C _r/ log 8.3x1012 [ ]

-17.44 x 40 = -7.61 = 51.6 + 40 2.0 x 10s poise

A more general form of the WLFequation reads l°gl0aT

-8.86(TTs) = 101.6 + (T

(2.41)

where Ts is the arbitrary reference temperature usually located about 50°C above Ta. The constants C1 and C2 now have different values, and the shift factor aT is expressed as a ratio of relaxation times, 7, at T and that is, aT = 7(T)/~’(Ts) (2.42) The relaxation time is a function of the viscosity and modulus (E) the polymer and, according to the Maxwell model, "r = rifE. (The viscosity, r?, has the units of dyne-s/cm2r and the modulus, E, has the units of dyne/cm2, so ~- has the units of time.) Since the modulus is much less temperature dependent than the viscosity, we can write aT = (r/T/r/s),

ChainDimensions,Structures, and Transitional Phenomena

111

which demonstrates the equivalence of the empirical equation (2.39) with the Eq. (2.64) derived from the free volume theory. ComparingEq. (2.40) with Eq. (2.38), B/2.303fg fg/af

= 17.44

(2.43)

= 51.6

(2.44)

Here, three unknowns and two equations are shown, which can be solved by assuming for the constant B a value of unity [41], consistent with the viscosity data of Doolittle. Then fg -4 = 0.025, and o~.f = 4.8x10 °K-1. Sharma et al [37] found ay = 3.2×10° °K-a.

Problem 2.26 For the polymers for which af = 4.8x10-4 °K-*, estimate a value of Tg assumingthat these polymersalso obey the S-B relationship given by Eq. (2.24). Answer: Since the free volume is assumedconstant below Tg, the volumechange with temperature in the glassy region is only due to the occupied volume, while the volumechange in the robbery region above T9 has contributions from both the occupied volumeand the free volume. Therefore, one can write, approximately, aR -- aa ~ a I = 4.8×10-4 OK-1 Substituting this in Eq. (2.24), Ta ~ 0.113/(4.8×10 -4 °K-I) = 235 °K -- - 38°C. This value is in the range of the Ta’s observed for manypolymers.

The finding of fg = 0.025 is significant. It thus assigns a value of 2.5% for the free volume at To. This is low compared with the S-B estimation (11.3%). Other values ~ of 8 percent from the "hole" theory of Hirai and Eyring [31] and 12 percent calculated by Miller [43] from the heats of vaporization and liquid compressibilities, illustrate the uncertainty surrounding the magnitude of this free-volume parameter. For numerical results, the WLFequation has been found to be gi~od for the temperature range Tg to (Tg + 50). The merit of the equation lies in its generality as no particular chemical structure is assumedother than a linear amorphouspolymer above r~. For polymer scientists and rheologists over the last four decades, the WLFequation has provided a mainstay both in utility and theory. While the WLFequation is based on the free-volume theory of glass transition which is concerned with the introduction of free volume as a requirement for coordinated molecular motion, the equation also serves to introduce some kinetic aspects into the quantitative theory of glass transition.

112

Chapter

Problem2.27 The experimental the value of Ta dependson the time or frequency fiame of the experiment. Calculate from the WLF equation the change that would be expected in the Ta value if the time frame of an experimentis decreased by a factor of 100. Answer: FromEq.(2.40):

r-.% \T - T~] -17.44/51.6 = -0.338 Since the time frameof experimentis decreasedby a factor of 100, the shift factor aT is 1/100. Therefore, T - Tg - -0.338

"~ 6°K

So the glass transltlon temperature would be raised by about 6°C. This is in agreement with experiment.

Effect of Molecular Weight on Tg The value of Tg depends on the methodof the experiment but it is also found to be a function of the polymer chain length. At high molecular weights T~ is essentially constant when measured by any given method, but is found to decrease as the molecular weight of the sample is lowered. This behavior would be expected since a polymer chain end requires more free volume in which to move about than a segment in the chain interior and, being connected only at one end, the end units are able to rotate more readily with increasing thermal energy than the rest of the chain. Thus the more chain ends a sample has (i.e., the lower the molecular weight) the greater the contribution to the free volume when these begin moving; consequently T9 decreases as the. molecular weight of the sample is lowered. This behavior can be approximated to an equation of the form [44,45]

T~ = T~,oo - K/A¢

(2.45)

relating Tg to molecular weight M; Tg,5o in this equation is the value of Tg for a polymer sample of infinite molecular weight and K is a__constant. In a polymer sample of density p and molecular weight /~/n, the number of chains per unit volume is given by pNA,/Mn, where NA, is Avogadro’snumber, and so the number of chain ends per unit volume is 2pNAv/Mn. If/9 is the contribution of one chain end to the free volume then the total fractional free volumedue to chain ends fc is given by

fc = 2pNAvO/Mn

(2.46)

113

Chain Dimensions,Structura%and Transitional Phenomena

o 376 37Z~ ~ 372 370 [

0

~

Figure 2.28 Plot of Tg against reciprocal molecular weight (Problem 2.28). It can be argued that if a polymerwith this value of fc has a glass transition temperature of Tg then fc will be equivalent to the increase in the volume on expanding the polymer thermally between Ta and T9,o~. This means that fc

= c~f(T9,~

- Ta)

(2.47)

where o~f is the thermal expansion coefficient of the free volume. Combining Eqs. (2.46) and (2.4"7) and rearranging leads to the equation

:

2pNAO

(2.48)

which is exactly of the same form as Eq. (2.45) when K 2pNAO/o~f. The observed dependence of the Tg upon molecular weight is thus predicted from considerations of free volume.

Problem 2.28 A polydisperse polystyrene sample was fractionated into four componentsof various molecular distributions and Tg of each fraction was measured:

Chapter 2

1 I4

Component 1 2 3 4

Wt. fi’action (w) 0.05 0.41 0.39 0.15

M~ 1.5x106 4.8x105 1.2x105 3.7x104

Tg (°K) 378.9 378.5 377.2 373.3

(a) Obtain a relation between 9 and molecular w eight. (b) Calculate the T9 value of the polydisperse polystyrene. Answer: (a) T~ is plotted against 1/~, in Fig. 2.28 according to Eq. (2.45), intercept and slope of the straight line plot,

From

Ta,~ = 379°K, K = 2.1x105 °K g mo1-1 (b) From the definition

1

W 1

of number average molecular weight (see Chapter 4): ’//9,~

W 3

W 4

Using Eq. (2.45),

K

K

K + ~3(T.,~

K

- Tg,3)

+

K

Simplifying, T¢ = w,T9,1 + w2T~,2 + w3T~,3 + w,~T~,4 = (0.05)(378.9 °K) + (0.41)(378.5 °K) + (0.39)(377.2 °K) + (0.15)(373.3 = 377.2 °K

Effect of Branching on T~ A small number of branches on a polymer chain are found to reduce the value of Ta. This also can be explained using the free volume concept. Since branches give rise to chain ends, the above analysis of the effect of molecular weight on T~ can be extended to branching. Thus, the glass transition temperature of a chain possessing a total number of y ends per chain is given by an equation of the form

T~ : Tg, oo

ypNAvO o~ f M~

(2.49)

where Tg, oo is again the glass transition temperathre of a linear chain of infinite molecular weight. Since a linear chain itself has two ends, the

Chain Dimensions,Structures, and Transitional Phenomena

115

number of branches per chain is (y -- 2). Equation (2.49) is, however, valid only when the numberof branches is low. A high density of branching will have the same effect as side groups in restricting chain mobility and hence raising T a. Problem 2.29 The Tg of a linear polymer with ~]~ = 2500 was found to be 120°C,but it increased to 150°Cfor a sampleof the samelinear polymerwith M~ = 10,000. A branched version of the same polymer with M~= 6,000 was found to have a Tg of 114°C. Determine the average numberof branches per chain of the branched polymer. Answer: If pNA~,O/oqis assumedto be constant (K) over the temperature range defined by the T~’s, Eq. (2.49) can be written Ky Ta = Tg. oo --~,~ Linear

polymer:

~ =2500gmo1-1, T~ =393 °K, y =2 2K 393°K = Tg,~- (2300 gmo1-1)

Linear polymer : ~ = 10,000gmo1-1, Tg = 423 °K, y = 2 2K 423°K = Ta,~ -

(10000g

mo1-1)

(P2.29.1)

(P2.29.2)

SolvingEqs. (P2.29.1) and (P2.29.2) simultaneouslygives -1----- 4.48x104g °Ktool K Tg, ~ =432 OK Branched polymer: _~--], = 6,000 g tool -1, Ta = 387 °K, y = ? (4.48 x 104 g °K mol-1)l 387°K = 432°K -- y i~-0~ g--m-~-i---fy J y = 6.02 Therefore, the average numberof branches = 6 - 2 = 4

Effect of Cross-linking on Tg The effect of cross-linking on 7’~ can be treated along similar lines to the effects of molecular weight and branching. Whentwo polymer chains are connected by introducing a crosslink at intermediate points, the chains are pulled closer together at these points and the free volumeis decreased. This reduction in the free volume raises ra.

116

Chapter 2 A semiempirical equation relating T9 to cross-linking [46] is

ATg,c = Zu (2.50) where the change in the glass temperature due to cross-linking, ATg,c, is equal to the cross-link density, u, times a constant Z. According to this equation, Tg will increase linearly with u. It is, however, applicable only at sufficiently low ~. For very muchhigher degrees of cross-linking Tg is found to rise more rapidly than the cross-linking density [45]. Effect

of Diluents

on Tg

The effect of adding a low-molecular-weight substance to a polymer is to lower its glass transition temperature. The effect is well known, and is exploited in practice to obtain flexible products of poly(vinyl chloride) adding relatively nonvolatile diluents such as dibutyl phthalate or tricresyl phosphate as plasticizers to the polymer. Since the free volume of a low-molecular-weight liquid is very large compared to that of a polymer at the same temperature and pressure, the addition of even a small amount of diluent has a large effect on the total free volume in the polymer phase, which is reflected in a significant drop in Tg. Empirically Tg of a polymer-diluent mixture could be expressed in terms of the glass transition temperature of the pure polymer Tgp and that of the pure diluent rgd by

Tg

=

+

+

-

(2.51)

where wp and Wdare the weight fractions of polymer and diluent, respectively, and b is an empirical parameter for the polymer-diluent system. A relation between T~ and composition of a polymer-diluent mixture can be derived [47] in a relatively straightforward manner from the freevolume concept by postulating that the free volumes of the polymer and diluent are additive in the mixture, and that the free volumefraction has a critical value fo, which is the same for the pure polymer, the diluent and their mixtures-~at their respective glass temperatures. The composition of polymer-diluent mixtures is conveniently expressed in terms of the volume fractions of polymer ep and diluent ed. According to the free-volume theory, as we have seen earlier, the freevolume fraction f in a polymer at a temperature T above Tg can be expressed in a linear form [cf. Eq. (2.33)]: f = f~ + (~R -- ~)(~ -- T~) (~.5~) where fg is the free volume fraction at the glass transition; O~Rand o~Gare respectively the coefficients of thermal expansion above and below ~g. Considering 1 cm3 of polymer-diluent mixture comprising ep cm3 of polymer and ed cm3 3of diluent, the free volume f associated with ep cm

117

ChainDimensions,Structures, and Transitional Phenomena of polymer at the glass temperature Ta of the mixture is, therefore,

fp = ep[fg + - G)(Ta- Tap)]

(2.53)

Similarly for ed. cm3 of the diluent the free volumefd is fd = ed[fg

+ ad(Tg

--

Tad)]

(2.54)

since for a simple liquid the whole of the thermal expansion may, to a first approximation, be regarded as contributing to the free volume. Since for 1 cm3 of the mixture the free volumefraction at Tg, i.e., fa, is given by (fp q- fd), adding Eqs. (2.53) and (2.54) and rearranging gives

=

epTap(aR

_

--

aG) + edTgdad

+

(2.55)

Since (aR -- aG) is roughly constant for many polymers - a value 2.2×10 -4 °K-1 being reported by Sharma et al [37] - Eq. (2.54) can be used to predict the Ta value of a polymer-diluent mixture of given composition, provided Ta values of the polymer and the diluent and ad of the diluent are known.

Problem2.30 In plasticization of poly(methylmethacrylate) (Tg --= 105°C) diethyl phthalate (Tg = - 65°C, ad = 10×10-4 °K-l), how muchof plasticizer wouldhave to be added in order for Ta to be 50°C ? Answer: Substituting ep = 1 - q~d; T~ = 105°C = 378 °K; °K-1 " Tad = -- 65°C --= 208 °K " ad = 10-3 °K-1 " Eq. (2.55) becomes

-4 aR -- aa = 3.2x10 Ta = 50°C =- 323 °K,

-a 323 = (1 - ed) X 378 × 3.2 × 10-4-4 + ed × -208 X 10 (1 - ed) x 3.2 x + ed X 10 ~ Solving, qtd = 0.13. So, 13%(by vol.) of plasticizer polymer.

Effect

of Copolymerization

is to be added to the

on Tg

The glass transition behavior of copolymers can also be analyzed using the free volumeconcept. Block and graft copolymers usually have multiple glass transition temperatures that are near to the values of Tg for each constituent homopolymers.In contrast, random or statistical copolymers usually have a single Tg between that of the corresponding homopolymers. If a aeries of such copolymers are produced from two monomersby varying their relative

118

Chapter2

proportions and Tg is pl~otted against composition, then it is found that the Tg of the copolymerlies on or (more usually) below a straight line joining the T~’s of the two homopolymers. The dependence of the Tg of a random or statistical copolymer upon composition can be predicted if it is assumed that each type of monomer unit has a characteristic free volumeand that it is the same in a copolymer or homopolymer. Denoting the two types of monomerunits in a copolymer by subscripts 1 and 2, Eq. (2.33) can be written

fl = f~,l + (T- T~,1)~,1 f, = fg,~ + (T- T,,,)~<,2

(2.56) (2.57)

and for the copolymer

f = L + (T- Tg)~

(2.58)

where f is the fractional free volume at the glass transition temperature and o~f is the thermal expansion coefficient of the free volume. If it is assumed that in the copolymer the free volumes add in proportion to the weight fractions Wl and w2 of comonomers 1 and 2, then

(2.59) and f = wlfx

+ w~f2

(2.60)

If the thermal expansion coefficients of the free volumealso add in a similar manner, then (2.61) O~I 2 -~ WlO~f,1 ~- W20~f, combining Eqs. (2.56)-(2.61) and rearranging leads Tg(W~ + t3w~) ---where fi = ~],2/~,1.

wlTg,,

+ flw2Tg,~

(2.62)

If fl ~ 1 then

T~= ~T~,~+ ~T~,~

(2.6~)

which gives a straight line relationship between Tg of the copolymer and the copolymer composition. This equation usually predicts Tg too high. By assuming, as above, that each Wpeof monomer unit retains its characteristic free volume in the copolymer above Tg and using the isofree-volume criterion for the glass transition, the following relationship has been derived 08] 1 1 (2.64) [ w~ Bw~]

~. - (~1 + ~) + ~. ,~ ]

Here B is a constant for the pair of monomers.It is specified by theo~ but is difficult to predict numerically; it is however never far ~omuniW. When

ChainDimensions,S~ructures, and Transifional Phenomena

119

B is exactly unity, Eq. (2.64) reduces to the very simple, but nonlinear form 1 w1 w 2

= Tg,--; + Tg,2

(2.65)

This is the familiar Fox equation [49]. This equation predicts the typically convex relationship obtained when Tg is plotted against Wl or w2. Equation (2.64) can be rearranged to a more convenient form:

l(Tg - T.,1) + k 2(Tg- = 0

(2.66)

where k = B(Tg,1/Tg,2). This is knownas the Woodequation, which fits data for manycopolymers very well. ThoughEqs. (2.63) to (2.66) are derived for random copolymers, these are also applicable to those cases of polymer blending and plasticization with low molecular weight compounds where phase separation does not take place and one phase is retained.

Problem2.31 The following data were obtained for Tg of compatible blends of polyphenyleneoxide*(PPO) and polystyrene (PS) as a function of weight fraction (wl) of PS wl 0 0.2 0.4 0.6 0.8 1.0 T9 (°K) 489 458 431 413 394 378 By fitting the data to a suitable theoretical curve estimate the value of Tg for a PPO/PSblend with PS weight fraction of 0.3. Answer: The Tg values of the PPO/PSblends are plotted against the weight fi-action wl of PPOin Fig. 2.29. The curves for the linear equation (2.53), the Fox equation (2.55) and the Woodequation (2.55) are drawnon the same graph. Best fit seen to be obtained for the Woodequation with k = 0.579. From Eq. (2.55): WlTg,1 + kw2Tg,2 wl + kw2 With wl = 0.3, w2 = 0.7, T~,a = 378 °K, Tg,2 = 489 °K, Tg = 0.3× (378 °K) + 0.679×0.7× (489 0.3 ÷ 0.679 × 0.7 ---- 446 OK Problem 2.32 Howmuch of a low molecular weight plasticizer with T~ = 80°C should be added to a film of nylon-6,6 in order that Tg is reduced from 50°C to 25°C ?

120

Chapter 2

Assumingsingle phase is retained on plasticization, equations derived for random polymers can be used Ta,I=50°C =323 °K, Ta,2 =- 80°C= 193 °K, Ta = 25°C-=298 °K From Eq. (2.63)

(298°K)

(1 - ~)(323°K)+ ~o2(193 0.19

From Eq. (2.65) 1

(298OK) W 2

323 °K) 193 °K)’

0.17

The difference is due to higher Ta value predicted by the linear equation.

490 470

~

Linear Eq: (2-63)

450

.65)

~

430 410

--/ WoodEq. (2.66) 390 370 0

~ I

f

0.2

1

I

I

0.4

I

0.6

I

I

I

0.8

.0

~O 1 Figure 2.29 Plot of T. against composition for PPO/PSblends according to (a) the linear equation, (b~) the Fox equation, and (c) the Woodequation. Circles represent experimental points (Problem2.31).

ChainDimensions,Structures, and Transitional Phenomena RELAXATION PROCESSES POLYMERIC STATE

IN

121

AMORPHOUS

The amorphous or molten polymer is a conglomeration of badly packed interlacing chains, and the extra empty space caused by this random molecular arrangementis called the free volume, which essentially consists of all holes in the matrix. Whensufficient thermal energy is available, the vibrations can cause a segment to jump into a hole by cooperative bond rotation and a series of such jumps will enable the polymer chain eventually to change its position. While the translation of a polymer chain can proceed by means of a series of segmental jumps (explained above) involving short kinetic units, the complete movementof a chain cannot, however, remain unaffected by the surrounding chains since considerable entanglement exists in the melt and any motion will be retarded by other chains. According to Bueche [50}, the polymer molecule may drag along several others during flow and so the energy dissipation is due to friction between the chain plus all those which are entangled and the neighboring chains as they slip past each other. The length of the polymer chains in the sample would thus be expected to play a significant role in determining the resistance to flow. The effect of chain length on melt viscosity (r/), measuredat low shear rates ensure Newtonianflow, is illustrated in Fig. 2.30. The plot [51] shows the dependenceof melt viscosity on chain length Z defined as the weight average number of chain atoms in the polymer molecules. (For vinyl polymers is twice and for diene polymers four times the weight-average degree of polymerization.) Below the critical chain length Zc the melt viscosity is given by 1"° (2.67) ~7 = KLZ and above Zc, the melt viscosity is given by 3"4 (2.68) r] = KHZ where lr(L and KHare constants for low and high degrees of polymerization. They are temperature dependent. While the 1.0 power dependence in Eq. (2.67) represents the simple increase in viscosity as the chain gets longer, the dependenceof the viscosity on the 3.4 power of the chain length as shown in Eq. (2.68) arises from entanglement and diffusion considerations. The critical chain length Zc is interpreted as making the dividing line between chains which are too short to make a significant contribution to viscosity from entanglement effects and those large enough to cause retardation of flow by entanglement with their neighbors. Typical values for Zc are 610 for polyethylene, 730 for polystyrene, and 208 for poly(methyl methacrylate). In general, Zc is lower for polar polymers than for nonpolar polymers.

122

Chapter 2

6.0

4.0

2.0

0

-2.0

3.0

4.0

5.0 [Og~O -~

6.0

Figure2.30 Dependence of melt viscosity on chain length Z, for polyisobutylene fractions

measured at low shear rates and at 217°C. (From Ref. 51.)

Problem2.33 A certain extruder for plastics was found to work best at a melt viscosity of about 20,000 poises. The vinyl polymer of DP,o = 750 and Tg ----- 80°C usually used with this extruder had this viscosity at 150°C. However,a batch of the same polymer received subsequently has DP,~ = 500. At what temperature should the extruder now be run so that the viscosity remains at optimum conditions ? Answer: For a vinyl polymer, Z = 2(D’~n). Since both samples have Z > 700, Eq. (2.68) can be used for both. For polymer with DP~ = 750 and at 150°C, 2 × 104 = KH(2 3. X 4750) -7 KH = 3.18 X 10 at 150°C~ For polymer with DP~ = 500, r/= 3.18×10-7(2x500) 3’4 = 6023 Poise at 150°C

Chain Dimensions,Structures, and Transitional Phenomena

i23

The WLFequation (cf. Eq. (2.40) can be written iog~T : -17.44(T - T~) ~;g 51.6 + (T Since at high molecularweightT~ is essentially constant, both the polymersamples maybe assumedto have the same T~. Therefore, 6023 -17.44(150 - 80) log -~g 51.6 + (150 - 80) ~?a = 6.6 x 1013 Poise Applying the WLFequation again 2 x 104 -17.44(T - 80) log 6.6 × 1013 -- 51.6 + (T - 80)"

The Reptation

Solving, T -- 142°C

Model

The large difference in viscosity variation below and above Zc suggests a very clear-cut distinction between the movementof chains of length less than Zc and the relative immobility of the entangled chains with lengths greater than Zc. Since independent chain mobility is not possible for these longer chains after the onset of entanglement, a modified modelis required to account for the ability of 10ng chains to translate and diffuse through the polymer matrix. De Gennes [52] introduced his theory of reptation of polymer chains. His model, illustrated in Fig. 2.31, consisted of a single polymeric chain, P, trapped inside a three-dimensional network of entangled chains, described as a ’gel.’ The gel itself maybe considered to constitute a set of fixed obstacles as shownin Fig. 2.31. The chain P is not allowed to cross any of the obstacles; however, it may movein a snakelike fashion amongthem [52]. This snakelike motion is called reptation. In the de Gennes approach, the polymer chain is assumed to be contained in a hypothetical tube [Fig. 2.32(a)] which is placed initially in three-dimensional network formed from other entangled chains. Although for simplicity these network"knots" are shownin Fig. 2.31 as fixed obstacles around which the chain under consideration must wriggle during translation, in practice these obstacles would also be in motion. The contours of the tube are then defined by the position of the entanglement points in the network. Mechanistically, reptation can be regarded as the movementof a kink in the chain along its length [see Fig. 2.32(b)] until it reaches the end the chain and leaves it. Motion of this kind translates the chain through the tube and successive defects movingthe chain in this way will eventually take it completely out of the tube. The time required for a chain to movecompletely out of the tube is the relaxation time T. Since the time necessary to move a certain distance

124

Chapter 2

P

(a)

0

Figure 2.31 A model for reptation.

The chain P moves among fixed obstacles,

O, but cannot cross any of them.

(b) Figure 2.32 (a) Schematic representation of a polymer chain restrained hypothetical tube. (b) Movementof a "kink" along the chain.

in

ChainDimensions,Structures, and Transitional Phenomena

125

depends on the square of the distance, the time necessary for the chain to reptate out of the tube depends on the length of the tube, L, squared. Then, ~- ~- L2 / Dt (2.69) Here Dt is the diffusion coefficient within the tube and can be expressed as the frictional coefficient for the chain within the tube confines, i.e., Dt = ]~T/ft. Since the reptation is assumed to occur by migration of a segmental kink along the chain, the force needed to do this is applied one segment at a time and so it is essentially proportional to the number of atoms (Z) in the chain. Thus, r ~ L~Z (2.70) Since L is proportional to Z, 3 ~- o¢ Z (2.71) Equation (2.71) shows that the relaxation time is proportional to the cube of the chain length. This is the fundamental result of the reptation model. The viscosity of the system is given by r/ = ~-E (Maxwell model) and according to the reptation model, the modulus .E depends on the distance between obstacles and does not depend on the chain length. Therefore r/ e( Z3 (2.72) Equation (2.72) should be compared with Eq. (2.68). While the power dependenceis not quite correct in this simple derivation, it illustrates the principal molecular weight dependence of the viscosity. Using scaling concepts, de Gennes [52] found that the diffusion coefficient, D, of a chain in the entangled polymer matrix depends on the molecular weight M as D c( -2 (2.73) Numerical values of the diffusion coefficient in bulk systems range from 10-2 to 10-6 cm2/s. For example, polyethylene of 1 x 10~1 g/tool has a value of D near lxl0 -8 cm2/s at 176°C and polystyrene of lx105 g/tool has a diffusion coefficient of about 1 x 10-12 cm2/s at 175°C. The temperature dependence can be determined either through activation energies (Ea 90 kJ/mol for polystyrene and 23 kJ/mol for polyethylene) or through the WLFequation [Eq. (2.40)]. The diffusion law of Eq. (2.73) holds for the "welding" of polymers at an interface which can be explained by reptation. Whentwo blocks of the same polymer are brought and held at a temperature just above the T9 for a short time t, interdiffusion of the chains takes place from each block across the interface thereby joining the blocks together. The strength of the junction formed will depend on time t.

Problem 2.34 Bulk polymeric materials being pressed together under-molding conditions require diffusion of the order of 100 ) to producea significant number

126

Chapter 2

of entanglements, thereby fusing the interracial boundary. At 170°C, the diffusion coefficient of polystyrene was found to depend on .the weight-average molecular weight as [54] D -= 8 × 10-3(~w) -2 cm2/s In commercial welding of polystyrene of weight-average molecular weight of the order of l0 S a temperature of 170°C may be used. Calculate the flux of polymer molecules across the interface in such a welding process. Consider diffusion over a 100 _)1 distance for this calculation. (Density of polystyrene = 1.0 g/cm3.) Answer: of polymer -- (1.0 gcm-3) = 10_8 -3 mol cm (10s -1) g tool Concentration gradient over 100 )1. or 10-8 cm distance from the bulk concentration to a zero concentration = [(10 -8 - 0) mol cm-3]/(10 -~ -4 cm) = 1 mol cm Bulk concentration

D = 8×10 -3 ×10 -1°

cm2s-1

= 8×10 -13

cm2 -1 s

= (8 × 10-13 cm~ s-1)(1 tool cm-4)(6 × 1023 molecules -1) = 48 × 101° molecules cm-2 -1 s Comparing the number of atoms displaced by the flUX with the average density of atoms in the polymer it is obvious that the original boundary will be obliterated in a few seconds under the welding conditions. Flux

The diffusion law of Eq. (2.74) is, however, not applicable if the blocks being welded are composed of two different, though miscible, polymers. For example, it has been found that if a block of poly(vinyl chloride) is brought in contact with a block of polycaprolactone, at temperatures above rg, then D is higher than that expected from Eq. (2.73) and is proportional (l/M). This has been interpreted as being a consequence of the negative enthalpy of mixing in the system which acts as an additional driving force for the chains on either side of the boundary to cross into other matrix. This driving force will be proportional to the number of monomer units in the chain or the molecular weight, hence the change in the diffusion law. The reptation theory can also be applied to polymer dissolution processes.

REFERENCES 1. K. H. Meyer and H. Mark, Ber. Dtsch. Chem. Ges., 61, 593 (1928). 2. J. L. Koenig, Chemical Microstructure of Polymer Chains, Chaps. 6-8, Wiley Interscience, NewYork (1980). 3. H. Staudinger, Die HochmolekularenOrganischen Verbindung, Springer, Berlin (1932). 4. E J. Flory, J. Macromol.Sci. Phys., B12(1), 1 (1976). 5. R. S. Stein, J. Chem. Ed., 50, 748 (1973).

¯ Chain Dimensions, S~ructures, and Transitional Phenomena

127

6. E J. Flory, Faraday Discuss. Chem. Soc., 68, 15 (1979). 7. P. J. Flory, Pure Appl. Chem., Macromol.Chem., 8, 1 (1972). 8. W. Pechhold, M. E. T. Hauber, and E. Liska, Kolloid Z. Polym., 251, 818 (1973). 9. P. H. Lindenmeyer, J. Macromol.Sci. Phys., 8, 361 (1973). 10. V. P. Privalko and Yu. S. Liptov, Makromol.Chem., 175, 641 (1974). 11. E Rodriguez, J. Chem. Educ., 45, 507 (1968). 12. E. Guth and H. Mark, Monatsch. Chem., 65, 93 (1934). 13. W. Kuhn, Kolloid Z., 68, 2 (1934). 14. T. A. Orfino, Polymer, 2, 305 (1961). 15. B. H. Zimmand W. H. Stockmayer, J. Chem. Phys., 17, 1301 (1949). 16. W. W.Graessley in Chracterization of MacromolecularStructure (D. Mclntyre, ed.), pp. 371-388, National Academy of Sciences Publication No. 1573, Washington, D.C. (1968). 17. E A. Bovey, Polymer Conformation and Configuration, Academic Press, New York (1969). 18. E A. Bovey and G. V. D. Tiers, J. Polym. Sci., 44, 173 (1960). 19. B. Wunderlich, Macromolecular Physics, vols. 1, 2, 3, Academic Press, London (1973, 1976, 1980). 20. H. D. Keith, E J. Padden, and R. G. Vadinsky, J. Polym. Sci., 4A(2), 267 (1966). 21. E P. Price in R. H. Doremus, B. W. Roberts, and D. Turnbull, eds., Growth and Perfection in Crystals, p. 466, Wiley, NewYork (1958). 22. J. N. Hay, Brit. Polym. J., 3, 74 (1971). 23. G. V. Vinogradov, E. A. Ozyara, A. Y. Malkin, and V. A. Grechanovskii, J. Polym. Sci. A2(9), 1153 (1971). 24. J. J. Aklonis, J. Chem. Educ., 58(11), 892 (1981). 25. R. E Boyer, J. Appl. Phys., 25, 825 (1954). 26. R. G. Beamen,J. Polym. Sci., 9, 470 (1952). 27. J. Brandup and E. H. Immergut, eds., Polymer Handbook, 2nd, 3rd eds., Wiley Interscience, NewYork (1975, 1989). 28. J. D. Ferry, Viscoelastic Properties of Polymers, 3rd ed., Chap. 1, Wiley, New York (1980). 29. M. C. Shen and A. Eisenberg, Prog. Solid State Chem., 3, 407 (1966); Rubber Chem. Technol., 43, 95 (1970); [bid 43, 156 (1970). 30. W. Wrasildo, "Thermal Analysis of Polymers," Advances in Polymer Science, voh 13, p. 3, Springer Verlag, NewYork (1974). 31. H. Eyring, J. Chem.Phys., 4, 283 (1936). 32. T. G. Fox and P. J. Flory, J. Appl. Phys., 21,581 (1950). 33. T. G. Fox and P. J. Flory, J. Polym. Sci., 14, 315 (1954). 34. R. Simha and R. E Boyer, J. Chem. Phys., 37, 1003 (1962). 35. R. Simha and C. E. Weil, J. Macromol. Sci. Phys., B4, 215 (1970). 36. R. E Boyer and R. Simha, J. Polym. Sci., Bll, 33 (1973). 37. S. C. Sharma, L. Mandelkern, and E C. Stehling, J. Polym. Sci., BIO 345 (1972). 38. A. K. Doolittle, J. Appl. Phys., 22, 1.471 (1951). 39. D. Turnbull and M. H. Cohen, J. Chem. Phys., 34, 120 (1961). 40. E Bueche, J. Chem. Phys., 30, 748 (1959). 41. M. L. Williams, R. F. Landel, and J. D. Ferry, J. Am. Chem..Soc., 77, 3701 (1955). 42. J. D. Ferry, Viscoelastic Properties of Polymers, 3rd ed., Chap. 1, Wiley, New York (1980).

C]~apter 2 43. A. A. Miller, J. Chem. Phys., 49, 1393 (1968); J. Polym. Sci., A-2(6), 249, 1161 (1968). 44. T. G. Fox and P. J. Flory, J. Appl. Phys., 21,581 (1950). 45. T. G. Fox and P. J. Flory, J. Polym. Sci., 14, 315 (1954). 46. K. Ueberreiter and G. Kanig, J. Chem. Phys., 18, 399 (1950). 47. E N. Kelley and E Bueche, J. Polym. Sci., 50, 549 (1961). 48. L. Mandelkern, G. M. Martin, and E A. Quinn, J. Res. Natl. Bur. Stand., 58, 137 (1957). 49. T. G. Fox, Bull. Am. Phys. Soc., 1, 123 (1956). 50. E Bueche, Physical Properties of Polymers, Interscience Publishers, NewYork (1962). 51. T. G. Fox and P. J. Flory, J. Phys. Chem., 55, 221 (1951). 52. P. G. de Gennes, J. Chem. Phys., 55, 572 (1971). 53. P. G. de Gennes, Phys&sToday, 36(6), 33 (1983). 54. P. J. Mills, P. E Green, C. S. Palmstrom, J. W. Mayer, and E. J. Kramer, Appl. Phys. Lett., 45(9), 957 (1984).

EXERCISES 2.1. Polymer A contains m freely jointed segments each of length l,~, and polymer B contains n freely jointed segments each of length lb. If one end of A is tied to an end of B, what will be the average end-to-end distance of the new molecule ?

[~n,. (mZ~ + nZ~)’/2] 2.2. (a) Calculate the root mean square end-to-end distance and the radius gyration for a macromolecule in molten polypropylene of molecular weight 105. [Data: carbon-carbon bond length = 1.54×10-8 cm; tetrahedral bond angle = 109.5°; steric parameter, a = 1.6 at 140°C] (b) Howextensible is the molecule ? (That is, what ratio does its extended length bear to the average chain end separation ?) [~Ins. (a) 2.4×10-6 cm; 9.8×10-7 cm. (b) 24 times] 2.3. Assuming that the RMSend-to-end distance is an approximation to the diameter of the spherical, coiled polymer in dilute solution, compare the volume occupied by one molecule of polyisobutylene of molecular weight l0 s (a) as a solid at 30°C (density = 0.92 g/cm3) and (b) in th etasolvent. Take the value of the steric parameter a for the polymer as 2.0 and the carbon-carbon bond length as 1.54×10-8 cm. [Ans. (a) 1.81×10~ J43 (b) 2.92×10s ~l. 3 ] 2.4, For c/s- and trans-polyisoprene the RMSend-to-end distance is 2.01x/~ ~4 and 2.90V~ ,~1, respectively, where n is the total number of bonds in each real chain. Comparethe chain stiffness of the two polymers. [Ans. Trans-polyisopreneis 1.4 times stiffer than c/s-polyisoprene.]

Chain Dimensions, Structures,

and Transitional Phenomena

129

2.5. From the data given in Exercise 2.4 calculate the values of ~r and C’~ for c/s- and trans-polyisoprene, given that the length of an isoprene repeating unit is 4.60×10-1° m. [Ans. C~-polyisoprene: ¢r -= 1.2; ~’co = 3.1. trans-polyisoprene: o" = 1.8; C~ = 6.4] 2.6. For trans-polyisoprene the RMSend-to-end distance < r2-’o" 1/2 is found to be (2.90V/-~)10 -8 cm, where rz is the total number of bonds in the real chain. The length of an isoprene repeating unit is 4.60x10-8 cm. Using these dimensions to characterize an equivalent freely jointed chain, calculate the number of monomerunits a trans-potyisoprene molecule has per equivalent random(freely jointed) chain link.

~4.~.1.591 2.7. Drawprojection diagrams (planar zigzag) for (a) Isotactic 1,2-polyisoprene (b) Syndiotactic 3,4-polyisoprene. 2.8. DrawFisher projections (planar) for (a) Isotactic poly(3,4-isoprene) Co) Syndiotactic poly(1,2-isoprene) 2.9. What kind of tacticity do you expect in poly-l-butene? Which form is likely to be less crystalline? What kind of isomerism do you expect in polychloroprene ? Whichisomer should be most crystalline ? 2.10. The a-methyl resonance in poly(a-methyl styrene) is found to be split into three peaks which are assigned to isotactic, heterotactic and syndiotactic triads. Fractions of the polymers in the three configurations determined by the area of these peaks are given below for poly(a-methyl styrene) prepared with two different catalysts [S. Brownstein, S. Bywater, and D. J. Worsfold, Makromol. Chem., 48, 127 (1961)]: Polymer

Catalyst

Fraction of polymer Iso Hetero Syndio

BFa

-

0.11

0.89

Na-Naphthenide

0.13

0.48

0.39

Determine the probability P,~ and test for consistency with Bernoullian statistics. What do the values of Pmobtained in the two cases signify ? [Arts. Polymer 1: P,,~ = 0.06; Polymer 2: Pm = 0.4 ] 2.11. The crystalline density of polyethylene is 0.996 g/cm3 and the density of amorphous polyethylene is 0.866 g/cm3. Calculate the percentage, of crystallinity in a sample of linear polyethylene of density 0.970 g/cm3 and in a

130

Chapter 2 sample of branched polyethylene of density 0.917. Whydo the two samples have considerably different crystallinities ? [Ans. 42.6%; 82.1% ]

2.12. Compare and comment on the extent of hydrogen bonding possible for nylon-6,6 and nylon-7,7 considering the fully extended chain structures. 2.13. Chemically both wax and polyethylene can be described as polymethylene -(-CH2-),~-, but while the former is a brittle solid the latter is a tough plastic. What are the reasons for this difference in mechanical behavior ? 2.14. Draw a logE versus temperature plot for a linear, amorphous polymer and indicate the position and namethe five regions of viscoelastic behavior. How is the curve changed if (a) the polymer is semirrystalline, (b) the polymer is cross-linked, and (c) the experiment is run faster 2.15. To be a useful plastic, an amorphous polymer must be below its Tg at the ambient temperature, while crystalline polymers may be either above or below their T~’s. Why? 2.16. Poly(vinyl chloride) used in the manufactureof plastic raincoats is plasticized, until Ta is below ambient, to make the material flexible. One might expect that if the coat was hung on a hook (i.e., subjected to a tensile load), would eventually flow onto the floor after prolonged expansion. Whydoes this not happen ? 2.17. Both propylene oxide and styrene oxide can be polymerized to give isotactic polymers which can crystallize. Which of the two polymers do you expect to have the higher T,~ and/or higher Tg, and why ? 2.18. Explain the following: (a) By dipping into liquid nitrogen, an adhesive tape loses its stickiness. (b) A hollow rubber ball when cooled in liquid nitrogen and thrown hard against the wall, breaks into pieces. (c) A dinner bell coated with latex paint and kept in a freezer makes louder noise than the one coated and kept at room temperature. (d) Moldingor extrusion of plastics too close to T~ can result in a stiffening of the material. (e) The Tg of a semicrystalline polymer is often higher than the same polymer in a completely amorphous state. (f) In emulsion polymerization of styrene, often carried out at 80°C, the reaction does not proceed quite to 100%conversion. (g) Whennylon shirts are washed and hung up to drip-dry, creases staighten out by themselves. (h) The postage stamp, which is coated with linear poly(vinyl 91cohol) adhesive, needs to be moistened with water (or saliva) before applying.

131

Chain Dimensions, S¢ructures, and Transi~iona2 Phenomena

2.19. What are the requirements in respect of T9 and T,~ of fibers used for clothing purposes ? A number of polyesters and polyamides are cited below along with their T9 and T,~ values. Assuming that these polymers meet the requirements of a fiber, determine their suitability for clothing by considering their Tg and T~ values. Polymer Poly(ethylene succinate) Poly(ethylene terephthalate) Poly(1,4-butylene adipate) Poly(1,4-butylene terephthalate) Polycaprolactam (Nylon-6) Poly(undecanoamide) (Nylon-11) Poly(lauryllactam) (Nylon-12) Poly(hexamethylene adipamide) Poly(hexamethylene sebacarnide)

(o¢) -1 81 -68 66 62 46 37 45 40

2.20. Account for the differences in glass transition polyesters and polyamides listed in Exercise 2.19.

108 265 54 227 228 198 178 267 240 temperatures among the

2.21. Whyare glassy polymers generally tougher than inorganic glasses, both are brittle ?

though

2.22. Define the first- and second-order phase transitions and give one experimental method to differentiate between them. 2.23. Whenpolyethylene is chlorinated, chlorine replaces hydrogen at random. The softening point of this chlorinated product depends on the chlorine content. It is found that small amounts of chlorine (10 to 50 wt% CI) lowers the softening point while large amounts (,-~ 70%) raise the softening point. Rationalize this observation on the basis of intermolecular forces. 2.24. Account for the fact that the coefficient of volume expansion below Ta is smaller than that above Ta. Whyis it said that in the glassy state the polymer is not in a true thermodynamicequilibrium ? 2.25. The preliminary evaluation of a newly synthesized thermoplastic polymer shows that the polymer has a T~ of ll0°C and a melt viscosity of 1.2x106 poises at 140°C. The polymer is found to degrade at temperatures above 160°C. Can this polymer be processed with an extruder which works best at 2x 103 poises? If not, what can be done to the polymer to increase usability? [Ans. Cannot be used as r/160oC > r/ of processability. Plasticizer may be added.] 2,26. The following data were obtained [L. Mandelkern, G. M. Martin and F. A. Quinn, J. Res. Natl. Bur. Stand., 58, 137 (1957)] for the values of glass transition temperature Ta of poly[(vinylidene fluoride)-co-chlorotrifluoroethylene as a function of the weight fraction ’/ill of vinylidene fluoride.

Chapter 2

132 I 0 0.14 0.35 0.40 0.54 1.0 T~ (°K)I 319 292 270 265 258 235 Wl

By fitting these data to a snitable theoretical curve estimate the value of Tg for a copolymer with wl = 0.75. [Ans. 245 OK] 2.27. Twopolystyrene standards (monodisperse) of molecular weight 1.8 x 106 and 1.6x105 were found to have Tg values of 105.9°C and 104.7°C, respectively. Calculate the T~ of a polydisperse polystyrene made up of four low polydisperse components as shown below: Component 1 2 3 4

Wt. fraction 0.05 0.41 0.39 0.15

(w)

_ M~ 1.5 × 106 4.8 × 105 1.2 x ~ I0 3.7 x 104

[Ans. 104.3°C] 2.28. A new polymer with a weight average degree of polymerization of 1400 and five atoms in the repeating unit has a melt viscosity of 1500 poises at 190°C. What will be the viscosity at the same temperature if its molecular weight is doubled ? [Ans. 15834 poise ] 2.29. A vinyl polymer with a Z value of 200 was found to have a melt viscosity of 100 poises. What is the viscosity of this polymer when Z = 800 ? [Ans. 798 poise ] 2.30. A polymer with a Ta of 105°C and a Z value of 400 was found to have a melt viscosity of 500 poises at 170°C. What is the melt viscosity at 150°C Combine the DP dependence when Z = 8007 Assume Zc = 600. (Hint: with the WLFequation.) JAm. 7.5×105 poise ] 2.31. Give reasoning for the fact that a strip of rubber warms on stretching and cools on being allowed to contract. (This can be easily confirmed by student using a rubber band. The rubber is brought into contact with the lips and stretched rapidly, constituting an adiabatic extension. The warming is easily perceived by the temperature-sensitive lips.)

Chin Dimensions, S~ruc~ures, and Transitional

APPENDIX

Phenomena

133

2.1

END-TO-END DISTANCE OF A FREELY JOINTED

CHAIN

The simplest mathematical model of a polymer chain is the freely jointed chain. It has n links, each of length l, joined in a linear sequence with no restrictions on the angles between successive bonds. The length of the chain along its backbone is knownas the contour length and is given by nl. However, for linear flexible chains, it is more usual, and more realistic, to consider the dimensions of the molecular coil in terms of the distance between the two chain ends, that is the end-to-end distance r [Fig. A2.1(a)]. Whenconsidering an isolated polymer molecule, a unique value of r cannot be assigned because the conformation of the molecule (and hence r) changes continuously due to rotation of backbone bonds. However, whilst each conformation has a characteristic value of r, some of the conformations give rise to the same value of r. Hence, some values of r can be more probable than others. This probability distribution of r can be represented by the root mean square (RMS)end-to-end distance, < ~2 >1/2, where the angular brackets indicate that the enclosed quantity is averaged over time, i.e., averaged for the manyconformational sizes available to the same chain. The analysis of this model is similar to that of the well-known random-walk model, which was first developed to describe the random movement of molecules in an ideal gas. The only difference now is that for the freely jointed chain, each step is of equal length l. To analyze the model one end of the Chain may be fixed at the origin O of a three-dimensional rectangular coordinate system, as shownin Fig. A2.1(b), and the probability, P(x, y, z), of finding the other end within a small volume element dx.dy.dz at a particular point with coordinates (x, y, z) may be calculated. Such calculation leads to an equation of the form P(x, y, z) = W(x; y, z) dxdydz

(A2.1)

where W(x, y, z) is the probability density function, i.e., probability per unit volume. Whenn is large and r << nl, W(x, y, z) is given by W(x, y, ~) ----

(Z/~’/~) ~ exp[- Z2(x~ + y2 + z~)l

(A~.2)

wherefl = [3/(2nl=)]l/LSincer z = x2 + y2 + z~, Eq. (A2.2)simplifies W(x, ~, z) = (fl/Trl/=) ~ exp(- fl~r =) (A2.3) This equation shows that W(x, y, z) is Gaussian di stribution fu nction th at ha s a maximumvalue at r = 0, as shown in Fig. A2.2(a) for a large chain of carboncarbon links. The value of W(x,y, z) corresponds to an end-to-end distance r in a particular direction specified by the set of coordinates (x, y, z). However,there may be many such coordinates each of which gives rise to the same end-to-end distance r but in a different direction. Thus, a more important probability is that

134

Chapter 2

Z

(a) Figure A2.1 (a) Schematic representation of a coiled polymermolecule showingthe endto-end distance. (b) Diagramshowinga coiled polymermolecule of end-to-end distance r in a rectangular coordinate systemwith one chain end fixed at the origin.

of finding out chain end at a distance r in any direction from the other chain end located at the origin. This is equal to the probability W(r)dr of finding the chain end in a spherical shell of thickness dr at a radial distance r from the origin. Since the volumeof the spherical shell is 47rr2dr and the probability density function is W(x, y, z), the probability W(r)dr is given by W(r)dr = W(x, y, z) 47rr2dr

(A2.4)

Substitution of W(x, y, z) from Eq. (A2.3) ~hen gives the following expression for the radial distribution function W(r)

w(r) = 4~(/3/~1/2)3r 2 e~(-/32~2)

(A~.S)

This function plotted in Fig. A2.2(b), has a maximumvalue at a certain value of r. It can be shown by simple differentiation that the maximumoccurs at r = 1//3. It is also possible to showthat W(r) normalizes to unity, i.e., fF W(r)dr = 1 (A2.6) Equation (A:2.5) also highlights a deficiency of the theory since it implies W(r) is finite even whenr >> nl, a condition which physically is not possible (i.e. the end-to-end distance cannot exceed the contour length). However, since W(r) is small for r > nl, the errors introduced are negligible.

Chain Dimensions, Structures,

0

0

I 100

and Transitional

~ ~ ~ 200 300 400

135

Phenomena

0 0

100

200

300 &O0

0

r(A)

r(A)

Figure A2.2 Plot of (a) Eq. (A2.3) and (b) Eq. (A2.5) for a chain 4 l inks, each of length 2.5 .~. The mean square end-to-end distance, < r 2 >, is the second moment of the radial distribution function and so is defined by the integral

~ < r: >=J’o

(A2.7)

Substituting for W(r) from Eq. (A2.5) and integrating leads to < v= 2) >= 3/(2/~ Since f12 = 3/(2nl2), < r2

>)/2~_

nil2

l

(see Problem .422.1)

(A2.S)

the RMSend-to-end distance is given by (A2.9)

where the subscript f indicates that the result is for a freely jointed chain. Equation (A2.9) reveals that < r 2 ..1/2 .-f is a factor of n~/~ smaller than the contour length. Since n is large, this result highlights the highly coiled nature of flexible Gaussian polymer chains. A polymer chain in dilute solution can thus be pictured as a coil, continuously altering its shape and size due to conformational changes under the action of random thermal motions. The angular brackets of < r 2 > denote averaging over all possible conformational sizes available to polymer molecules of the same molecular weight. Size differences are further accentuated for polymer samples which are polydisperse and contain chains of variety of chain lengths. This calls for another averaging. Chain dimensions of polydisperse linear

136

Chapter 2

polymers are thus given by the average RMSend-to-end distance < r ~ >1/2, where the bar indicates averaging due to chain polydispersity, whereas < r2 >1/2 without bar represents the RMSend-to-end distance of a single chain or a monodisperse polymer sample.

Problem A2.1 Showby integrating that the radial distribution function W(r) given by Eq. (A2.5) normalizes to unity, and derive equations for (a) most probable value of (b) the root-mean-square value of r, and (c) the meanvalue of Answer : The integral to be evaluated is

Uponsubstitution for W(r) from Eq. (A2.2), I

= 4~r

~

r 2) dr

exp (-

(PA2.1.2)

Note that I(b)

= f~ b e xp(- ax2) dx

(PA2.1.3)

Therefore, (PA2.1.4)

_~(9) = (~)Trl/2 a-3/2 Using this solution to the standard integral, Eq. (PA2.1.2) becomes

(PA2.1.5) =1 Thusthe radial distribution W(r) normalizes to unity. (a) The most probable value of r corresponds to the maximum in W(r), i.e. ~rr

47r

~ r 2 exp(-~r

~)

=0

(PA2.1.6)

This leads to 2r ex-p(--fl2r 2) -- 2r3 fl2 exp(--~2r2) = 0 Solving,r ----- l/ft.

(PA2.1.7)

Chain Dimensions~ S~ructures, and Transitional

137

Phenomena

Since ~ = {31(2nF)]m the most probable value of r is given by v = (2/3~ I/2 n1/2 I = 0.816nl/21

(PA2.1.8)

(b) The meansquare value of r is defined ~y r~W(r)dr < r 2 >~ f~W(r)d~"

(PA2.1.9)

which on substitution for W(r) from Eq. (A2.5) reduces (PA2.1.10) bemuse, as shown above, f~ W(r)dr = Note that for b = 4, Eq. (P~.l.3) redu~s

Using this solution to the standard integral, Eq. (PA2.1.10)becomes

(PA2.1.11) it followsthat

(c) The meanvalue of r is defined < r >=

fy rW(r)dr f~ W(r)d~

(PA2.I.I2)

Since the denominatoris unity, substitution for W(r) from Eq, (A2.5) gives < r >= 47r ~--~

3exp(-l~2r2)dr r

Makinguse of the following solution to the standard integral Eq. (PA2.1.3)

(PA2.1.13)

138

Chapter 2

Eq. (PA2.1.13) reduces ----4~

-4 ~ /3

2 (PA2.1.14) Since ~ ~ = [3/(2nl~)]V Eq. (PA2.1.14) reduces

< v >= \~T/ = 0.92nV~l

na/2 l

Chapter

3

Polymers in Solution INTRODUCTION The important structural details of macromolecules, such as molecular weight, chain length, branching, and chain stiffness, are best studied when the individual molecules are separated from each other. Such studies are therefore made with dilute solutions of polymers. However,the dissolution of a polymer also brings with it a host of new problems. For a correct interpretation of the behavior of polymersolutions it is essential to understand the thermodynamicsof polymer-solvent interaction. Wewill therefore explore some of the basic underlying thermodynamicprinciples of polymer solutions in this chapter. A characteristic feature of a dilute polymersolution is that its viscosity is considerably higher than that of either the pure solvent or similarly dilute solutions of small molecules. The magnitude of the viscosity increase is related to the dimensions of the polymer molecules and to the polymersolvent interactions. Viscosity measurementsthus provide a simple means of determining polymer molecular dimensions and thermodynamic parameters of interactions between polymer and solvents. These aspects will also be considered in a later part of this chapter. Studies of various kinds of equilibria provide a wealth of information about polymer systems. Classical thermodynamics, which is concerned with the macroscopic properties of a system and the relations that hold between themat equilibrium, form a sufficient basis for description of these equilibria in polymer systems. Weshall consider in a major part of this chapter methods of study of polymer solutions that deal with equilibria and can be fully described by thermodynamicrelations. These include vapor pressure, osmotic pressure, and phase separation in polymer-solvent systems. Before proceeding with a discussion of the equilibrium thermodynamics of polymer solutions, it is, however, useful to briefly review some relevant 139

140

Chapter3

aspects of thermodynamictheory and terminology.

THERMODYNAMICSOF LIQUID MIXTURES Like any other system, a solution is characterized by thermodynamicparameters such as volume V, internal energy U, entropy S, Gibbs free energy G, and enthalpy H. In dealing with solution processes, it is customary to determine the difference between the thermodynamic functions of the solution and the same functions of the componentsbefore dissolving. Thus, the flee energy of mixing in the formation of a solution /k(~mix is given by AGmix

= G-

(3.1)

~G~

where (~ is the free energy characterizing the state of the solution and (~ represents the free energies of the individual (pure) componentsi forming the solution. The enthalpy of mixing A/r/mi x and the entropy of mixing ASmi x can be similarly defined: AHmix -- °H - ~ H~ (3.2)

ASmix ---- S - ~ S~

(3.3)

The properties of real solutions are nonadditive, e.g., V 7~ ~iV/°; 3H ~ ~;iH/°; S ¢ ~2iS~, etc. (For example, addition of 50 cm of water to 50 cma of ethanol at 20°C and 1 atm gives a solution whose volume is only 96 cm3 at 20°C and 1 atm. This is because intermolecular interactions in the solution differ from those in pure components.) Hence, the volume, enthalpy, entropy, etc., of components in the dissolved state differ from their values before dissolving. This makes it necessary to use the concept of partial molar or partial specific functions to characterize the thermodynamic behavior of components in solution.

Partial MolarProperty For each extensive property, there is a corresponding partial molar and partial specific property. Consider any thermodynamicextensive property, such as volume, free energy, entropy, energy content, etc., the value of which, for a homogeneoussystem, is completely determined by the state of the system: e.g., the temperature, pressure, and the amountsof the various constituents present. Thus, denoting by Y any extensive thermodynamic property, it can be represented by Y = f (T,P, nl,n2,

....

,n~, ....

)

(3.4)

141

Polymersin Solution

where nl, n2, ...., ni ...... are the numbers of moles of the respective constituents, 1,2 ..... , i, ...., of the system. If there is a small changein the temperature and pressure of the system, as well as in the amounts of its constituents, the change in the property Y is given by OY

OY

dnl +"" + T,P,n:,...

dni +. ¯ (3.5) T,P, nj¢i

(The constancy subscript nj¢i indicates that the number of moles of all components, except component i, are kept constant when evaluating the partial derivative.) The derivative (OY/Oni)T,p, nj¢i is called the partial molar property for the constituent i, and it is represented by writing a bar over the symbol for the particular property, so that

T,P,n2 ,na,...

T,P,nl,na....

or generally,

Chemical

__ Y,

= (0~nY/)T,P,ni#i

(3.6)

PotentlM

The partial molar quantities most commonlyencountered in the thermodynamics of polymer solutions are partial molar volume Vi and partial molar Gibbs free energy Gi. The latter quantity is of special significance since it is identical to the quantity called chemical potential, Izi, defined by P,i

~

= ~i (3.7) P,T,n~¢i Since any partial molar property of a pure substance is simply the corresponding molar property, the chemical potential of a component i in pure form, denoted by #], is evidently equal to the molar Gibbs free energy G~ of pure component i at the same temperature and pressure. The chemical potentials are the key partial molar quantities. The I~i’s determine reaction and phase equilibrium. Moreover, all other partial molar properties and all thermodynamicproperties of the solution can be found from the/z i’s if we knowthe chemical potentials as functions of T, P, and composition. For example, the partial derivatives of /z with respect to r

142

Chap~;er3

and P give Si and IV’i, respectively. The use of #i = Hi - TSi then gives Hi. Once we knowthe partial molar quantities #i, Si, Vi, etc., we get the solution properties as G = ~iniGi, ,.~ = EiniSi, V -=- EiniVi, etc. Note that knowing V as a function of T, P, and composition means we knowthe equation of state of the solution. Recognizing that chemical potential #i is equal to the partial molar Gibbs free energy, as shown by Eq. (3.7), and that the chemical potential of pure componenti, /z~ is equal to simply the molar Gibbs free energy of pure componenti, i.e., G~ at the pressure .and temperature of the system, it is easy to show from Eq. (3.1) that

o {O(ZXGm ) P,T,nj#,

(3.8)

The partial derivative on the right hand side of Eq. (3.8) is the parfal molar Gibbs free energy change, AGi. It maybe noted that all knownthermodynamicrelations are also valid for the partial molar functions. Thus, the relation G = H - TS in terms of partial molar functions is

(3.9)

Gi = Hi - TSi Therefore, from Eq. (3.8)

#i-

#~ = AGi = AHi-

T-ASi

(3.10)

For ideal solutions that form with a zero heat effect, AHi -- 0. Hence for ideal solutions

~ - ~ = /XGi = -T/XSi

(3.11)

It will be shown later (see p. 141) that ideal entropy of mixing of two components, say, 1 and 2, is given by

ASmix = -R(nl

lnxl + n21nx2)

(3.12)

where ni and xi are, respectively, number of moles and mole fractions of component i. Partial ideal entropies of mixing can be obtained by differentiating F.q. (3.12). Thus for componenti, in general, ASi-

0(ASmix)

_

.R]_nzl

(3.13)

Substituting in Eq. (3.11), O

Izi - #i = t~Tlnxi

(3.14)

Thus, the change in chemical potential of the ith componentin an ideal solution depends only on the mole fraction of that component in solution.

Polymersin Solution

I43

In the case of a real solution, p~ t~i

(3.15) p.O $ where Pi and p~ are the partial vapor pressures of the ith component above the solution and above the pure component, respectively. Since for a thermodynamicallystable solution Pi ( P~, it follows from Eq. (3.15) that /k#i < 0 or /zi < ¢z~, i.e., the chemical potential of each componentin the solution is smaller than that of the pure component. Thus, dissolution is a spontaneous process. As more and more of the component i passes into the phase with the lower chemical potential, the value of the latter in that phase increases until /k/zi = 0, which corresponds to equilibrium. ComparingF~qs. (3.14) and (3.15) we obtain - Iz~

= RTln

;~/p7

= ~

(3.16)

Relation (3.16) was obtained empirically by Raoult and is called Raoult’s law. Hence in ideal solutions Raoult’s law holds over the entire range of compositions, this being represented by a straight line in the vapor pressure composition curve (Fig. 3.1). In reality, most solutions do not obey Eqs. (3.14) and (3.16). Such solutions are called nonideal, or real. The deviation from ideality may be positive (,Pl/P~ > zl) or negative (Pl/P~ (see Fig. 3.1). Negative deviations from ideality are due to good interaction between the componentsin solution resulting in a decrease in the number of molecules of the given component in the vapor phase. Positive deviations, on the other hand, are due to poor interaction between the components, as a result of which the molecules tend to pass into the vapor phase in larger amounts, increasing the partial vapor pressure p~. Polymer solutions characteristically display sharp negative deviations from ideality, as can be seen in Fig. 3.2. A method of estimating the deviation of solutions from ideality, suggested by Lewis, consists in substituting a certain function called activity for the mole fractions in all thermodynamicequations which are not valid for real solutions. Thus, instead of Eqs. (3.14) and (3.16) we may write and

#i-

lz~

= Api

pi/p =

= RTlnai

(3.17)

(3.1s)

Whenthe dependence of A(~ of solutions on the independent variables is known, all the basic thermodynamicproperties of solutions are derived easily using standard thermodynamic manipulations. For example, expressions for the activities’of the componentsof a system can be obtained by combining Eq. (3.17) with expressions derived by partial differentiation an appropriate relation for ~(~mix.

144

Chapter 3

3

Xl=O x2=1

Molefraction

Xl= x2=O

Figure 3.1 Dependence of solvent vapor pressure

over a solution on mole fraction of components in solution. Curve 1: ideal solution, Pl/P~ = Zl. Curve 2: negative deviation from ideality, Pl/P~ < :~1- Curve 3: positive deviation from ideality, pl/p ° > xl.

Xl=O X2=1

Figure 3.2 Dependence of vapor pressure fraction of components.

over a polymer solution

on mole

Polymersin Solution

145

Most theoretical procedures for deriving expressions for A(~mix start with the construction of a model of the mixture. The model is then analyzed by the techniques of statistical thermodynamics. The nature and sophistication of different models vary depending on the level of the statistical mechanical approach and the seriousness of the mathematical approximations that are invariably introduced into the calculation. The immensely popular Flory-Huggins theory, which was developed in the early 1940s, is based on the pseudolattice modeland a rather low-level statistical treatment with manyapproximations. The theory is remarkably simple, explains correctly (at least qualitatively) a large numberof experimental observations, and serves as a starting point for manymore sophisticated theories. Flory-Huggins theory is based on splitting AGmixinto an enthalpy term and an entropy term according to Eq. (3.19) and evaluating these two terms separately (3.19) A(~mix = AHmix - TASrnix The entropy of mixing ASmix is computed from the number of possible arrangements of the molecules on a lattice, while the enthalpy of mixing, AHmi x, is calculated as change in interaction energies among molecular surfaces during the process of mixing. Before turning to Flory-Huggins theory, it is convenient to develop first the theory for mixtures of low molecular weight solvents to get a better grasp of the principles involved.

Low-Molecular-Weight Mixtures: van Laar Model Let us consider a mixture of two substances A and B satisfying the conditions that (i) molecules of A and B are identical in size and shape and (ii) the energies of like (i.e., A-Aor B-B) and unlike (i.e., A-B) molecular interactions are equal. Such mixtures form ideal solutions. Condition (ii) leads to athermal mixing (i.e., AHmix= 0), which also means that there are no changes in the rotational, vibrational, and translational entropies and in the entropy of intermolecular interactions of the components upon mixing. Thus, ASmixdepends only upon the configurational (or combinatorial) entropy change, ASconf, which is positive because the number of distinguishable spatial arrangements of the molecules increases when they are mixed. Hence A~mix of Eq. (3.19) is negative and formation of ideal solution is favorable. The methods of statistical mechanics can be used to derive an equation for A,.~conf by assuming that the molecules are placed randomly into cells that are of molecular size and are arranged in the form of a threedimensional lattice (represented in two dimensions in Fig. 3.3). The white circles in Fig. 3.3 represent solvent molecules and the black ones, solute

146

Chapter 3

O00!OO000

00000!000

0 ql-

o

0

ooooo

O00OOO00 (~)

00000000 (b)

Figure 3.3 Schematicrepresentation of quasicrystalline lattice modelfor solution. (a) Mixture of molecules of equal size. (b) Mixture of solvent molecules with a polymermolecule showingthe connectivity of polymersegments.

molecules or polymer chain segments tied with chemical bonds. It is assumed that solvent molecules can exchange sites with solute molecules and with polymer chain segments. The latter is possible only if the chain is capable of bending. An exchange of sites between unlike molecules results in an increase in the number of ways they can be arranged, and hence in an increase in entropy. By definition, ASmixis the difference between the entropy of the mixture and the entropies of the components in unmixed (pure) form. According our model, only configurational entropies need be considered. Therefore /~Smix is equal to ,~Sconf. The fundamental relation between the entropy S of an assembly of molecules and the total number Q of distinguishable degenerate (i.e., of equal energy) arrangements of the molecules is given by the well-known Boltzmannrelation : S = hlnf~ (3.20) where k is the Boltzmannconstant. Application of this equation to formation of an ideal solution thus gives ASconf = k[Inf~mix(lnf~z + lnf~B)] (3.21) where f~A, f~B and, f~mix are, respectively, the total numbers of distinguishable spatial arrangements of the molecules in the pure substance A, pure substance B and the ideal mixture. Since all the molecules of a pure

Polymersin Solution

147

substance are identical, there can be only one distinguishable spatial arrangement of them, that is, of placing N indistinguishable molecules in a lattice containing N cells. Thus f]A = 1 and ~B = 1 and Eq. (3.21) reduces to (3.22) /~Sconf ----- k In’mix For ideal mixing of NA molecules of A with NB molecules of B in a lattice with (]VA + NB) cells, the total number of distinguishable spatial arrangements of the molecules is equal to the number of permutations of (NA + NB) objects consisting of NA identical objects of type A and NB identical objects of type B, that is, ~mix

(3.23)

= (NA + NB)! NA!NB!

Substitution of Eq. (3.23) into Eq. (3.22) gives /~Sconf

-"

q]g In [(NA L ~v’~)v-~.~

(3.24)

and introducing Stirling’s approximation, lnN!

= NlnN

-

N

(3.25)

(for large N), leads /~,qconf

= --]g

NA

In

NA + NB

NB ln

(NA

~N~)]

(3.26) It is more usual to write thermodynamicequations in terms of numbers of moles, n, and mole fractions, z, which here are defined by nA = NA/NAv, nB= NB/NAv, n = nA +nB ZA = nA/(n A -~- T~B), XB --~ 1"LB/(T~A + ~B)

(3.27) (3.28)

where NAy is Avogadro’s number. Thus, Eq. (3.26) becomes /k~qconf = -- R [hA In ~A -[- T~B In

XB]

(3.29)

since the universal gas constant 2~ = kNAv.(It may be noted that since the configurational entropy of unmixedcomponents is zero, /kZconf is in fact equal to Sconf of the mixture.) Since for the formation of an ideal solution, /k~qmix = /kSconf and AHmix---- 0, Eq. (3.19)becomes AGmix = RT [hA lnxA

+ ns lnxB]

(3.30)

Equation (3.30) provides the basis from which standard thermodynamic relationships for ideal solutions (e.g., Raoult’s law) can be derived. However,

148

Ch~p~er3

since relatively few solutions behave ideally, Eq. (3.30) requires modification to make it more generally applicable. For small molecules, nonideality of solutions is invariably due to nonathermal mixing (i.e., AHmlx~ 0) on account of nonequivalent intermolecular interactions. Intermolecular interactions result from so-called dispersion forces. Electrons movingwithin molecules produce a fluctuating dipole in all molecules, whether polar or nonpolar. Polar molecules also possess permanent dipoles. Both types of dipoles induce other dipoles (called induced dipoles) in neighboring molecules. The induced dipoles are attracted to the original dipoles and this attraction lowers the energy of the system. The decrease in energy is called the interaction energy. This energy is larger for polar molecules than for nonpolar molecules and it is quite sizeable as it is responsible for the cohesion of liquids and molecular crystals. Since the dispersion forces decrease with the sixth powerof the distance between the two dipoles, they have significant values only when the two interacting entities-such as atoms, atomic groups, small molecules-are in direct contact. The amount of interaction energy per contact depends on the nature of the interacting groups. Thus, it is reasonable to ascribe a characteristic value of interaction energy to each intermoleeular contact, or better to each contact of atoms or atomic groups that are not bonded. In the liquid lattice defined above, each molecule has z contact points, z being the coordination number of the lattice. In a mixture there are three types of contact, namely, A-A, B-B, and A-B. They are assigned interaction energies O.)AA, O~BB , and wAS,respectively. The model implies that all contact points of a molecule are equivalent. The enthalpy of mixing is given by AHmix = H -- HA --. HB (3.31) where H, HA and HBrepresent the contributions of the contact energy to the enthalpy for the mixture, for pure component A, and for pure component B, respectively. HA and HB can be calculated from (3.32) HA ---ZNAWAA/2 (3.33) HB = ZNBWBB/2 where the factor 2 accounts for the fact that two contact points are needed to form a single contact. In a mixture, all three types of contacts are present; thus, the enthalpy of the mixture is a sum of three terms. Assuming that the number of heterogeneous contacts is NAB, we can thus write for H: H ---

NABWAB-{-

Substitution then yields

of HA,

(ZNA -- NAB)WAA/2 t- ( ZNB -- N AB)WBB/2 ( 3.34)

HBand H from Eqs. (3.32)-(3.34)

into Eq. (3.31)

Polymersin Solution

AHmix ----

149

NAB[~AB--

(~AA-t-~BB)/2]

----

NABAOJAB(3.35)

where /k~AB ---- O)AB- (WAA q- mBB)/2, representing the change in energy for the formation of an ABcontact, is the excess contact energy of an ABcontact. The number of h~terogeneous contacts NABcan be easily estimated. It is equal to the number of contact points associated with A molecules multiplied by a probability P13 that a neighbor of molecule A is molecule B, that is, NAB = zNAp~ (3.36) It is difficult, however, to estimate PBbecause the frequency of different types of contacts is governed by Boltzmann factors involving contact energies. As an approx/mation, it is therefore replaced by a probability that any molecule in the solution is a molecule B (not. necessarily a neighbor of molecule A). The use of this latter probability, which is equal to the mole fraction of component B in the mixture, is justified only when the arrangement of molecules on the lattice is completely random. Combiningl~qs. (3.35) and (3.36) and replacing PB by the mole fraction XB of component B which is equal to NB/(NA q- NB), one obtains AHm~x

(3.37)

= NAXSZAWAB

Substitution of/YA from I~q. (3.27) then yields AHmix

(3.38)

= xsnANAvzAwAB

The product NAvZ/kWAB has a well-defined physical meaning. It represents an enthalpy change accompanying transfer of 1 mol of A (or B) from its pure state to an infinitely dilute solution in B (or A). .Combining Eqs. (3.19), (3.29) and (3.38), and noting that ASmix ASconf, one obtains for AGmix, AGmix = RT (nA In ZA +nB In

xs +nAXBNAvZAWAB/RT ) (3.39)

The last term in this expression is usually simplified by introducing the interaction parameter XAB, defined as (3.40) XA~ = N~vZA~AB/RT = ZAWA~/kT The interaction parameter XAB, according to this model, is thus seen to be inversely proportional to temperature, whereas AwAB is assumed to be independent of temperature. Note that /~03AB , and hence XAB, can be either positive or negative. Comb/nationof lEq. (3.39) with I~q. (3.40) yields AGm~ -~

RT (ng

lnxA

q-

ns lnx~

-k

nAXBXA~)

(3.41)

150

Chapter3

Equation (3.41) describes the van Laar model of solvent mixtures and is applicable only to mixtures of low-molecular-weight components with apprwdmately the same molar volume.

Problem 3.1 Based on the lattice model for low-molecular-weight mixtures, determine the effect of non-athermalmixing on Raoult’s law. Answer: Applyingthe recipe for the calculation of chemicalpotentials [Eq. (3.8)] to the low molecular weight mixtures described by Eq. (3.41) and combiningwith Eq. (3.17) for real solutions, one obtains (cf. Problem3.4): ]-la -- #° A = RTlnam = RT(lnxA -~XABX2B) (P3.1.1) ttB -- ,u°~ = RTlnaB = RT(lnxB "q- XABX2A) (P3.1.2) For athermal mixing, AHm~x = 0 and XAB= 0; hence Eqs. (P3.1.1) and (P3.1.2) reduce to expressions valid for ideal solutions with activities ai equal to mole fractions and activity coefficients 7i equal to unity. Sincefor mixturesthat are not athermal, AHmi x ~ 0 and XAB~=0, activity and activity coefficients read a A = nAT A = (P3.1.3) Ae nxP(XA,x~B) with similar expressions for componentB. xFor endothermicmixtures (i.e., mixtures absorbing heat during mixing), AHmi and XABare positive. Suchmixtures therefore exhibit positive deviations from Raoult’s law. Similarly, exothermic mixtures with negative values of AHmi x and X, must have negative deviations from Raoult’s law. The failure of the van Laar model to give realistic predictions of the thermodynamicproperties of polymer solutions, arises from the assumption made in this model that the solvent and solute molecules are identical in size. However, Flory [1] and Huggins [2] proposed, independently, a modified lattice theory which takes into account the large differences in size between solvent and polymer molecules, in addition to intermolecular interactions.

Polymer-Solvent Mixtures: Flory-Huggins Model The theory sets out to predict AGmixfor the formation of polymer solutions based on lattice model. In order to place both the solvent molecules and polymer molecules onto the same pseudolattice, it becomes necessary to consider the polymer molecules to be chains of segments, each segment being equal in size to a solvent molecule. The number, o’, of these segments in

Pol>Tnersin Solufion

151

the chain defines the size of a polymer molecule and is given by the ratio of molar volumes Vp./V1 (from now on we will assign the subscript 1 to the solvent and 2 to the polymer solute). Hence cr is not necessarily equal to the degree of polymerization. The model assumes that each lattice cell is occupied by either a solvent molecule or a chain segment, and each polymer molecule is placed in the lattice so that its chain segments occupy a continuous sequence of ff cells (as indicated in Fig. 3.3(b)).

Problem 3.2 Calculate, in terms of the lattice model of the Flory-Huggins theory, the number of segments per polystyrene molecule of molecular weight 290,000 dissolved in (a) toluene and (b) methyl ethyl ketone (MEK)at [Data: polymer density = 1.083 g/cmz in toluene and 1.091 g/cma in MEK, calculated on the assumptionof additivity of volumes; toluene density -- 0.86~ g/cm3; MEK density = 0.799 g/cm3, all at 25°C.] Answer: (a) Polystyrene/toluene M2 2.90 x 105 g mo1-1 = 2.678 x 105 cm3 -~ mol -3 1.083 g cm P2 Wl = (92 g mo1-1)/(0.861 -3 g )cm = 106.852 cm3 -1 tool V (2.678 x 105 cm3 mo1-1) 2 = 2506 V~ (106.852 cma mo1-1) (b) Polystyrene/MEK: Vz

=

V~

=

~r

=

-1 (2.90x105 gmo1-1) = 2.658x 105 cma tool -3) (1.091 g cm (72 g mol-~) = 90.11 cm3 mo1-1 -a) (0.799 g cm (2.658 x 105 cm~ -~) tool = 2950 -~) (90.11 cma mol

In comparison, the degree of polymerization of the polymer = (2.90x105 mol-~)/(104 g mo1-1) = 2788.

The first stage in the development of the Flory-Huggins theory is to derive an expression for ASmix when AHmix = 0. As for mixing of simple molecules considered above, this involves application of Eq.

152

Chapter3

(3.21), but this time with ~2 > 1 because each molecule in an amorphous polymercan adopt manydifferent conformations (i.e., distinguishable spatial arrangements of the sequence of segments). Hence for polymer solutions (considering the configurational entropy change ASconf to be the only source of ASrnix) /XSma = ~1n(~12/~2)

(3.42)

The procedures used to evaluate ~12, ~2, and ASmixof Eq. (3.43) are relatively complex.The basis of the calculations, considered in outline below, is the formation of a polymer solution by mixing N2 polymer molecules, each comprising ~r segments in the chain, with N1 solvent molecules in a lattice containing N1 q- ~rN2 (= N) cells. The N2 polymer molecules are added one-by-one to the lattice before adding the solvent molecules. For adding a polymer molecule, the first segment of the chain can be placed in any empty cell. However, for the requirement of connectivity in a chain, each successive segment of the remaining cr -- 1 segments must be placed in an empty cell that is adjacent to the previously placed segment. The number of possible placements is thus calculated for each successive segment of the chain, and by multiplying these numbers together the total number, u, of possible conformations of the polymer molecule in the lattice is obtained. Similar calculations are carried out for each of the N2 polymer molecules as they are added one-by-one to the lattice. Whenadding a polymer molecule it is assumed, using a mean field approximation, that the segments of the previously added polymer molecule are distributed uniformly in the lattice. The N1 solvent molecules’are added to the lattice after the addition of _N2 polymer molecules. However, since the solvent molecules are identical, there is only one distinguishable spatial arrangement of them obtained by placing one solvent molecule in each of the remaining N1 empty cells. Thus, the total number of distinguishable spatial arrangements of the mixture is given by

where 1-I is the symbol for a continuous product, which in this case is ul × u2 × u3 × u4 × "’" × UN~. The factor of 1/N~! is introduced on account of the fact that the N2 polymer molecules are identical and therefore indistinguishable from each other. The resulting expression for given in a useful form by Flory is

=

(3.44)

Polymersin Solution

153

The quantity in the first of the square brackets represents the total number of conformations possible for _N~ polymer molecules when each one has the complete freedom of the lattice, that is, the lattice is assumed to be empty when adding each polymer molecule. The quantity in the second pair of square brackets represents the fractions of these conformations that are allowed when -N2 polymer molecules compete for cells in a lattice of crN2 cells corresponding to the pure amorphous polymer. The quantity in the third pair of square brackets is the factor by which the total numberof conformations of N~ polymer molecules increases on account of the greater spatial freedom provided by dilution with N1 solvent molecules. Evidently, the product of the quantities in the first two pair of square brackets of Eq. (3.44) represents f~2 and so Eq. (3.42) simplifies ASmix

= kln

[\~]

J

(3.45)

+ N~InqSz]

(3.46)

\~rN~]

which simplifies further to give A~qmix

=

-k[Nlln¢l

where ¢1 and ¢2 are the volume fractions of solvent and polymer given by the fractions of the total numberof lattice points occupied by solvent molecules and polymer segments, respectively: cfii

= N1/(N1

+ ~rN2) and ¢2 = o’N2/(N1

q-crN2)

(3.47)

In terms of number of moles, Eq. (3.46) becomes ASm~x = -R [nl

In ¢1 + n~ In ¢2]

(3.48)

This expression for polymer solutions is seen to be very similar to the corresponding expression for ideal mixing, i.e., Eq. (3.29). The only difference is that for polymer solutions, the mole fractions in Eq. (3.29) are replaced by the corresponding volume fractions. It may be noted that Eq. (3.48) a more general expression for athermal mixing and reduces to Eq. (3.29) when (r = 1.

Problem3.3 Using the entropy change equation of the lattice model, calculate the change in entropy whenequal amounts(100 g) of two componentsare mixed for each of the following systems: (a) toluene and methyl ethyl ketone (MEK); (b) toluene and polystyrene of molecularweigh~1.2x 10s; and (c) polystyrene poly(methyl methacrylate) each of molecular weight 10,000. Take the d~nsity each componentas 1.0 g/cma. Commenton the comparative entropy changes of

154

Chapter3

mixing for the three systems. Answer: (a) To use Eq. (3.29) let A and 13 represent toluene and MEK,respectively. Molar masses of A and B: MA (CrHs) = 92 g mo1-1 MB (C4H80) = 72 g mo1-1 Moles of A and B : (100 g) nA -- (92 g mol-~) -- 1.087 tool, Mole fractions of A and B: ZA = 1.087/(1.087 + 1.389)

nB -----

= 0.439,

(100 g) (72 g mo1-1) -- 1.389 tool

ZB = 1- 0.439 = 0.561

ASmix = --(8.314 J tool -1 K-1)[(1.087 mol) In 0.439 + (1.389 tool) In 0.561] -1 = 14.11 J K (b) Eq. (3.48) is used for calculating ASmixof polymer solution. rh = nA = 1.087 mol n2 = (100 g) / (1.2×105 g mo1-1) = 8.33×10-4 tool The polymer consists of cr segments, each of which can displace a single solvent molecule from a lattice site. Thus, cr is defined as O’--

p2~o

where M~is the molecular weight of the polymer that would have density P2 in the corresponding amorphous state at the solution temperature and V~ ° is the molar volume of the solvent. Using appropriate data the calculated values are (1.2 × 10~ -1) g tool (z = 1.301103 [’(92 g mol-’)] ](1.0g cm-Z)[(-~-.0 ;- c--~_3) (1.087 tool) ¢~ = [(1.087 mol) + 1.30 × 103(8.33 x -4 mol)] =

0. 50

q~2---- 1.0 -- 0.50----- 0.50 ASnaix = -(8.314 J tool -1 °K-~)[(1.087 tool) In 0.50 + (8.33110 -4 tool) In 0.50] -~ = 6.27 I K (c) Eq. (3.48) applies also if two polymers are being mixed. In this case,

155

Polymersin Solution

numberof segmentsai in the ith componentof the mixture is calculated from o’i = Mi/piVr where V~ is nowa reference volumeequal to the molar volumeof the smallest polymerrepeating unit in the mixture. The correspondingvolumefraction ¢i is For the given problem, nl = n2 = (100 g)/(10000 g mo1-1) = 0.01 Reference volume = molar volume of methyl methacrylate repeatingunit ( ------ C5H802) V~ -- MCsHsO2 - 100-3g tool -1 _- 100 cm3 -i mol Density 1.0 gcm (10000 g mo1-1) 3 to ol ~ = (z2 = (1.0 g cm-Z)(X00-~)

= 100

0.01 x 100 ¢1 = 0.01 x 100 + 0.01 × 100 = 0.5,

¢2 = 0.5

--(8.314 l mo1-1°K-I)[(0.01 mol)ln0.5 + (0.01 mol)ln0.5] -~ = 0.12 J K

ASmix =

Comment:The entropy of mixing of polymers with solvent is small comparedto that of micromoleculesbecause there are fewer possible arrangementsof solvent molecules and polymer segments than there would be if the segments were not connected to each other. The entropy gain per unit of mixture is muchless if two polymers are mixedthan if one of the componentsis a low-molecular-weight solvent, because nl is muchsmaller in the former case.

Having derived an expression for ASmix,the second stage in the developmentof Flory-Hugginstheory is to derive an expression for the effects of intermolecular interactions. In the original theory, this was considered only in terms of an enthalpy change, AHmix.Calculation of ZXHmi x for polymer solutions follows almost exactly the outline depicted for low-molecularweight mixtures in Eqs. (3.31)-(3.38). The only difference is in probability P2 [i.e., PB in Eq. (3.36)] that a given lattice point is occupied by a polymer segment. In this case, the probability must be approximated by the volume fraction ¢2, instead of mole fraction n2. Thus, for a polymer solution the expression for AHmixreads [cf. Eqs. (3.38) and (3.40)] AHmix

= nl¢2NAvZAQ2

= RTn~¢~XI~

(3.49)

where X12is the Flory-Huggins interaction parameter defined by Eq. (3.40).

156

Chapter 3

A lacuna in the above expression for AHmixis easily recognized. Thus, while all z contact sites of a low-molecular-weight solute are available for contact with solvent molecules, two of the contact sites of a polymer segment are permanently occupied by neighboring segments and only z - 2 sites are available. Thus, relation (3.49) is plausible only when the difference between z and z -- 2 is negligible, that is, in the limit of z approaching infinity. This limit is, of course, physically meaningless. A more rigorous derivation taking into account finite z’s reveals that X12should be concentration dependent. This dependence is a slowly varying function and the absolute value of Xr). increases with increasing ¢2A combination of the entropy term of Eq. (3.48) and the enthalpy term of Eq. (3.49) according to Eq. (3.19) gives the following expression AGmix: AGmix = RT (nllnea

+ n21n¢2

+ nl¢2X12)

(3.50)

This remarkably simple expression is the famous Flory-Huggins equation for the Gibbs free energy of mixing, which has been the cornerstone of polymer thermodynamics for more than five decades. Using Flory-Huggins theory it is possible to account for the equilibrium thermodynamicproperties of polymer solutions, particularly the fact that polymer solutions show major deviations from ideal solution behavior, as for example, the vapor pressure of solvent above a polymer.solution invariably is very much lower than predicted from Raoult’s law. The theory also accounts for the phase separation and fractionation behavior of polymer solutions, melting point depressions in crystalline polymers, and swelling of polymer networks. However, the theory is only able to predict general trends and fails to achieve precise agreement with experimental data. The deficiencies of the Flory-Huggins theory result from the limitations both of the model and of the assumptions employedin its derivation. Thus, the use of a single type of lattice for pure solvent, pure polymer and their mixtures is clearly unrealistic since it requires that there is no volumechange upon mixing. The method used in the model to calculate the total number of possible conformationsof a polymermolecule in the lattice is also unrealistic since it does not exclude self-intersections of the chain. Moreover,, the use of a mean-field approximationto facilitate this calculation, wherebyit is assumed that the segments of the previously added polymer molecules are distributed uniformly in the lattice, is satisfactory only whenthe volumefraction ¢2 of polymer is high, as in relatively concentrated polymer solutions. In dilute polymer solutions, the polymer molecules are isolated from each other by regions of pure solvent, i.e., the polymer segments are not uniformly distributed in the lattice. In view of this, the Flory-Hugginstheory is least satisfactory for dilute polymer solutions and only applies to concentrated solutions or mLxtures. Furthermore, the interaction param-

157

Polymers in Solution

eter X, introduced to account for the effects of polymer-solventcontact interactions, is not a simple parameter and should contain both enthalpy and entropy contributions. Additionally, as noted earlier, Xhas also been shownto be dependenton the solution concentration. Despite the drawbacks,Flory-Hugginslattice theory was a major step forward towards understanding the thermodynamicsof polymer solutions and is the basis of manyother theories. Since it wasfirst proposed,other workershave elaborated on it to improveon the assumptionsinvolved. Some of these more refined theories will be given brief consideration, but in general are beyondthe scope of the book. Flory-Huggins l~nctions

Expressions

for

Thermodynamic

Once the analytical expression for AGmixis known,the calculation of chemicalpotentials and other thermodynamic functions (activities, activity coefficients, virial coefficients, etc.) is straightforward.For polymersolutions, wemust apply Eq. (3.8) to Flory-Hugginsequation (3.50), keeping in mindthat volumefractions ¢i are functions of the numberof moles, as given by ¢1 = ~l/(nl -]- O’~Z2) (3.51) ¢2 = o’n2/(nl + o’n2) (3.52) [These relations result from Eqs. (3.47) by dividing numeratorand denominator by the AvogadronumberNAv.]Thuspartial differentiation of Eq. (3.50) with respect to nl (note that both ¢1 and ¢2 are functions of leads to the followingrelation for the solvent (see Problem3.4): (1 - % )¢2 -[- X¢~] (3.53) o and partial differentiation of Eq. (3.50) with respect to n2 similarly leads to the followingrelation for the polymer P,1- /A~ = AG1 : aT [112¢1 -[-

,2--#~

= AG~ : RT[In¢2

+ (1 - ~r)¢1

+ ox’¢~ ] (3.54)

where X = X12 -- ¢I(OX12/c~(fi2)P,T ! X : X12 -[- ¢2(OX12/~¢2)P~T

(3.55) (3.56)

Problem3.4 Starting from Flory-Hugginsequation (3.50) showthat for the formationof a solution froma monodisperse polymerthe partial molarGibbsfree energyof mixing,AGx,for the solventis givenby Eq. (3.53).

158

Claap~er3

Answer: The Flory-Huggins equation (3.50) AG~,:

= RT[nlln¢I

+ n~ln¢~

+

and from Eq. (3.7)

Since

¢~ = n~/(n~

+an~) and @~ = n2a/(n~

= RT {~[nllnn~

-nlln(n~

+ an~),

then

+ an~)]

+ ~[n~l~ - ~i~(~ + ~n~)]

AG[ RT

=

n~ + lnn~ - ln(n~ n~ + 0 +

n2

+ anz)

(n~ + n~) (m + ~n~)

+ an2x12[

(n~ + ~)~ +

n~ + an~

an~n~ dx,~ (n~ + ane)" d¢2 dn~

Simplifying and substituting

for d¢2/dnl,

(nl + n2)

~i U~I~2

dx12

[-~i + ~i+ ~1

--~2

(n~+ an~) de: ~ (n~+ an~)

RT

~"~-n~-.~ ~n~+ "~+(~1 ~.~) + ~¢I -

~T1~41+ (nl + z~2) ~ where X

+

(~1~

¢~¢~k~¢~

159

Polymersin Solution

In the simplest form of the Flory-Huggins theory, the parameter X12 is independent of concentration, and does not depend on q~l and q~2. Consequently, the functions X12, X, and Xp are equal and they are usually represented indiscriminately by the same symbol X. Further, replacing the factor a by (3.57) ~ = V~/V~ where V1 and Y~ are the molar volumes of solvent and solute (polymer), respectively, Eqs. (3.53) and (3.54) are rewritten ~1 --

~ ----

]~T

ln¢1

+

1 - ~ ¢2 + X¢~

(3.58)

Thc~c equations can b¢ combined with Eq. (3.17) to obtain cxprcssion~ for the corresponding activities of the components. For example, the acti~ of the solvent is given by ln a~ : ~1 -- ~R~ : ln(1- ¢2)~ (1 --

~V1)~

(3.60) It can be shown that in the limit of e~remely small volume fractions, the right-hand side of ~q. (3.60) is equal to in z1 and £q. (3.60) reduces to the equation valid for ideal solutions with activities ~i equal to mole fractions zi. This result is, of course, necessa~; even macromolecular solutions must obey Raoult’s law in e~remely dilute solutions. However, at higher concentration of polymer molecules and in the absence of interaction parameter X (that is, for vanishing enthalpy of m~ng, or athermal m~ing), the solution obeying ~q. (3.60) deviates from ideal solution behavior and the deviation increases with increasing concentration of the polymer (expressed as its volumefraction). Deviations are still greater for exothermic solutions (that is, producing heat during m~ing) with negative values of ~Hm~and hence of X (see ~q. 3.49). Colligative Parameter

Properties X

and Interaction

Those properties which depend only on the concentration of solute molecules and not on the nature of the solute are called colligative (from the Latin colligatus meaning"bound together"). In a solution at T and P, the partial molar Gibbs free energy (chemical potential) of the solvent (denoted component 1) is from Eq. (3.17), #1 = #~ RTlnax, where /z ~ is

Chapter 3

160

the molar Gibbs free energy of the pure liquid solvent at T and P and where al = q’lx~, "/1 and xl being the activity coefficient and mole fraction, respectively, of the solvent. Thus, #1 in solution differs from of pure A. This change in solvent chemical potential leads to a change in the vapor pressure, the normal boiling point, and the normal freezing point, and causes the phenomenonof osmotic pressure. These four properties are the colligative properties. Each involves an equilibrium between two phases. Thermodynamic equations relating #1 and al to various colligative properties are found in any textbook on thermodynamics. The following relationships are obtained on using the expression for lnP~

_

~1

--

~ __

lnal

(3.61)

(3.62)

lnal

RT

: -lna~

(3.63)

(3.64)

In these equations, AHf and AHb are the latent heats of freezing and vaporization; Tb and Tf are the boiling and freezing points of the solution; T~ and T~ are the boiling and freezing points of the pure solvent; Arb (= Tb -- T~) is the elevation in the boiling point; AT.f (= Ty - T~?)is the depression in the freezing point; pl is the vapor pressure at temperature T of the solution and p~ the vapor pressure over pure solvent at the same temperature; II is the osmotic pressure of the solution. CombiningEqs. (3.61)-(3.64) with Eq. (3.60) all the above colligative properties can be related to the interaction parameter X. For the vapor pressure, for example, ln

Pl = ln(1

--

¢2)

+ (1 --

--~)¢2

+ X¢~ (3.65)

The interaction parameter can thus be determined from experimental measurement of colligative properties. The interaction parameter X [note that X is used instead of X~2 in view of Eq. (3.55)] plays a very important role in the theory of polymer solutions. It follows from Eq. (3.49) that X is a dimensionless parameter equal to the ratio of the energy of interaction of the polymer with the solvent molecules to the kinetic energy/~T, which should not depend on

Polymersin Solution

161

the concentration of a solution. This parameter is empirical and can be determined only experimentally. The parameter X provides a measure of thermodynamic affinity of a solvent for the polymer, or a measure of the quality of the solvent. The smaller the X, the better is the solvent thermodynamically. For very poor solvents, X may be higher than unity, and for very good ones, X may be negative. Based on X values, solvents may be classified as: ideal solvent ½); good solvent (X < ½); and poor solvent (X (X Relative

Vapor Pressure

and X

Equation (3.65), in which p~/p~ is the relative vapor pressure of solvent over solution, can be used for the determination of X. For a polymer of very high molecular weight, V1/V~ is small and may be neglected. Then Eq. (3.65) becomes simpler: 2lnPl _ ln(1 - ¢2) + ¢2 + X¢~ (3.66) P~ Measuring the value of p~/p~ over a wide range of concentrations and plotting [ln(pl/p~) -- ln(1- ¢2) -- ¢2] against ¢22, we get a straight line the slope of which is equal to X.

Problem 3.5 Measurements were made of the vapor pressure of polystyrene (mol. wt. 290,000) solutions in toluene and methyl ethyl ketone (MEK)[3]. table belowshowsthe ratios of the vaporpressure of the solvent over the solution, Pl, to the vapor pressure of the pure solvent, p~, against the correspondingweight fraction wl of the solvent at 25°C. Toluene wl Pl/P~ 0.156 0.523 0.236 0.704 0.304 0.791 0.380 0.866 0.476 0.920 0.599 0.969 0.744 0.997 (Data fromRef. 3.)

Methylethyl ketone 0.108 0.153 0.206 0.208 0.298 0.401 0.529

0.610 0.726 0.824 0.833 0.920 0.968 0.990

For the calculation of volumefractions of the polymer,its density wasdeterminedin each solvent at 25°Cby measuringthe densities of solutions of knownconcentration and assumingadditivity of volumes. The determinations gave a polymerdensity of 1.083 g/cm3 in toluene and 1.091 g/cm3 in MEK.Densities of toluene and MEK

162

Chapter 3

at 25°C are 0.8610 and 0.7996 g/cm3, respectively. Determine the parameter X for polystyrene in the two solvents. Answer: Definitions : weights of solvent and polymer, respectively. molecular weights of the repeating unit (mer) in the polymer and of the solvent and polymer, respectively. Pl, P~ = densities of solvent and polymer, respectively. o- = number of segments in the polymer chain in solution. X = degree of polymerization of polymer, i.e., M2/Mo In the lattice theory, the volumes of polymer segments and solvent molecules are assumedto be of equal size, i.e., %1)i,W2 ---~

Mo, MLM2=

(M2/p2) Substituting X

M2 ---- XMoand rearranging, p~ M1

we get

oPl M0 The ratio X/o" can thus be evaluated from the measured densities: X 1.083 92 -..... 1.11 a 0.861 104 X 1.091 72 ¯ In MEK, -- = = 0.94 o0.7996 104 In toluene,

The size of the polymer segment is therefore very close to that of the repeat unit of the polymer in the present case. The number of segments in the polymer may thus be equated to the number of repeat units. Then from Eq. (3.52)

(~2/Mo) (Wl/M1) q- (w2/Mo) ~v2

(Mo/M1)~I+ Values of [ln(pl/p~) - ln(1 - ¢2) - ¢2] are thus calculated and plotted against ¢~ according to Eq. (3.55) in Fig. 3.4. From the slopes, X is found to be 0.45 for polystyrene-toluene and 0.70 for polystyrene-MEK. Thus, toluene is a good solvent and MEKis a poor solvent for polystyrene.

163

Polymersin Solution 0.6 0.5

0 0

0.2

0.4

0.6

0.8

Figure 3.4 Plot of Eq. (3.66) for data of Problem3.5.

Osmotic Pressure

and X

A relationship betweenthe osmotic pressure 1-I of a solution and the activity (al) of the solvent in it can be derived [4] I’-IVI = - .RT in a (3.67) I where V1 is the molar volume of the solvent in the solution. CombiningEq. (3.67) and Eq. (3.50) we obtain (note that the partial molar volume V1 of a pure solvent is the same as the molar volume V1):

H

RT = ---~-1 lnal = -

RT

In -77-_ (RT ~ "1 ~ ¢1 -Vl -v2)¢2

RT -.1 (3.68)

Assuming, as in the lattice model, that there is no change in volume on mixing, volume fraction ¢i and concentration c~ are related as

¢i = c//d~ where d~ is the density of component i. Similarly,

(3.69) molar volume V/ is

164

Chapter 3

related to molecular weight Mi by

(3.70)

Vi = Mi/di Nowdeveloping In Cx into a Taylor series

ln¢l

= ln(1-

¢2) = -¢2-

¢-~ - ¢-~ .... 2 3

and employing Eqs. (3.69) and (3.70) we can transform Eq. (3.68)

H c~

RTdlc~ 3Mld~

= =

RT RTdl (~ M~ +’~o - - X)C2 RT + RTA~c2

(3.71)

where A2 is the second virial coefficient (see later) represented dl

A2 -

(1

_ X)

(3°72)

M1~2

At low concentrations, the second term of LHSof Eq. (3.71) is small and maybe neglected. The graphical representation of the resulting equation, plotted as ~/c2 vs. c2, is a straight line (see MembraneOsmometryin Chapter 4) the slope of which is RTA2. Hence it is possible to determine A2and from it the value of X using the relation (3.72).

Problem 3.6 The osmotic pressure data for polystyrene of molecular weight 1.6 x 106yielded, accordingto Eq. (3.72), the followingvalues for the secondvirial coefficient: (a) 2.88x10-4 mol cm3 -4 g-~ in dichloroethane and (b) -0.37x10 mol cma g-2 in cyclohexane, both at 22°C. DetermineX for the polymerSsolvent systems. Whichis a better solvent ? [Data: Density of polystyrene = 1.05 g cm-3; density of dichloroethane = 1.24 g cm-3; density of cyclohexane= 0.77 g cm-3.] Answer: From Eq. (3.72) 1

x = -~--

A2MI~

a.x

Polystyrene-dichloroethanesystem Molar mass of C2H4C1~:M1= 2x12 + 4×1 + 2×35.5 = 99 g/tool 1 (2.88 x 10-4 mol cm~ ~ g-2)(99 g mo1-~)(1.05 g cm-3) X -- 2 (1.24 g -3) cm = .0.475

Polymersin Solution

165

Polystyrene-cyclohexanesystem Molar mass of C~H12: M1= 6×12 + 12xl = 84 g mo1-1 1 (-0.37 x 10-4 tool cm3 2g-~)(84 g moI-I)(1.05 g cm-a) X - 2 -3 0.77 gcm = 0.504 Thermodynamically, dichloroethane is thus a better solvent than cyclohexane for dissolving polystyrene.

Virial

Coefficients

Whenanalyzing the thermodynamic properties of polymer solutions, it is sufficient to consider only one of the components, which for reasons of simplicity is normally the solvent. For manyphysicochemical calculations, especially whendealing with dilute solutions, it is convenient to express the solvent activity as a power series in terms of polymer concentrations ¢2 in mass/volumeunits.

Problem3.7 Starting with Eq. (3.17), express the difference in chemicalpotential of the solvent in the solution and in the pure state, i.e., (#1 - #~), in terms the massconcentrationsof the solute in dilute solutions. Answer: Withthe solvent labeled as component1 and solute as component2, Eq. (3.17) gives for a bicomponentsolution: Izl - tz~ = RT In a1 = RTln’71~cl = RT (1n71 + lnxl) (P3.7.1) where71 is the activity coefficient and xx is the molefraction of the solvent in the solution. (For the pure solvent, "~1 = 1.) If the solute in solution is neither associatednor dissociated, then xl = 1 - x2, where xz is the solute molefraction. For dilute micromolecularsolutions it is generally a goodapproximationto take q,~ = 1 and 1n71= 0, that is, we assume an ideally dilute solution. Equation(P3.7.1) is accordinglyapproximated #1 - #7 = RT ln(1 - x2) ~_ 1~ (-z2 - z~/2 .... ) (P3.7.2) In dilute solution, the total numberof moles of solute and solvent in unit volume will approach Ca, the molar concentration of solvent. Then the mole fraction xz of solute can be expressed as

¯ ~ = c~/(ca + c~) ~_ c~/cl

0’3.7.3)

166

Chaplet 3

where C2 is the molar concentration of solute. If the mass/volumeconcentration of solute is c2 and M2is the molecular weight of solute, then c2 = C2M2 (P3.7.4) and x2 = c2/CaM2 (P3.7.5) If the molar volumeof solvent is V~, with the samevolumeunit as is used to express the concentration cl and C1 (e.g., liters), then for dilute solution C1 = l/V1 and x2 = c2V~/M2 (P3.7.6) Substituting in Eq. (P3.7.2) gives the following relation for the difference chemicalpotential of the solvent in the solution and in the pure state: #I - #I = -RT V~ ~ + ~ 4 + ~3M~) ~ +"" (P3.7.7)

Equation (P3.7.7) shows the difference in chemical potential of the solvent in the solution and in the pure state to be a power series in solute concentration. Such equations are called virial equations. Equation (P3.7.7) is the key to the application of colligative properties to polymer molecular weights. Thus, insertion of experimental values of V1, and (/Zl -- #~) into Eq. (P3.7.7) would provide a measure of the solute molecular weight M2and we have seen in the preceding section how the colligative properties are related to (/~1 -- #~), as in Eqs. (3.88) - (3.91). Equation (P3.7.7) derived above for ideal solutions is invalid for real solutions. Polymersolutions are rarely ideal even at the highest dilution that can be used in practice. It is, however,useful to retain the form of the ideal equation and express the deviation of real solutions in terms of empirical parameters. Thus, the usual practice in micromolecular thermodynamics is to retain Eq. (P3.7.7) but substitute fictitious concentrations called "activities" for the solute concentrations. In polymer science, on the other hand, solute concentrations as measured are used and deviations from ideality are expressed in the coefficients of the concentration terms. Thus, for a real solution Eq. (P3.7.7) is expressed in a parallel form ]~1--

~ = --RT

V1 [c__~

+ Azc~

+ Azc~

+’"]

(3.73)

where A2and A3 are the second and third virial coefficients. In a bicomponent solution the subscript 2, which refers to solute, is often deleted. Equation (3.73) may thus be written I~

#~

RT V1

+ 2c + A3c ~ + "’"

Polymersin Solution To obtain virial expressions for colligative properties, Eq. (3.74) may be combined with respective Eqs. (3.61)-(3.64). If, for example, osmotic data are used, Eq. (3.74) is combined with Eq. (3.64) to write the virial +A2c+A3c2+’’’]c equation ~=RT[-~ (3.75)

Unfortunately, there is no uniformity in the exact form of the virial equations used in polymer science. Alternatives to Eq. (3.75) include

and

II

RT

+ Bc + Cc 2 + -.. c M The three forms are seen to be equivalent if -

(3.77)

B = RTA2 = (t~T/M)F~ (3.78) Authors mayreport virial coefficients without specifying the equation to which they apply. However, this can be deduced by inspecting the units of the virial coefficient. Virfal

Coefflcfent

and X

In the lattice model for polymer solutions which assumes that there is no change in volume on mixing, volume fractions ¢i and concentrations c4 are related as 4i = c/vi (3.79) where the vi’s are specific volumes. Developing ln(1 -- ¢2) into a Taylor series and employing Eq. (3.79), we can transform Eq. (3.60) lnal

= A~z~_ =RT (-¢~-

¢~ -2 ¢~3 ....

)

(1_ = - ~ + ~t~ - x) + 3 + ... Noting that ~ = v2M2, we can modi~ this relation

[ IV12

further

(3.80) as

168

Chapter 3

or

where the second virial coefficient A2 given by A2 = (½ - X)v~/V1 (3.83) and the third virial coefficient A3 are defined as coefficients of the power series in Eq. (3.81). This equation gives the virial expansion of the activity and relative chemical potential of the solvent in a polymer solution. Interpretation of the second and third virial coefficients, A2 and A3, in terms of Flory-Huggins theory is apparent from Eq. (3.82), The second virial coefficient A2 evidently is a measure of the interaction between a solvent and a polymer. WhenA2 happens to be zero, Eq. (3.82) simplifies greatly and many thermodynamic measurements become much easier to interpret. Such solutions with vanishing A2 may, however, be called pseudoideal solutions, to distinguish them from ideal solutions for which activities are equal to the molar fractions. Inspection of Eq. (3.83) reveals that A2 1 vanishes when the interaction parameter X is equal to ft. Weshould also recall that X, according to its definition given by Eq. (3.40), is inversely proportional to temperature T. Sincelx is positive for most polymer-solvent systems, it should acquire the Value fi at some specific temperature. Moditlcatlon

of

F1ory-Huggil~S

Theory

It will be recalled that the first two terms on the right-hand side of Eq. (3.50), which was derived from statistical considerations, represent the combinatory part of the entropy of mixing and the last term represents the contact free energy of mixing. The original Flory-Huggins theory of the contact term essentially followed the van Laar theory for low molecular weight mixtures and assumed that X is independent of molecular weight and concentration. However, it was soon discovered that X depends on both these variables. For an analytical representation of the experimentally found concentration dependence, X is normally expanded into series in powers of X = XO q-

Xl(fi2

q-

X2gb2 2 q-

"’"

(3.84)

The coefficients X1, X2, etc., are determined from observed variation of X with ¢2. Very often X increases strongly with polymer concentration, particularly in the case of poor solvents. The systems benzene/polyisobutylene, cyclohexane/polystyrene, and methyl ethyl ketone/natural rubber constitute examples. In some cases, however, X see’ms to be independent of concentration, as proposed by the original Flory-Huggins theory. These findings mainly concern good solvents; some examples are benzene/polyisoprene

169

Polymersin Solution

and cyclohexane/polyisobutylene. In a few cases, mostly with highly exothermal systems, X decreases with concentration, as with toluene/polystyrene. The effect of molecular weight is qualitatively explained by excluded volumetheories, which are described later. Thoughthe Flory-Huggins relation also describes the temperature dependence of the thermodynamics of mixtures, it was later found that, because of its conceptual limitations, the theory is not satisfactory for treating temperature dependences. Flory therefore modified the theory in the following way. It will be recalled that the term1~ appearing in (~1 -- X), which is the key factor for the calculation of A2 from Eq. (3.83) originated from the Taylor expansion of ln(1 -- ¢2) and so could be traced back the entropy of mixing, while the X term was of enthalpic origin. However, in real systems, molecular contacts may change the contribution of individual molecules to entropy. Hence X in real systems has also an entropic component. Similarly, in arriving at the entropy term that led to the value 1

7, we included only the configurational entropy of the lattice model and neglected possible contributions from changes in the volume in mixing and from contact interactions. A modification of the theory therefore abandons the particular values for both terms and adopts more general unspecified values: ~ for the entropic term and t¢ for the enthalpic term. Recognizing that the enthalpic term must be inversely proportional to temperature, t¢ is replaced by ~b0/~T, where t9 is a new parameter with the dimension of temperature. The factor (½ -- X) is thus replaced by (~b - ~), that 1

~ - X = ¢ - ~ = ¢(1 - 19/T)

(3.85)

In terms of the new symbols, the second virial coefficient is then given by the expression [cf. Eq. (3.83)]: A~ = ¢(1

- 19/T)v~/V~

(3.86)

Thus, at some special temperature T = 19, A2vanishes and the solutions become pseudoideal. Such solutions are also called theta solutions. The second virial coefficient is positive at temperatures higher than /9 and negative at lower temperatures. Essentially what we have done above is to replace one interaction parameter X with twonew parameters ~b and 19, adding flexibility to our relations. However,by doing so we have effectively abandoned the premises of the pseudolattice model and adopted a phenomenological approach of modeling.

Problem 3.8 Osmotic measurements on a high-molecular-weight polystyrene samplein cyclohexaneat various temperaturesyielded the followingvalues for the

170

Chapter 3

second virial coefficient (A2): Temperature, 25 30 35 40

°C

A2 x 104, cm3 mol g-2 - 0.334 - 0.146 0.036 0.212

Determine for the polymer-solvent system, (a) the temperature at which theta conditions are attained, (b) the entropy of dilution parameter ¢ and (c) heat of dilution parameter ~ at 27°C. [Specific volume of polymer = 0.96 cm~/g; molar volume of cyclohexane at 27°C = 108.7 cm3/mol.] Answer: According to Eq. (3.86), the plot of A2 vs. lit should be linear over a narrow temperature range and A2 should be zero at T = /9. The plot shown in Fig. 3.5 thus gives 0 = 34°C. From Eq. (3.86),

¢ = v (1-A2V~ 0/T) -4 At T = 27°C ~- 300°K, g-2. Therefore, (-0.26

A2 (from

Fig.

× 10.4 cm3 mol g-2)(108.7

Theta

= 0.13 "¢0 (0.13)(307 °K) .... T (300 °K)

Temperature

for

= -0.26x10

cm3 mol

3 mo1-1)

(307°K)]

(0.96 cm" g-’)-~

n

3.5)

0.133

Polymer-Solvent

Systems

Just as real gases do not necessarily obey ideal gas laws, except at low pressures, real solutions do not behave as ideal solutions certain limiting circumstances. We define an ideal solution chemical potentials of all components are given by

#~ - #’~ = RTlnx~ To take into account deviations into Eq. (3.87) and write

from ideal

behavior

#~ - I~’~ = RTlnx~ + RTlnTi

except under such that the

(3.87) we add another

term

(3.88)

Polymersin Solution

1 71

0.5

0.5

0.3 = 3/_,°C E

OA

E

0

U

-0.1 ~o 123

< -0.3 ! _0.5

l 3.1

~ 3.3 -r-ix 103 -1) (K

~

-0.5 3.5

Figure 3.5 Plot of secondvirial coefficient vs. reciprocal temperature(Problem

where ")’i is termed the activity coefficient of componenti and is in general a function of the composition of the solution. Comparison of Eq. (3.88) with Eq. (3.17) showsthat the product zi"/i is equal to the activity of the component. Whenanalyzing thermodynamic properties of polymer solutions it is sufficient to consider only one of the components, which for reasons of simplicity normally is the solvent. It is convenient to separate Eq. (3.88) for solvent (i = 1) into ideal (or__excess) (/~1 and -- nonideal #~)ideal /~Tln~contributions by defining 1 (3.89) (#,t~) E = RTln’yl (3.90) where the superscript E indicates an excess contribution. For dilute polymer solutions (< 20 g/L, depending on polymer molecular weight), nl >> crn2 and 1 >> n2, which l ead t o d?2 = o’n2/(nl x2 = n2/(nl

+ o’r~2) "." on2/nl + n~) "~ n2/nl

(3.91) (3.92)

1 72

Chapter 3

Hencefor dilute polymer solutions both ¢2 and x2 are small and related by x2 = ¢2/~r. The logarithmic terms In ¢1 and In xl in Eqs. (3.53) and (3,89) ~n be appro~matedby series expansion to give InCa = ln(1 and

- ¢2) = -¢2 - ¢~/2 - ¢~/3 ....

(3.93)

lnx~ = ln(1 - x2) = -x2 - z~/2 - z~/3 ....

(3.94)

which upon substitution

lnz~ = -(ff~/~)

of ~2 = ~2/ff gives

- (ff~/~)2/2 - (ff~/~)z/3

(3.95)

Since ~2 is small and ff is large, only the first ~o terms of Eq. (3.93) and the first te~ of Eq. (3.95) need to be retained. Using these appro~mationsin Eqs. (3,53) and (3.89) leads ~ - ~ o= -RT¢~/~ + RT (x~~¢2 (3.96) and (Ul -- ~)ideal = nT¢2/ff (3,97) Compa6son of these ~o equations showsthat for dilute polymersolutions, Flo~-Huggins theo~ predicts

the excess contribution

-

to be

= (x

(3.98) ~)¢~ in which the quanti~ -- ~0~/2 arises from the connectivi~ of polymer chain segments and the quanti~ ~X~from contact interactions. When X = ~, these ~o effects compensate each other, (~1 -~)E = 0 and the dilute polymer solution behaves ideally, i.e., as if ~x = 0 and the polymer chain segments were not connected. Flo~ used the te~ them condi6onsto describe this ideal state of dilute polymer solutions and developed the concept defining the excess pa~ial ~E furthe?~y molar enthalpy, ~H~, and entropy, &~ of m~ng as A~

= RTa¢~

=

(3.99)

(3. 00)

where~ and ¢ are enthalpic and entropic parameters, respectively. Recalling that chemicalpotentials are partial molar free energies, that is,

=

:

-

(3.101)

and makinguse of relations (3.99) and (3.100) weobtain ~ (3.102) (.1 - /~)E RT(~¢) ¢2 Comparisonof Eqs. (3.98) and (3.102) showsthat [cf. Eq. (3.86)] n--

¢ = X- ½

(3.103)

1 73

Polymersin Solution

Theta conditions occur at a particular temperature T = 0, known as the theta (or Flory) temperature, for which (/zl -- /-t~) E = 0. It means that both terms on right hand side of Eq. (3.101) compensate each other at T = 0 such that A~f = 0A~f (3.104) and hence ~ ---- ¢ (3.105) Equation (3.104) provides an alternative definition of the theta temperature, as being the proportionality constant relating /k~l E to E. A~I According to Eq. (3.105), the theta (or Flow) temperature can defined as the temperature at which the condition t~ ----- ~b prevails. Substituting Eqs. (3.99) and (3.100) into Eq. (3.104)

which when substituted into Eq. (3.102) leads

A temperature function of X is established and (3.107)

by combining Eqs. (3.98)

x -- ¢[(O/T) - 1] + 1/2

(3.10S)

Mixing is promoted by a negative value of (it1 -- p~)E, which according to Eq. (3.98) corresponds to X < ~1and according to Eq. (3.107) corresponds to T > 0 for positive ~b and T < 0 for negative ~b. These equations also show that under theta conditions X = ½, T = 0 and the dilute polymer solution then behaves as an "ideal" solution. Values of 0 and ~b for a number of polymer-solvent systems [5] are given in Table 3.1. It maybe noted that the theory presented above for dilute polymer solutions is based upon the Flory-Hugginsequation (3.50), whichstrictly is not valid for such solutions because of the mean-field approximation and so Eqs. (3.99) and (3.108) do not accurately predict (it1 -- E. Nevertheless, these equations are of the correct functional form as shownby the fact that the relationships

(.1 - (x - ½)

(a.109)

and

are also obtained from the more realistic excluded-volumetheories discussed later. It may thus be stated that the Flory-Huggins theory leads to correct

174

Cl~ap~er3

Table 3.1 aThermodynamic Parameters Polymer bPolyisobutylene

bPolystyrene

cPoly(acrylic acid) Polymethacrylonitrile aFrom Ref. 5.

for Some Polymer-Solvent

Solvent Benzene Toluene Ethylbenzene Cyclohexane rz-Heptane Cyclohexane Benzene Toluene Ethyl acetate Methyl ethyl ketone Dioxane c Butanone

0 in °K 297 261 251 126 0 307 100 160 222 0 302 279

~/, 0.15 0.14 0.14 0.14 0.035 0.13 0.09 0.11 0.03 0.006 -0.310 -0.630

Systems ~ at 25°C 0.15 0.12 0.117 0.059 0 0.134 0.03 0.06 0.02 0 -0.314 -0.590

bThermodynamic parameters are calculated from intrinsic viscosities and their temperature coefficients. Thevalues are significantly lower than those obtained by other methods, such as osmotic method(cf. Exercise 3.10) and phase separation studies (see Problem3.16) CThermodynamic parameters are derived from phase separation studies.

qualitative conclusions about the general thermodynamic behavior of dilute polymer solutions, especially with regard to the prediction of ideal behavior when X = ½ and T = 0. The accuracy of the Flory-Huggins theory improves as theta conditions are approached, i.e., as X --~ 1 ~ and

Significance

of ~b and t¢

The thermodynamic parameters ~b and ~ introduced above, pertaining to polymer-solvent interactions in dilute solutions, may be determined from thermodynamic studies of dilute solutions of the polymer, e.g., from osmotic pressure or turbidity measurements at different temperatures. These parameters may also be determined, at least in principle, from viscosity measurements on polymer solutions (see Frictional Properties of Polymers). The parameter ~b, which is a measure of the entropy of mixing, appears to be related to the spatial or geometrical character of the solvent. For those solvents having cyclic structures, which are relatively compact and symmetrical (e.g., benzene, toluene, and cyclohexane), ~ has relatively higher values than for the less symmetrical acyclic solvents capable of assuming a number of different configurations. Cyclic solvents are thus more favorable

Polymersin Solution

175

from the standpoint of the entropy than acyclic ones. A solvent for which the entropy of mixing of a given polymer (as measured by ~b) is high is a "geometrically good" solvent for the polymer. In such a solvent, the viscosity of the polymer solution will be high, or will approach a high value at high temperatures. In a "geometricalIy poor" solvent (where ~b is small) the viscosity will be relatively low. A "thermally good" solvent, on the other hand, is one for which the heat of mixing (as measured by ~) is low, while for a "thermally poor" solvent ~ is relatively large. In an athermal solvent (~ = 0), the volume expansion of the polymer molecule will be independent of temperature, while in a thermally poor solvent it will increase with temperature. The change of viscosity with temperature will thus be low in a solvent with low ~ and relatively high in a solvent with relatively large t~. There appears to be no correlation between ~b and heat of mixing [5]. Thus a geometrically good solvent for a polymer may be thermally good or thermally poor. For’ example, both benzene and cyclohexane are geometrically good solvents for polyisobutylene (Table 3.1), but the former is thermally poor while the latter is thermally good. However, the role of the two solvents is seen to be reversed for polystyrene (Table 3.1). For polystyrene in methyl ethyl ketone, ~b is nearly zero; since X = ½ + t~- ~b, this solvent is therefore a poor one for polystyrene not because of an adverse energy of interaction but because of the low entropy. It will be observed even for the limited data in Table 3.1 that entropies of dilution (as indicated by ~b) are highly variable from one polymer-solvent system to another and from one solvent to another for the same polymer depending on the geometrical character of the so/vent. This is contrary to the theory developed from consideration of lattice arrangements according to which ~b should be approximately 1~ and nearly independent of the system. It maybe noted that theories of polymer solutions fail to take into account the specific geometrical character of the solvent in relation to the polymer segment. This is a serious deficiency which must be borne in mind in applications of these theories. Methods to Determine 7~heta Solvents Theta (0) solvents are solvents in which, at a given temperature, a polymer molecule is in the so-called theta-state. The temperature is knownas the theta-temperature or the Flory temperature. (Since R J. Flory was the first to show the importance of the theta-state for a better understanding of molecular and technological properties of polymers, theta temperatures are also called "Flory temperatures.") In the theta-state, as explained above, the solution behaves thermodynamically ideal at low concentrations.

1 76

Chapter 3

Long-range interactions (interactions with solvent and excluded volume effect, described later) are not present in the theta-state. At the theta temperature, the polymer molecule thus exhibits its unperturbed dimensions, i.e., dimensions influenced only by short-range interactions between neighboring groups and by skeletal effects (bond distances, valence angles). The suitability of a solvent as theta-solvent thus depends on both the polymer (constitution, configuration) and the solvent (constitution), and temperature, because all these factors influence long-range interactions. Several methods can be used to determine theta solvents. These include phase equilibria studies (see Phase Equilibria in Poor Solvents), determination of second virial coefficient (see Problem 3.8), viscosity-molecular weight relationship, and cloud point titration. The viscosity method makes use of the fact that the exponent, a, in the Mark-Houwinkequation (see Frictional Properties of Polymer Molecules a, is equal to 0.5 for a random coil in a in Dilute Solution), r/ = KM theta-solvent. A series of polymers of the same type with widely different knownmolecular weights is used to determine intrinsic viscosities [r/] at different temperatures and hence a at different temperatures. The thetatemperature can thus be determined either by direct experiment or, if it is not in the measurable range, by calculation. In the cloud point titration method, polymer solutions of different concentrations are titrated with a nonsolvent until the first sign of cloudiness (see Phase Equilibria in Poor Solvents). The logarithm of the nonsolvent concentration at the cloud point is then plotted against the logarithm of the polymer concentration at the cloud point and extrapolated to 100%polymer. The solvent/non-solvent mixture corresponding to the 100%polymer is a theta-mixture. Flory-Krigbaum

Theory

One of the deficiencies of the Flory-Huggins theory, as we have noted earlier, is the use of a mean-field approximation whereby it is assumed that the segments of polymer molecules are distributed uniformly in the lattice. This assumptionis valid only for relatively high concentrations of the polymer where the polymer molecules interpenetrate each other, making the solution appear more or less homogeneouson a macroscopic scale. At very low concentrations, the molecules lie in regions as far separated from each other as possible; thus making the solution inhomogeneousas illustrated in Fig. 3.6. Then the aforesaid approximation is invalid. However, in such solutions, the density of segments within an elemental volume in the polymer domain can still be assumed to be constant, and the results of the Flory-Huggins theory applied to it. Flory and Krigbaum [6] have analyzed

177

Polymersin Solution

(a)

(b)

Figure 3.6 Schematic illustration solutions.

of (a) dilute and (b) concentrated polymer

dilute solutions in this mannerand found that this leads, on proper integration, to an additional factor _b’(X) in the second term of Eq. (3.96): ]A1 __ idlo = RT [__¢2 + (X ½)F(X)¢[ + -. .] (3.111) 2where

X F(X)

= 1 2!

23/2

X + 3!

33/~

0 M~/2 X =

4CM1b(1-~)

....

(3.112)

(3.113)

is a parameter depending on the properties of the polymer-solvent system and o~s is an expansion parameter. Both the terms are defined in the following section on Excluded Volume Theories. At T - 0, X becomes zero, F(X) becomes equal to 1, and the Flory-Krigbaum theory reduces to the Flory-Huggins theory. /~M

Problem 3.9 For polystyrene of molecular weight 100,000, calculate approximately the solution concentration below which the Flory-Krigbaumtheory should be used. Use the following relationship between the RMSradius of gyration ($2) 1/2 and the polymer molecular weight M: (S2) 1/~ ~- 2 x 10-gx/--~ cm. Answer: The radius of a sphere enclosing the segments of a polymer molecule is taken

1 78

Chapter3

approximatelytwice the value of ($2)~/~, that is, R "~ 4 × 10-9V/--~ cm The volume, V, of the solution containing n polymermolecules is given by V ~ n(~)Tr.t~

3

In terms of the weight, w, of the polymer, n maybe written as -3 molecules cm n = (w/M) NAv and so 3 _ 0.16w-v/-~cm V ~ (~)NAv(~)TrR 3 ~ The concentration at the point of overlap is, therefore, Coverla p = w/V = (0.16x/--~) -1 = (0.16 lx/~) -1 -3 = 0.02 g cm For polystyrene of molecular weight 105 the overlap therefore occurs above about 2%concentration. Belowthis concentration, one should therefore use the Flory-Krigbaumtheory.

Excluded

Volume

Theories

As noted above, the basis of derivation of the Flory-Huggins equation is satisfactory at higher concentrations of the polymer, but is invalid for dilute solutions, in which the domains inhabited by the individual polymer molecules are far apart. Yet whe~awe calculated in the preceding sections, chemical potentials and their virial expansion, we employed precisely the forbidden assumption of low concentration of the polymer. The result had to lead to discrepancies. For example, according to Eq. (3.83), the second virial coefficient A2should be independent of molecular weight of the polymer, but experiments showthat it decreases significantly with increasing molecular weight. Evidently, to cope with solutions, a different approach is required and for this reason we invoke the concept of excluded volume. The excluded volume of a solute molecule is the volume that is not available (because of exclusion forces or for other reasons) to the centers of mass of other similar solute molecules. As an example, let us consider the excluded volume of a spherical particle of radius, R. The position of a sphere is fully described by coordinates of its center. It is apparent from Fig. 3.7(a) that the center of one solid sphere cannot approach the center of another solid sphere closer than two radii (2R). Hence, the volume excluded by one sphere equals ~Tr(2R)3, that is,eight times its actual volume. The excluded volume of asymmetric particles cannot be calculated so easily. This is because the distance between their centers of mass when they are in contact depends on their orientation. Nevertheless, it has been

179

Polymersin Solution

(a) ..

being added ~_~Molecute

~ /;)"

(.b)

votume

,

/

Motecute being added

~ ,- "~-’"~

_

~

\ /--Exctuded

I

Figure 3.7 The excluded volumefor solid spherical particles. (a) Nooverlap excluded volumein very dilute solutions. (b) Anoverlap of excluded volume more concentrated solutions. calculated that the excluded volume of a rod of diameter d and length L 1 2. It is also apparent from Fig. 3.7(b) that at higher concentrations is ~d/~ the volume excluded by two or more particles may overlap with the result that the total volume excluded by all particles will be less than the sum of the volumesexcluded by individual particles. Hence, the theory will run into complications at higher concentrations.

Compact Molecules In very dilute solutions, the molecules of the solute are completely separated from each other and are in contact 0nly with solvent molecules. If more solvent is added to such a solution, the number of solute-solvent contacts does not change. Hence no enthalpy change accompanies this process. Solute molecules occupying some location exclude this location from the volumeavailable to other molecules. This steric interference influences entropy but not enthalpy [7].

180

Chapter 3

Calculation of the entropy of mixing is based on the concept that the contribution of a solute molecule to entropy depends on the number of ways in which we can place the molecule into solution. The latter quantity is assumed to be proportional to the total volume of the system minus the excluded volume of solute molecules placed in solution previously. Based on this consideration one derives [7]: o,( -~1 ;z, - #1 = -RTc~V1

UNAv~ +) c2-~-~9.2

(3.114)

where u is the excluded volume of each solute molecule, NAv is Avogadro’s number, and other terms are as defined earlier. Comparison of Eqs. (3.82) and (3.114) yields for the second virial coefficient UNAv A2- 2M22 (3.115) For compact particles of a given shape, the molar excluded volume, uNAv, must be proportional to the molecular weight of the particles, M,2. Hence, according to the excluded volume model the second virial coefficient of particles of a given shape decreases with increasing molecular weight, being inversely proportional to the latter.

Problem 3.10 Osmotic pressure measurement was made at 27°C on an aqueous solution of a globular protein believed to be spherical in shape. For a solution of 3 3the osmotic head was 1.64 mmwater and for 5 mg/cm concentration 1 mg/cm it was 8.31 mmwater. The numberaverage molecular weight of the protein is 1,56,500 and its specific volumeis 0.75 cm3/g. Calculate the excludedvolumeof the protein and compareyour result with the partial molar volume. Density of 3. water at 27°C is 0.996 g/cm Answer: The secondvirial coefficient (A2) can be calculated experimentallyfromvalues 1]/c (omitting the subscript 2 for solute) measuredat two concentrations cl and c2 of dilute solution using [cf. Eq. (3.71)]: A2 = (Y~/c)2 - (H/e), RT(c2 - cl) Pressure of h cm columnof water at 27°C = (h cm)(0.996 g -3) --: (0.996h g cm-2)/(1033 -2 atm-1) or 9.6 4 × 1 0- 4h atm -2) (latm = 1033gem

Polymers in Solution

181

cl = 1 mg/cm-3 = 1 g/L-l; 1-[1 = 0.164x9.64×10 -4 or 1.58×10 -4 atm c2 = 5 mg/cm-3 = 5 g/L-~; 1-[2 -- 0.831×9.64×104 -4 or 8.01×10 -4 atm (8.01 × 10-4 atm)/(5 g -~) - (1.58 x 10-4 at m)/(1 g -~) A2 = (0.082 L atm mo1-1 °K-1)(300°K)(5 -1 - 1 g L- ~) = 2.24x

10 -8 molLg-2 (= 2.24x

10 -~ molm3 -~) kg

Excluded volume per mol, UNAv,from Eq. (3.115): = 2A2M~ = 2(2.24×10 -s tool L g-2) (1.565x10 ~ 2g tool-l) 3 t ool = 1.1×103 L tool -~ (= 1.1 -1) Partial

molar volume, V~

= (0.75×10 -3 I. g-l) (1.565×10 ~ -~ g )tool -~ -1) 3 = 0.12×103 L tool (= 0.12 t ool The excluded volume (per moO of the globular prgtein calculated above is about 9 times the molar volume. This compares well with the case of spherical pa~rticles for which the excluded volume equals eight times the particle volume.

Polymer

Coils

Since most polymer molecules have relatively flexible backbone, they tend to be highly coiled and can be represented as random coils. A molecular coil, being very loose, inhabits a region significantly larger than the actual volume of the coil. In fact, the actual macromolecule accounts for only a few percent at most of the region of the coil. It may thus appear that two polymer coils can easily interpenetrate and the excluded volume and its effects are very small. However, even as approximate calculations for a fictitious case in Problem 3.11 show, this impression is totally untrue.

Problem 3.11 Consider two polymer coils each consisting of 1000 segments homogeneously distributed in their respective spheres. If the segments occupy 1% of the sphere’s volume, what is the probability that the two spheres can be superimposed so that all segments find an unoccupied location simultaneously? Answer: Let us place the center of the two spheres on the same point so that the spheres overlap fully. For the first segmentin the first sphere, the probability that it does not occupy the same location as any segment in the second sphere is 0.99. For every other segment, the probability is the same. Therefore, the probability that all

182

Chapter3

segmentsfind an unoccupiedlocation simultaneouslyis 0.99l°°°, -5, or about 4× 10 aveD, small numberindeed.

While for compact particles (see Fig. 3.7) a given location is either excluded or it is not, i.e., the degree of exclusion is either zero or one, for loose particles (polymer coils) the degree of exclusion may, however, have any value betweenzero and one; and it is possible to find the effective total excluded volumeas an integral of the degree of exclusion over the region of coil. Thoughin the exampleof Problem3.11, the distribution of segments is considered to be uniform, in the actual models the distribution of segments is chosen so as to conform more closely to actual coils. One model assumes Gaussian distribution of densities in a particle of spherical symmetry.In fact, this relatively simple modelrepresents fairly the actual distribution averaged over a long time. Other models which take into account the continuity of the polymer chain and the distribution of intersegmental distances, usually require more advanced mathematics for treatment [7,8]. The above spatial problem is intertwined with the phenomenon of intersegmental interactions. Various theories differ mainly in the way in which they model the spatial problem and in the rigor of mathematical treatment, but they treat the problem of intersegmental interactions in the same way. If the formation of segment-segment contact lowers the overall energy (this corresponds to higher X12 in the Flory-Huggins theory), then interpenetration of coils is enhanced. At some stage the physical exclusion and the intersegmental attraction balance each other exactly and the coils then interpenetrate freely. The theories consider this state of balance as the perfect reference situation and calculate the perturbation of this perfect state that results if the balance is not complete. Accordingly, these theories are called perturbation theories. They are mathematically rather complicated, and we will be satisfied by considering only their main results. The final results of many perturbation theories (often called twoparameter theories) contain the combinations Nb2 and N2fl as the only sample dependent quantities. Here N is the number of segments, each of length b, in an equivalent freely jointed polymer chain, related by (r2)o = 2 (s ee p. 64), and fl, the so-called excluded volume inte gral, is the volume excluded by one chain segment to another. It is convenient to introduce a new parameter z, called the excluded volume parameter and defined as

183

Polymers inSolution

Introduction of the parameter z allows us to to express most of the results of perturbation calculations in a relatively simple form. For example, the expression for the second virial coefficient is A2 = ~,

(3.117)

2M~ ] h(z)

where NAyis Avogadro’s number and M2is the molecular weight of the polymeric solute. The exact form of the function h(z) has been the subject of extensive theoretical search. Thoughthis has resulted in manydifferent relations, most researchers agree that h(z) can be developed into a Taylor series : h(z) = 1- 2.865 z + ... (3.118) This series converges rather slowly; consequently, the results are useful only for small values of z and /~. It can be seen that for small values of z, the perturbation second virial coefficient given by Eq. (3.117) is equal to the factor in parenthesis. Since N is proportional to M2, the virial coefficient should be independent of molecular weight in the limit of small z. This is the same result we have derived earlier from the Flory-Huggins theory. In the limit of small z, Eq. (3.117) is frequently combinedwith Eq. (3.86) to yield

Thisrelation provides a usefullinkbetween ]V2~andF]ory’s constants ~ and0. Combining thisrelation withEqn.(3.116) we alsogetan alternative definition of the parameter z:

z:

M~/2

(3.120) The parameter A~ defined as above is frequently employed in experimental work and is independent of molecular weight. Combining Eqs. (3.117)-(3.120) yields for the second virial coefficient A~ ---

(v~2~

~b (1-

~-)(1-

2.865A~M~/2+

"")

(3.121)

Thus, for a given polymer-solvent system, A2 decreases with increasing molecular weight in agreement with experiments. The linear relationship (3.121) between A2and M~/2 is, however, valid only within a limited range of z values.

184

Chapter 3

Anexpression for the apparent excluded volumeof a coil can be obtained by comparingthe perturbation second virial coefficient [Eq. (3.117)] with the one from excluded-volume theory for compact molecules [Eq. (3.115)]. This gives u = N2flh(z) (3.122) Wemay now consider the effect of excluded volume on dimensions of polymer coils in solution. It maybe recalled that the mathematical modelfor evaluating (r2)10/2 is that of a series of connected vectors (representing backbone bonds) that are restricted locally to certain allowed orientations relative to each other. The model, however, places no restrictions upon the relative positions of two bond vectors widely separated in the chain. Thus, the model does not exclude the possibility of remotely connected bond vectors intersecting each other or even occupying the same volume in space. In a real polymer molecule, however, each part of the molecule excludes other more remotely connected parts from its volume. These 1/2, long-range steric interactions cause the RMSend-to-end distance, (r2} to be greater than (r2)lo/2. -4(re) Chain lo/2 dimensions which correspond to are unperturbed by the effects of volume exclusion and so are called the unperturbed dimensions. Similarly the RMSradius of gyration for perturbed dimensions in real chains, ($2) 1/2, is greater than that for unperturbed dimensions, (.$2)1o/2. The perturbed dimensions will differ from unperturbed dimensions by the expansion o~ of the molecule arising from the long-range effects. Thus, we may write <~.2>1/2 <$2> 1/2

= 0~r1o/2 = O/s<S2>1o/2

(3.123) (3.124)

No simple relationship exists between the two expansion parameters o~r and as. However,for small expansions (i.e., small z) the following relationships are obtained :

1 + (4/3)z

O~ s 2

1 + (134/105)z

(3.125) (3.126)

and so o~r > o~s. Since o~r and o~s are close (for small z), we shall use average expansion factor o~ instead of o~r and o~s for simplicitY. Equations (3.125) and (3.126) together with Eq. (3.120) show that expansion factor depends significantly on two molecular parameters. The first is molecular weight. At conditions far removed from unperturbed conditions, o~ increases without limit as the square root of the molecular weight. The second parameter determining o~ is A~, which chracterizes polymer-solvent interaction. Under theta conditions at which z becomes zero, a becomes unity. Physical measurements made under these conditions will reflect the characteristics of the unperturbed molecule. The overall dimensions of such a molecule will be determined solely by bond lengths

Polymersin Solution

185

and bond angles in accordance with the freely jointed chain model modified by fixed bond angle and restrictions upon bond rotation (see p. 63). Expansion

Factor

The second virial coefficient A2, whichis related to the Flory dilute solution parameters by Eq. (3.121), is a measure of solvent-polymer compatibility. Thus, a large positive value of Asindicates a good solvent for the polymer favoring expansion of its size, while a low value (sometimes even negative) shows that the solvent is relatively poor. The value of A2 will thus tell us whether or not the size of the polymer coil, which is dissolved in a particular solvent, will be perturbed or expanded over that of the unperturbed state, but the extent of this expansion is best estimated by calculating the expansion factor o~. As defined by Eqs. (3.123) and (3.124), o~ represents the ratio of perturbed dimension of the polymer coil to its unperturbed dimension. In a dilute solution, polymer molecules do not interact with each other and chain expansion depends upon the balance between the intramolecular segment-segment interactions (volume exclusion) and intermolecular segment-solvent interactions. Whena polymer is dissolved in a good solvent, the isolated polymer molecules expand from their unperturbed dimensions so as to increase the segment-solvent contacts. In a poor solvent, however, the segment-solvent interactions are weak and their Gibbs free energy of interaction maybe positive (i.e., unfavorable); the polymer chains therefore contract in order to reduce the numberof segment-solvent contacts. In poor solvents, isolated polymer chains are thus subjected to two opposing effects: (i) expansion due to unfavorable segment-segmentinteractions (i.e., volume expansion) and (ii) contraction due to unfavorable segment-solvent interactions. Under certain conditions in a poor solvent, these two effects may just balance each other and each polymer molecule then assumes its unperturbed dimensions. The conditions correspond to the theta conditions, introduced earlier to describe the situation when a dilute polymer solution behaves ideally. The solvent under these conditions is said to be a theta solvent for the polymer and measurement of dilute solution properties in such a solvent can yield unperturbed chain dimensions. According to Flory [9], the change in Gibbs free energy with increasing extension, i.e., with increasing c~ at highly elastic deformation, is given by the equation

Apart from this excluded volumeeffect, coil size in solutions is affected by interaction between a coil and a solvent. Solvent molecules penetrate

186

Ghapte~3

inside the coil, which thus swells and increases in size. The change in Gibbs free energy with coil size increase, caused by swelling, is expressed [9] by the equation OAG~ 6CMkT¢(1 - O/T)M~/2 (3.1~.8) ~(~ (~4 where k is the Boltzmann constant; CMis a parameter depending in a complicated way on properties of the polymer-solvent system, and is given by -3 27 ( /2 v~ I ((r2)o~ (3.129) \NAvVy] where v2 is specific volume of polymer and (r2)o is mean square end-toend distance of polymer molecule in ideal solvent. Note that since (r:Z)o directly proportional to number of backbone bonds in the polymer chain (see Problem 2.8) and hence to molecular weight M~. of polymer, CM independent of polymer molecular weight. The total change in Gibbs free energy accompanying the change in coil size is, from Eqs. (3.128) and (3.127): 6CMkT¢(1 At equilibrium,

- O/T)M~/~ 4o~

0AG/00~ = 0, and hence 2CM¢(1 --

O/T)M~/2 4o~

Rearranging, oP - oP = 2CM¢(1

- I~ O/T)M~

(3.132)

It follows from Eq. (3.132) that the size of a real coil increases with increasing molecular weight of the polymer. It also follows that at a temperature equal to the Flory temperature (T = 0), 0, i.e., ce = 1. Thus, there is a temperature for each dilute polymer solution, at which it behaves like an ideal solution, the factor ce being equal to unity, i.e., long-range interactions have no effect on the size of the macromolecules. The coil is then in an unperturbed state and the chain size depends only on short range interactions that occur between neighboring atoms and groups, and are usually forces of steric repulsion caused by overlapping of electron clouds. All the above reasoning applies only to very dilute solutions in which the chains do not interact with one another. Problem 3.12 Deduce from Eq. (3.132) that the molecular expansion factor a should increase with increase in temperature in a poor solvent, decrease with

Polymers in Solution

187

increase in temperature in a very good solvent, and be independent of temperature in an athermal solvent. Answer: From Eq. (3.106), 0 In a poor solvent, where both n and ¢ generally are positive, 0 also will be positive. Therefore according to Eq. (3.132), s - a3, and he nce a, should increase with increase in temperature. In a very good solvent, where the heat of dilution (~) is negative and ¢ normal (i.e., positive), 0 will be negative, and according to Eq. (3.132), ~ should increase with increase in temperature. In an athermal solvent, ,~ ---- 0, 0 = 0, and hence a should be independent of temperature. Problem 3.13 For a fractionated polyisobutylene sample of molecular weight 1.5 x 106 the intrinsic viscosity was measuredin cyclohexane at 30°C (good solvent) and in benzene at 24°C (theta solvent). The ratio of these two intrinsic viscosities yielded the factor az, by which the volume of the molecule in cyclohexane at 30°C is enlarged relative to the volume of the unperturbed molecule, as 4.03. Evaluate the total thermodynamic interaction (¢ - n) for polyisobutylene-cyclohexane 30°C. Use the observed relation for polyisobutylene between molecular weight (-~I2) and the unperturbed root-mean-square end-to-end distance (r2)1o/2, given by (r2>1o/2 = 0.75 × 10-8 M~°’5 cm to calculate the parameter CMas required. [Polymer density = 0.92 g/cm3; density of cyclohexane at 30°C = 0.772 g/cm3.] Answer:

~ = (~)i/~ = (4.03)i/~ = 1.591 v2 = 1/(0.92 gcm-z) = 1.087 cm3 g-1 -~ M~ (for C6H~2) = 84 g mol V~ = (84 g mo1-1)/(0.772

g -3) = 10 8.8 cm 3 mo1-1

From Eq. (3.129) CM = ~ (6.02

× 1023

mo1-1)(108.8

3 mol -~)

(0.75 x 10-8 cm mol1/2 g-V2)-3 = 0.0366 tool V2 g-V~ From Eqs. (3.106) and (3.132):

¢-~ = ¢(1-0/T)

188

Chepter 3

0/5 --

3 O~

22C,M! (1.591)5 - 4.03 2(0.0366tool 1/2 °.5 g-1/2)(1.5 x 106 g mol-’) = 0.068 This quantity maybe resolved into its entropy and energy components,if the temperaturecoefficient of the intrinsic viscosity is known[9]. Problem 3.14 Showthat the perturbed dimensions of highly expanded polymer coils are proportional to n3/5, where n is the numberof backbonebonds in the polymerchain. Answer: and so from Eq. (3.132): a~ ,~, KM~/2, For large expansion, where K is a constant for the polymer-solvent system. Therefore, o~ 0¢. M~/~° and hence, a o~ r~V10. From Eq. (2.2): Unperturbed dimension, ~/2 From Eq. (3.123): Perturbed dimension, (r~) 3/~n. (r2)1/2 o<

PHASE

EQUILIBRIA

IN

POOR SOLVENTS

The characteristic features of phase equilibria in polymer-solvent mixtures will be considered in the present section, the discussion being confined to systems having both phases completely liquid. If the solvent chosen for a given polymer becomes progressively poorer as the temperature is lowered, eventually a temperature may be reached below which the polymer and solvent are no longer miscible in all proportions. At each lower temperature, polymer-solvent mixtures over a certain composition range will separate into two phases in equilibrium. (Separation of a polymer solution into two phases may also be brought about by the addition of a poorer solvent to the mixture.) The conditions for equilibrium between two phases in a binary system can be expressed by stipulating equality of the chemical potentials in the two phases; that is, ’ = ~1" or A~I’ = A#I " (3.133) /~1 and

’ " #2 = #2

or

A#2 2 ’

= A# "

(3.134)

Polymersin Solution

189

where the single and double primes are used to indicate, respectively, the dilute and concentrated phases in thermodynamicequilibrium and subscripts 1 and 2 indicate solvent and polymer solute, respectively. Proceeding from these conditions we may examine the requirements for the occurrence of incomplete miscibility and deduce the concentrations of the phases in equilibrium. Equations (3.53) and (3.54), derived previously, express the chemical potentials/~1 and/z2 as functions of the volumefraction q~2 of the polymer (note that q~l = 1 -- ~b2) and a single parameter X (note that X and ~ are equal) occurs in these fractions. Fulfillment of the conditions represented by Eqs. (3.133) and (3.134) requires that there be two concentrations at which the chemical potential /z 1 or A/z~ has the same value, and so also the chemical potential /z~ or A/z2. Since /~ and /z2 are derived by differentiating the same free energy function, Eq. (3.50), it suffices consider either chemical potential alone.

Problem3.15 A polymersolution was cooled very slowly until phase separation took place to give two phases in equilibrium. Analysis of the phases showedthat the volume fractions of polymer in the two phases were ~b~ = 0.01 and q~ = 0.89, respectively. Using the Flory-Hugginsequation for A#I [cf. Eq. (3.53)], together with the equilibrium condition A#~= A#~’, calculate an estimate of the polymer-solventinteraction parameterfor the conditions of phase separation. Answer: In order to use the Flory-Hugginsequation for A#I, the number-averagevalue of g (the numberof segments per polymer molecule) should be known.However, for most polymers, ~r is su~ciently large for (1 - 1/~r) "~ 1.0. Makingthis assumption and applying the condition A#~= A#’I~ leads to 2 In(1 - 0.01) + (1.0 x 0.01) + x(O.01) :~ = ln(1- 0.89) + (1.0 × 0.89) + X(0.89) yielding,

X = 1.66

Figure 3.8 shows plots of the relative chemical potential of the solvent, --(/~1 -- lz~)/-~T, versus the volume fraction of the polymer solute, ~b;, calculated according to Eq. (3.53) for a polymer of size o- = 1000 and for several values of X- It is seen that for small values of X, the relative chemical potential decreases [i.e., -(pl - #~)/RT increases] monotonically with ~b2 throughout the concentration range, indicating total miscibility. However, as the value of X is increased, either by decreasing the temperature or by altering the nature and compositionof the solvent, a critical value Xc (in this

190

Chapter3

case 0.532) is reached at which the previously monotonic curve begins to showa point of inflection. Its appearance signifies the onset of incomplete miscibility of polymer and solvent and is exhibited by an incipient phase separation (opalescence). Since both the first and second derivatives of a function are zero at the inflection point, the conditions for incipient phase separation can be stipulated as (OJZl/O~)2)T, 2 2 (0 #l/O¢2)T,e

P = 0 = 0

(3.135) (3.136)

It is readily seen from Fig. 3.8 that for each value of X above Xc there are two different values of ¢~ at which the chemical potential of the solvent in the two phases is the same. This implies that solutions with concentrations defined by these two values of ¢2 can be in thermodynamicequilibrium for X > Xc. Moreover, it implies that a solution with an intermediate value of ¢2 will spontaneously separate into two stable liquid phases defined by these two concentrations; this will occur with a concomitant decrease in the free energy. Such phenomenaare in fact observed with solutions of flexible polymers for values of X above a critical value (Xc). Thus, if X is increased by decreasing the temperature, an initially totally miscible system at higher temperatures is transformed to one of limited miscibility by lowering the temperature; at some critical temperature Tc, incipient phase separation (as indicated by the onset of an opalescence) is encountered, followed by separation into distinct liquid phases at still lower temperatures. Application of the critical conditions to the chemical potential as given by Eq. (3.53) yields 1/(1-

¢2)

--

(1-

1/~)

--

2X¢2

and

1/(1 -- ¢2) 2 -- 2X ---- 0 Eliminating X, there is obtained for the critical composition

(3.137) (3.138)

(3.139) ¢2c = 1/(1 + v/-~) Substituting Eq. (3.139) in either of the equations (3.137) and (3.138), we obtain (3.140) Xc = (1 + x/’-J)2/2~r For large o-, Eqs. (3.139) and (3.140) reduce ¢2c and

Xc ----

= 1/V~

(3.141)

1/2 + 1/~/~

(3.142)

The critical concentration at which phase separation first appears on passage to the the two-phase region is thus predicted by Eq. (3.14.1) occur at a very small fraction of polymer; for a polymer having ~r = 1000,

191

Polymersin Solution

0

0.02

O.OZ~

Figure 3.8 Plot of Eq. (3.53) for various values of the interaction parameter, X, as indicated in the diagram. A value of 1000 has been assumedfor a, the numberof segments in the polymer. for example, q52c ~ 0.032. The critical value of X, according to Eq. (3.142), will exceed 1/2 by a small increment depending on the chain length. Thus, for a polymer with ff = 1000, Xc is 0.532 and at infinite molecular weight it must equal 0.50. The first definition of X given by Eq. (3.40) indicates that it is inversely proportional to the temperature. According to the more general interpretation, in which (X -- ½) is replaced by %b(O/T - 1) [cf. Eqs. (3.98) and (3.107)1, X should be a linear function lIT . If (½ - X is replaced by %b(1 -- O/T), we see that there is a critical temperature, To, corresponding to the critical value of X- It is readily shownthat ¢(O/Tc1/Tc = (1/0)[1

1) = 1/x/~ + 1/(¢)(1/v/-~

-b 1/2cr + 1/2~r)]

(3.143) (3.144)

192

Chapter 3

which for large ff can be written as

= (1/0)[1

(3.145)

The critical temperature, To of a given polymer in a given solvent may be determined by observing the precipitation temperature as a function of the volume fraction of the solute. The precipitation temperature, (Tp), is that temperature at which detectable turbidity can first be observed visually upon slow cooling (1-2°C per 10 rain). The critical temperature, To, is determined as the maximumpoint of the plot (phase diagram) the precipitation temperature versus the volumefraction of the solute (Fig. 3.9). The curve shown in Fig. 3.9 is also called the cloud point curve or the precipitation curve. If Tc is thus determined for a series of fractionated polymers with different degrees of polymerization (~r), a plot of 1~Toversus (1/v/-~ ÷ 1/2~r) or 1/V/-~ (for large o’)should be linear. The slope intercept (at infinite ~r) of such plots should yield values of %band 0 which should depend only on the nature of the polymer and the solvent and not on the degree of polymerization. For polymer-solvent systems exhibiting limited miscibility, values of 0 obtained in this manner agree within experimental error (<1°C) with those derived from osmotic measurements, taking 0 be temperature at which the second virial coefficient is zero.

Vo[.

fraction

of polymer,

~2

Figure 3.9 Determinationof the critical temperature, To, from a cloud-point or precipitation curve.

193

Polymersin Solution

Inasmuch as cr = Mv2/V’~, where v2 is the specific volume of the polymer of molecular weight Mand V~ is the molar volume of the solvent, Eq. (3.145) can be written 1/Tc = (1/0)(1

b/ Vr--~)

(3.146)

where the constant b is given by b = (~(-~/v2)

(3.147)

Theory thus predicts that the reciprocal of the critical temperature (in °K ) for the onset of opalescence should vary linearly with the reciprocal of the square root of the molecular weight in a given polymer-solvent system. Furthermore, 0 may now be identified as the critical miscibility temperaturein the limit of infinite molecularweight. The relations developed above have been subjected to numerous experimental tests using a variety of flexible polymersin a variety of solvents. In general terms, the results have been in qualitative but not quantitative agreement with the theory. This is not unexpected considering the extreme simplicity of the model on which the theory is based. The discrepancy serves as a reminder that the simple thermodynamictheory, though generally reliable in predicting the form of observed results, often is inaccurate numerically.

Problem3.16 A series of solutions of polystyrene samples of varying molecular weight and dissolved in cyclohexane were slowly cooled to determine the precipitation temperatureat whichthe first detectable turbidity appeared. Fromthe maximum point of the precipitation temperature versus volumefraction of polymer plot for each polystyrene samplethe critical temperatureTc was determined. The values of Tc together with the polymermolecular weight are given below. To, (°c) 31.3 27.8 23.9 19.5

M 920,000 182,000 53,900 31,500

Assumingthat the volume of one monomerunit is the same as that of a cyclohexanemolecule, estimate the theta temperature0 and the entropic dilution parameter ¢. Answer: Onthe assumption that the volumeof met (repeating unit) is the same as that of a cyclohexanemolecule, a is equal to the degree of polymerization,

194

Chapter 3

where Mois the molar mass of the mer, i.e., 104 g/tool. (Alternatively, a can be taken as the ratio of the molar volume of the polymer to that of the solvent, the molar volume in each case being obtained from molar mass and density.) From the plot of 1/T~ versus (1/V’-~ + 1/23) in Fig. 3.10: 1 - = Intercept = 3.254 x 10-z °K-1 0 0 = 307.3 °K -- 34.3°C 1/0~b = ~b =

Slope = 2.876 x 10 (3.254 x 10-3 °K-1)/(2.S76

× -3 °K-l)

---- 1.131 [This value is considerably higher than the value obtained by osmotic measurements (cf. Exercise 3.10) or viscosity measurements (see Table 3.1)]

3.5 3./-.

_ 1/0 3.1 --

/

I~

0

I

I

I

,

1

Figure 3.10 A plot of the reciprocal

I.,

I

2

3

1.

h

I

I

1

5

I

6

of the critical temperature against the molecular size function occurring in Eq. (3.144) from data in Problem 3.16.

195

Polymers in Solution

A useful relationship [9] between the two phases that separate out on cooling a binary system consisting of a polymer fraction in a single solvent component may be derived from Flory-Huggins equations (3.53) and (3.54) by applying the equilibrium conditions [Eqs. (3.133) and (3.134)]. Denoting the polymer volume fraction in the more dilute phase by ¢~ and that in the more concentrated phase by ¢~, the following approximate relationship ! between 45 and the ratio 3’ = ¢2/42 # may be derived by substituting Eq. (3.53) on either side of Eq..(3.133) and Eq. (3.54) on either side Eq. (3.134): 4~ ~ {--(7-t-1)9

+ [(7+1)292

+4(7-1)ag]

1/2}

/ 2(3’-1)

3 (3.148)

where g = (12/o)[(’T The corresponding X = (7Substitution

+ 1)(In’T)/2

- (’T

- 1)]

value of X is given by 1)(1

-

1/o’) 1)

of ~b and 0 values in 1

2- 1)

(3.150)

~bO/T

(3.151)

+ (ln7)/¢~o-

gives numerical correlation of the theoretical parameter X and the absolute temperature T. This process represents, in effect, a calibration of the X scale in terms of temperature based on the change in the observed critical temperature Tc with molecular weight.

Problem3.17 Use the ¢ and 0 values obtained in Problem 3.16 to determine from theory the precipitation curves and the critical temperatures (To) for the four polystyrene fractions of Problem 3.16 in cyclohexane solution. Answer: On the assumption that the volume of polystyrene repeat unit is the same as that of a cyclohexane molecule, ~r is taken to be equal to the degree of polymerization. By substituting several chosen values of the ratio 7 in Eq. (3.148), together with the corresponding values of 9 calculated from Eq. (3.149) for given value of ~r, one may calculate ¢; and then ¢~ from ¢’~ = "/¢;. The corresponding values of X may be calculated from Eq. (3.150) and the conversion of X to the precipitation temperature of the ordinate scale can be accomplished by the use of Eq. (3.151). The precipitation temperature plotted against the composition for a given a gives the corresponding precipitation curve and the maximumin the curve represents the critical consolute temperature (To), below

I96

Chapter 3

which two stable liquid phases may coexist. The results obtained by this procedure are shown in Fig. 3.11. Note that at temperatures not much below the critical temperature To, even the more concentrated phase contains much more solvent than solute (polymer). These characteristics are more marked the higher the molecular weight.

1. TC= 31. Z,oC; 2. Tc =28.0°C ; 3. Tc=23.3°C ; 4. Tc=19.1°C ;

M = 920,000 M=182,000 M= 63,900 M = 31,500

29 .~ 25

5

0

0.05

0.10

Votume fraction Figure 3.11 Theoretical

I

0.15

I

0.20

0.25

of potymer, q~2

precipitation curves for four polystyrene (molecular weights indicated) in cyclohexane (Problem 3.17).

fractions

197

Polymers ~ Solution

Upperand LowerCritical Solution Temperatures Cloud-point curves or precipitation curves for different polymer-solvent systems have different shapes (Figs. 3.12 and 3.13). The maximaand minima on these curves indicate the upper critical solution temperature (UCST)and the lower critical solution temperature (LCST),respectively. As indicated Figs. 3.12 and 3.13, the phase diagram of a polymer solution has two regions of limited miscibility: (i) below UCSTassociated with the theta temperature (see Problem 3.16) and (ii) above LCST. The thermodynamic affinity of a solvent to a polymer changes with temperature, and this predetermines the type of critical solution temperature. If the positive values of the second virial coefficient A2 and the negative values of AGmixdecrease as temperature drops, it may be predicted that the system will separate into two phases on cooling, i.e., UCST is observed (e.g., solution of polystyrene in cyclohexane). If the positive values of A2and the negative values of AGmixdecrease as temperature rises, the system will separate on heating, i.e., it possesses LCST(e.g., solution of polyisobutylene in pentane). Different polymer-solvent systems may have completely different phase diagrams. For some systems, such as polystyrene-cyclohexanone, UCST < LCST[Fig. 3.13(a)] but for others, e.g., highly polar systems like polyoxyethylene-water, UCST> LCSTand closed solubility loop is found [Fig. 3.13(b)].

One liquid

phase Two [iquid

/

UCST

LEST One tiquid Vot. fraction of potymer,~ 2 (a)

phase

Vot. fraction of potymer, qb 2

(b)

Figure 3.12 Schematic phase diagrams for polymer-solvent systems showing (a) UCSTand (b) LCST.

198

Chapter3

Two tiquid

/

LCST

~Cloud -point U_.CST _~ \ closed curve

-~~p

One tiquid

phase

~ One LCST Vo[.fraction

of polymer,~2 (a)

Vot.fraction

tiquid phase

of po[ymer~ ~2

(b)

Figure 3.13 Phase diagrams for polymer-solvent systems where (a) UCST< LCST and (b) UCST> LCST.

SOLUBILITY BEHAVIOR OF POLYMERS Knowledgeof the compatibility (i.e., miscibility) of specific polymer-solvent systems or of the solubility of specific polymersin different solvents assumes importance in manypractical situations. Thus, it is important in assessing the chemical resistance of polymers and their application potentialities in the fields of paints, spinning fibers and casting films. These applications involve the selection of solvents for dissolving a particular polymer. A knowledgeof polymer-solvent compatibility is necessary for the specification of an elastomer that will not swell (i.e., absorb liquid) in applications knownto involve contact with certain liquids, and the prediction of possible environmental crazing of a polymer during service. Important also is the knowledge of the solubility of other materials, such as plasticizers and extenders in the polymer, especially since this has an important bearing on plastics formulation. The underlyng reason that one material can act as a solvent for another is the compatibility of the materials-i.e., the ability of the molecules of the two materials to coexist without tending to separate. If we denote the force of attraction between the molecules of one material A by FAA, that between the molecules of another material B by EBB, and represent that between one A and one B molecule as FAB, then the. system will be compatible and a solution will result if FAB > FBB and FAB > FAA. On. the

190

Polymersin Solution

other hand, if FAAor -FBB > I~AB, the system will be incompatible and the molecules will separate, forming two phases. In the absence of any specific interaction (e.g., hydrogen bonding) between solvent and solute, we can reasonably assume the intermolecular attraction forces between the di.ssimilar molecules to be approximately given by the geometric mean of the attraction forces of the corresponding pairs of similar molecules; that is, _NAB = (FAA.FBB)1/2. Consequently, if FAA and FBB are equal, EAB will also be similar and the materials should be soluble. A measure of the intermolecular attraction forces in a material is provided by the cohesive energy, which approximately equals the heat of vaporization (for liquids) or sublimation (for solids) per mol. The cohesive energy of a volatile liquid can be estimated from the work required to vaporize unit amount of the material. In this process, the molecules are transported from their equilibrium distances in the liquid to an infinite separation in the vapor. The cohesive energy density (sum of the intermolecular energies per unit volume)is at its equilibrium value in the liquid state and is zero in the vapor. By this reasoning, the cohesive energy density in the liquid state is AE~/V, in which AEvis the molar energy of vaporization and V is the molar volume of the liquid. The square root of this cohesive energy density is knownas the solubility parameter (6), that is, 6 = (/~Ev/V)1/2

(3.152)

If the vapor behaves approximately like an ideal gas, Eq. (3.152) can be written as

6=

- RT)IV

]1/2

:

[(mg v __

RT )p/M ]1/2 (3.153)

where AHv is the molar enthalpy of vaporization and /9 is the density of liquid with molecular weight M. For a volatile liquid cohesive energy density and, hence 6, can be determined experimentally by measuring AHv and p.

Problem 3.18 Calculate an estimate of the solubility parameter for water at 25°Cfrom its heat of vaporization at the same temperature, given by H20 (~ = H20 (g),

AH2soc= 10.514 kcal

Answer: FromEq.(3.153): × 103 cal mo1-1) -- (1.987 cal tool -1 °K-1)(298 °K)] (1.00 g cm-Z)/(lS.O g mo1-1) -a -~ =, 551.2 calcm _= 2.3×109 Jm

62= [(10.514

200

Chapter3

6 = 23.5 (cal cm-3) -= 48.0 × 103 (J m-3)1/2 1/2 or Pa 1/2 = 48 MPa [Conversion factors: 1 cal cm-3 = 4.184×106 J m-3 = 4.184×106 Pascal (Pa) = 4.184 MPa. Hence, 1 (cal cm-3)1/2 a/ = 2] 2.045 MPa

The solubility parameter approach was first developed by Hildebrand [11] for calculating estimates of the enthalpy of mixing, AHmix,for mixtures of liquids. The equation employed (for derivation see Appendix3.1) AHmix : Vmix¢l¢2(51 -- 52) 2 (3.154) where Vmix is the molar volume of the mixture, and 51 and 52 are the solubility parameters of components1 and 2, respectively. Thermodynamictheories assume that a necessary requirement for solution and compatibility is a negative or zero Gibbs free energy change (AGmix) when the solution or blend components are mixed. That is say, AGmix = AHm~x -TASm~ <_ 0 (3.155) Since the ideal entropy of mixing, according to Eq. (3.29), is always positive (the In of a fraction being negative), the componentsof a mixture .are assumed to be miscible only if AHmix_< T/kSmix. Thus, in this analysis, solubility depends on the existence of a zero or small value of AHmix. Note that this theory allows only positive (endothermic) heats of mixing, as in Eq. (3.154). Therefore, miscibility or solubility will predicted if the absolute value of the (51 -- 52) difference is zero small [13]. Specific effects such as hydrogen bonding and charge transfer interactions can lead to negative AHmixbut are not taken into account by Eq. (3.154) and a separate judgment must be made to predict their effect on miscibility and solubility [14]. Solubility parameters of solvents can be correlated with the structure, molecular weight, and density 6f the solvent molecule. According to the additive method of Small [15], the solubility parameter is calculated from a set of additive constants, F, called molar attraction constants, by the relationship

5 - Mp ZF

(3.156)

where ~]/7’ is the molar attraction constants summedover the groups present in the compound; p and Mare the density and the molar mass of the compound. The same procedure is applied to polymers and Eq. (3.156) is used, wherein p is now the density of the amorphous polymer at the solution temperature, ~/wi the sum of all the molar attraction

Polymers in Solution

201

constants for the repeat unit, and M is the molar mass of the repeat unit. Values of molar attraction constants for the most common groups in organic molecules were estimated by Small [15] from the vapor pressure and heat of vaporization data for a number of simple molecules. A modified version of a compilation of molar attraction constants [16] is reproduced in Table 3.2. An example of the use of the tabulated molar attraction constants is given in the problem worked out below.

Problem3.19 Calculate an estimate of the solubility

parameter for the epoxy resin DGEBA (diglycidyl ether of bisphenol A) having the repeat unit structure 3. as shown below and density 1.15 g/cm CH 3

CH 3

OH

Answer: M (for repeating

Group -CH3 -CHz>CH- C-O- (ether) -OH -CH~ (aromatic) -C= (aromatic) 6-membered ring Para substitution

unit) = 284 g mo1-1 3)1/: (cal cm tool 148.3 131.5 85.99 32.03 114.98 225.84 117.12 98.12 -23.44 40.33

No. of groups 2 2 1 1 2 1 8 4 2 2

(cal cm3) 1/2 mol 296.60 263.00 85.99 32.03 229.96 225.84 936.96 392.48 -46.88 80.66 2496.64

6 = (1.15 g cm-~)(2496.64 1/2 cma/2 mo1 -1) (284 g mo1-1) = 10.1 (cal cm-3) 1/~ U2 = 20.7 MPa

2O2

Chapter 3

Table 3.2 Group Molar Attraction Group -CH3 -CH2>CH-CCH.~=(olefin) -CH= (olefin) >C= (olefin) -CH= (aromatic) -C= (aromatic) -O- (ether, acetal) -O- (epoxide) -CO0>C=O -CHO (C0)20 Structure feature Conjugation Cis Trans 4-membered ring 5-membered ring Source: Ref. 16.

Constants

Molar attraction, F (cal cm3)l/2/mol 148.3 131.5 85.99 32.03 126.54 121.53 84.51 117.12 98.12 114.98 176.20 326.58 262.96 292.64 567.29 23.26 -7.13 - 13.50 77.76 20.99

Group -OH--4 -OH aromatic -Nt-I: -NH-N-C-~N -N=C=O -SCI~ -CI (primary) -CI (secondary) -CI (aromatic) -Br -Br (aromatic) -F Structure feature 6-membered ring Ortho substitution Meta substitution Para substitution

Molar attraction, F (cal cma)l/2/mol 225.84 170.99 226.56 180.03 61.08 354.56 358.66 209.42 342.67 205.06 208.27 161.0 257.88 205.60 41.33 -23.44 9.69 6.6 40.33

Tables 3.3 and 3.4 list solubility parameters for some commonsolvents and polymers [17,18]. The units of ~ are in (energy/volume) 1/2 and those V2 tabulated in cal 1/~ cm-a/2 are called hildebrands. The SI value in MPa may be obtained by multiplying the fi value in hildebrand by 2.045. Most tabulated solubility parameters refer to 25°C. It is believed that the temperature dependence of d~ can be neglected over the range normally encountered in industrial practice. While the solubility parameter of a homopolymer can be calculated from the molar attraction constants as illustrated in Problem 3.19, the solubility parameter of random copolymers, ~c, may be calculated from

tic = ~ (5~ w~

(3.157)

where ~i is the solubility parameter of the homopolymer that corresponds to monomer i in the copolymer and wl is the weight fraction of repeating unit i in the copolymer [17] .

203

Polymers in Solution

Table 3.3 a’b Solubility

Parameters

for Some CommonSolvents

(ii) Moderately hydrogen-bonded (i) Poorly hydrogen bonded (generally esters, ethers, ke(generally hydrocarbons and derivatives containing halotones) gen, nitrate and cyano groups) .,, Solvent 6 Solvent (cal/cm3)l/2 (cal/cm3)l/2 7.4 n-Hexane 7.2 Diethyl ether 7.8 Carbon tetrachloride 8.6 Isoamyl acetate Toluene 7.9 8.9 Dioctyl phthalate Benzene 9.2 Methyl isobutyl ketone 8.4 Chloroform 9.3 Tetrahydrofuran 9.1 9.3 Methylene chloride 9.7 Methyl ethyl ketone 9.9 Carbon disulfide 10.0 Acetone Nitrobenzene 10.0 1,4-Dioxane 10.0 Nitroethane 11.1 Diethylene glycol Acetonitrile 10.2 11.9 Monomethyl ether Nitromethane 12.7 Dimethyl sulfoxide 12.0 (iii) Strongly hydrogen-bonded (generally alcohols, amides, amines, acids) Piperidine 8.7 Diethylene glycol 12.1 Tetraethylene glycol 9.9 Propylene glycol 12.6 Acetic acid 10.1 Methanol 14.5 Meta-cresol 14.6 10.2 Ethylene glycol t-Butanol 10.6 Glycerol 16.5 1-Butanol 11.4 Water 23.4 aData from Refs. 17 and 18. 1/2 is obtained by multiplying the value in (cal/cma)1/2 by 2.045. bsl value of 6 in MPa

Problem3.20 Calculate the solubility

parameter for a methyl methacrylatebutadiene copolymer containing 25 mol % methyl methacrylate. The solubility parameter values for poly(methyl methacrylate) (PMMA)and polybutadiene homopolymers,calculated from molar attraction constants, are, respectively, 9,3 and 8.4 (cal 1/2. cm-3) Answer: M (butadiene) Weight fraction

= 54 g mol-1; M (methyl

methacrylate)

of methyl methacrylate in copolymer -~) 0.25 (100 g mol -I) 0.25 (100 g mo1-1) + 0.’/’5 (54 g tool ~- 0.38

= 100 g -1

204

Chapter 3

From Eq. (3.157) = = =

0.38(9.3 cal 1/2 cm-3/2) + (1 - 0.38)(8.4 1/2 cm- 3/’) 1/2 8.7 (cal cm-3) V2 17.8 MPa

a’b Table 3.4 Solubility cPolymer

Parameters

for Some CommonPolymers 6 (cal/cmZ)l/~ 6.2 8.0 8.3 8.3 8.4 8.5 8.5 9.0 9.3 9.4 9.4 9.6 9.7 10.1 10.2 10.3

dH-bonding group

Polytetrafluoroethylene Poor Polyethylene Poor Polypropylene Poor Poor Polyisobutylene Poor Polybutadiene Poor Polyisoprene Poor Poly(butadiene-co-styrene) (75/25) Poor Polystyrene Medium Poly(methyl methacrylate) Poly(butadiene-co-acrylonitrile) (75/25) Poor Medium Poly(ethyl acrylate) Medium Poly(vinyl acetate) Medium Poly(vinyl chloride) Medium Poly(methyl acrylate) Medium Polyformaldehyde Strong Ethyl cellulose Poly(vinyl chloride-coMedium 10.6 vinyl acetate) (87/13) Strong Cellulose diacetate 11.4 Strong 12.7 Poly(vinyl alcohol) Poor 12.7 Polyacrylonitrile 13.7 Strong Nylon-6,6 aData from Refs. 17 and 18. 1/2 is obtained by multiplying the value in (cal/cm3)1/~ by 2.045. bsI value of 6 in MPa CCompositionsof copolymersare in parts by weight. dThe hydrogen-bondinggroup of each polymerhas been taken as equivalent to that of the parent monomer.(The hydrogen-bondingtendency can be assigned qualitatively alcohols > ethers > ketones > aldehydes > esters > in the order: hydrocarbons.)

Polymersin Solution

205

Alternating copolymers can be treated by taking the copolymerrepeating unit as that of a homopolymer.No satisfactory method exists, however, for assigning values to block or graft copolymers. The solubility parameter dimix of a mixture of two liquids of solubility parameters 51 and 52, respectively, can be approximated from 5mix : ¢151 -~-

¢2(~2

(3.158)

where ¢1 and ¢2 are volume fractions. It is not unusual to find that a mixture of two liquids, which themselves are nonsolvents of a given polymer, is a solvent for the polymer. This occurs if the 5 value of one nonsolvent is higher and that of the other nonsolvent is lower than the solubility parameter of the solute. Solubility can be expected if the absolute value of ((51 -- 52) is less than about unity and there are no strong polar or hydrogen-bonding interactions in either the polymer or the solvent. Whenspecific interaction comes into play in the dissolution process, depending on the solubility parameter alone for predicting solubility relations might be quite misleading, because solvents promoting different type and extent of interactions can act differently, even if they happen to have similar solubility parameters. To allow for the influence of hydrogen-bondinginteractions, it has been found useful to characterize solvents qualitatively as poorly, moderately, or strongly hydrogen bonded. The solvents listed in Table 3.3 are grouped according to this scheme. The practice of matching both solubility parameter and hydrogen-bonding tendency for predicting mutual solubility is useful if used with caution. For example, polystyrene, which is classed as poorly hydrogen bonded and has a 5-value of 9.0 (cal cm-3)1/2 is highly soluble in the poorly hydrogen-bonded solvents benzene (6 = 9.2) and chloroform (6 = 9.3), both of which have matching solubility parameters. The polymer can also be dissolved in methyl ethyl ketone (6 = 9.3, mediumhydrogenbonding), but the latter is not nearly as good a solvent as either benzene or chloroform. (This is revealed by the fact that the intrinsic viscosity of polystyrene of given molecular weight is higher in chloroform or benzene than in methyl ethyl ketone.) On the other hand, poly(methyl methacrylate), which has practically the same5 as polystyrene, is classed, unlike the latter, as medium hydrogen bonded. But poorly hydrogen-bonded benzene and chloroform do not seem to be weaker solvents for poly(methyl methacrylate) than moderately hydrogen-bonded methyl ethyl ketone. Problems such as these probably reflect the use of an oversimplified view of hydrogenbonding, in general. Numerous attempts have been made to improve the predictive ability of the solubility parameter method without complicating the procedure too greatly. The basis of these modifications is the recognition that intermolecular forces causing cohesion of molecules can involve, besides dispersion,

206

Chapter 3

interactions of dipole-dipole, dipole-induced dipole, or acid-base type, all of which cannot be taken into account by a single t5 value as used in the simple solubility parameter model. It may be noted that the single parameter model of Hildebrand is based on the assumption that dispersion forces provide the only significant cohesive energy. The most comprehensive approach to resin solubilities has been that of Hansen [19] in which the solubility parameter is divided into three components. The basis of this three-dimensional solubility parameter system is the assumption that the energy of evaporation, i.e., the total cohesive energy A/~t which holds a liquid together, can be divided into contribution from dispersion (London) forces AEd, polar forces AN:0, and hydrogenbonding forces AEh. Thus, AEt = AEd -tAEp + AEh (3.159) Dividing this equation by the molar volume of a solvent, V, gives

AEtv - AEdv +--~- +AEp

AEhv

(3.160)

or

(3.161) (St~ = ~ + (~p2 + where 5d = solubility parameter due to dispersion forces, tSp = solubility parameter due to dipole forces, and (Sh = solubility parameter due to hydrogen-bonding (or in general due to donor-acceptor interactions). method was developed by Hansen [20] for the determination of three parameters (Sd, (Sp and (Sh empirically on the basis of many experimental observations for a large numberof solvents (Table 3.5). Hansen’s total cohesion parameter, tSt, corresponds to the Hildebrand parameter (5, although the two quantities may not be identical because they are determined by different methods. Once the three component parameters for each solvent were evaluated, the set of parameters [201 for each polymer could be obtained (Table 3.6). Solubility was ascertained by visual inspection polymer-solvent mixtures at concentrations of 10%wA,.

Problem 3.21 Calculate the composition of a blend of n-hexane, 1-heptane, and dioctyl phthalate that wouldhave the samesolvent properties as tetrahydrofuran. (Take appropriate data from Table 3.5.) Answer: Since 6d values do not vary greatly, at least amongcommon solvents, only 6p and 6h values maybe matched.Let (}1, (}2 and (}3 be the volumefractions of n-hexane, 1-butanoland dioctyl phthalate, respectively, in the mixture. By definition of volumefractions, (}3 = 1 - (}1 -

207

Polymers in Solution From Eq. (3.158) and data in Table 3.5: 6p(tetrahydrofuran)

= 2.8 = ~bl(0) + ~b2(2.8) + (1- ~bl- ~b2)(3.4)

6h(tetrahydrofuran) = 3.9 = ~bl(0) + ~b~(7.7) + (1- ~bl- ~b~)(1.5) Simplifying, 0.6 = 3.4~b~ + 0.6q~z and 2.4 = -1.5q~ + 6.2~b9_ Simultaneous solution of these two equations yields: q~l = 0.10, ~b2 = 0.41. Hence, ~bz = 0.49 Check on 6d of the mixture: 0.10(7.3) + 0.41(7.8) + 0.49(8.1) From Table 3.5: 8d (tetrahydrofuran) = 8.2

Table 3.5 Hansen Parameters a’b for Solvents at 25°C Liquid

6d 2(cal/cmZ)V 6.8 6.3 7.9 9.0 7.8 5.4 8.0 8.0 8.1 6.5 4.9 6.5 4.5 7.3 7.8 7.4 7.8 8.6 8.6 8.2 8.0 5.9 8.1

6p 2(cal/cm3)V 6.0 4.8 4.2 4.0 2.8 6.7 1.5 4.9 3.4 4.2 7.4 2.6 7.5 0.0 4.4 6.0 5.6 6.8 4.9 2.8 3.9 11.1 3.5

5h ~ (cal/cma)V 9.2 5.4 2.0 0.0 7.7 3.1 0.0 3.9 1.5 4.3 14.6 2.7 15.3 0.0 2.5 10.9 4.9 0.0 3.7 3.9 0.8 19.7 1.2

Acetic acid Acetone Benzene Bromobenzene 1-Butanol Chloroform Cyclohexane 1,4-Dioxane Dioctyl phthalate Ethyl acetate Ethylene glycol Ethyl ether Glycerol n-Hexane Methyl ethyl ketone Methanol Morpholine Nitrobenzene Pyridine Tetrahydrofuran Toluene Water m-Xylene aData from Ref. 21. V2 is obtained by bsI value of a parameter in MPa multiplying the value in 1/~ (cal/cm3) by 2.045

208

Chapter3

Whenplotted in three dimensions, the Hansen parameters provide an approximately spherical volume of solubility for each polymer in did, dip, 6h space. The scale on the dispersion axis is usually doubled to improve the spherical nature of this volume. The distance of the coordinates (di~, di~, di~) of any solvent i from the center point t’diJ, d di~, di~) of the solubility sphere of polymer j is d : [4(6~ - diJd) 2 + (di~ -- dijp)9. + (dii h _ 5jh)2],/2 (3.162) This distance can be compared with the radius R of the solubility of the polymer (Table 3.6), and

sphere

d
Problem 3.22 Using Hansen parameters (Tables 3.5 and 3.6) determine polystyrene is expected to dissolve in a solvent mixtureof 60/40 v/v methylethyl ketone/n-hexane. Answer: Denote the solubility parameter componentsof MEK,n-hexane and polystyrene using superscripts i, j and k, respectively. FromTable 3.5: 1/2 MEK:6~ = 7.8, 6~ = 4.4, 6~ = 2.5 all in (cal cm-3) 1/~ n-hexane: 5~ = 7.3, 6~ = 0, ~ = 0 all in (cal cm-3) The Hansenparameters are combinedon a 60/40 volumefraction basis: j5~ = 0.6x7.8 + 0.4x7.3 = 7.6 ~/2 (cal cm-3) 6~J = 0.6x4.4 + 0.4x0.0 = 2.6 V2 (calcm-3) (cat cm-3) 6~J = 0.6x2.5 + 0.4x0.0 = 1.5 V~ FromTable 3.6: Polystyrene: 5d~ = 10.4, 6~ = 2.8, ~h~ ~/~ = 2.1, R = 6.2 all in (cal cm-3) From Eq. (3.162): d = [4(10.4 - 7.6) 2 + (2.8 - 2.6) 2 1/ + 2(2.1 - 1.5)2] ~/2 = 5.6 (cal cm-3) As this value is less than the radius of the polymersolubility sphere (6.2 calV~ cm-3/2), the polymeris expected to be soluble. This is found to be the case in 10%way solution.

Polymersin Solufion

209

Table 3.6 Hansen Parameters and Interaction b resinsa, Polymer

Radius of some Polymers and

R 6~t 6p 6h (cal/cma)l/2 (cal/cm3)l/2 (cal/cm3)~/~ (cal/cm3)l/2

Acrylonitrile-butadiene elastomer 9.1 4.3 2.0 4.7 Cellulose acetate 9.1 6.2 5.4 3.7 Epoxyresin 5.6 6.2 10.0 5.9 Furfuryl alcohol resin 10.4 6.6 6.2 6.7 Nitrocellulose 7.5 7.2 4.3 5.6 9.7 Phenolic resin 11.4 3.2 4.0 4.7 Polyamide,.thermoplastic 8.5 -0.9 7.3 Polyisoprene 8.1 0.7 -0.4 4.7 Poly(methyl methacrylate) 9.1 5.1 3.7 4.2 Polystyrene 10.4 2.8 2.1 6.2 Poly(vinylacetate) 10.2 5.5 4.7 6.7 Poly(vinylchloride) 8.9 3.7 4.0 1.7 Styrene-butadiene elastomer 8.6 1.7 1.3 3.2 Urea-formaldehyderesin 10.2 4.0 6.2 6.2 aDatafrom Ref. 22. V2by 2.045 bsI value in MPa1/2 is obtainedby multiplyingthe value in (cal/cm3) Three-dimensional presentations of solubility parameters are cumbersome and it is more convenient to transform the Hansen parameters into fractional parameters as defined by [23]: fd = 5d / (Sd + 6p + 6h)

(3.163)

fp = 5p / (Sd + v +5h) fh = 5h / (6d q- 6p q- 6h)

(3.164) (3.165)

The fractional parameters represent, in effect, the quantitative contribution of the three types of forces to the dissolving abilities for each solvent and can be represented more conveniently in a triangular diagram to provide a visual presentation of the nature of the solvating powers of liquids, including in one picture such diverse liquids as water, alcohols, organic acids, and hydrocarbons [23]. The triangular solubility chart can be used conveniently for the prediction of solubility of polymers. A chart can be constructed for a given resin, identifying each solvent coordinate point to indicate either complete, partial or lack of solubility. In general, solvents which provide clear solutions are grouped in a reasonably well defined area of the chart thus forming a loop and solvents which tend to swell the resin

210

Chapter3

lie near the border line of such loops. Since ~Sd values do not vary greatly, at least amongcommonsolvents, it has been suggested that a plot of ~p versus ~h should be sufficient for most practical purposes. Thus, another and probably more efficient two-parameter representation is made possible by defining [24] 5v = (5~ + 6~2) 1/z

(3.166)

The procedures outlined above have a practical use, but it should be realized that the parametric models are almost entirely empirical. Experimental uncertainties are also involved since solubility measurementsare not very accurate. Solubility loops described by the models only indicate the limits of compatibility and always include doubtful observations. FRICTIONAL MOLECULES

PROPERTIES OF POLYMER IN DILUTE SOLUTION

Whena polymer molecule movesin a dilute solution it undergoes frictional interactions with solvent molecules. The nature and effect of these frictional interactions depend upon the size and shape of the polymer molecule. Thus, the chain dimensions of polymer molecules can be evaluated from measurements of their frictional properties [25]. In a laminar flow at a definite shear rate, different parts of the polymer molecule moveat different rates depending on whether they are in the zone of rapid or relatively slow flow, and as a result the polymermolecule is under the action of a couple of forces which makes it rotate in the flow. Rotation and translational movementof polymer molecules causes friction between their chain segments and the solvent molecules. This is manifested in an increase in viscosity of the solution comparedto the viscosity of the pure solvent. In a dilute solution the long flexible polymer molecule curls up into a coil. Twoextremes of the frictional behavior of polymercoils in solution can be identified, namely free-draining and nondraining. A polymer molecule is said to be free-draining whensolvent molecules can pass through the coilform macromolecule, flowing past each segment of the chain with equal ease, while in the other extreme, a polymer molecule is said to be nondraining, whenthe molecule in coil-form retains a definite amountof solvent enclosed by it and moves together with this solvent. These two extremes of behavior lead to different dependencesof the frictional coefficient, fo, of a polymer molecule on chain length. A free-draining polymer molecule, referred to as the free-draining coil, is considered by dividing it into identical segments each of which has the same frictional coefficient ft. Since solvent molecules permeate all regions of the polymer coil with equal ease (or difficulty), each segment makes the same contribution to fo which therefore is given by

Polymersin Soluf~on

2]~

(3.167) fo = n~ where n is the number of segments in the chain. A nondraining polymer molecule, also referred to as the impermeable coil, can be represented by an equivalent impermeable hydrodynamicsphere of radius /~h- The frictional coefficient of this sphere which represents the frictional coefficient of the non-draining polymer coil can thus be written, according to Stokes’ law, as fo = 67rrloRh (3.168) wherer/o is the viscosity of the pure solvent. With the reasonable assumption that the unperturbed root-mean-square radius of gyration IS21~o/2 multiplied by the expansion factor ce (which accounts for osmotic swelling of the polymer coil by solvent-polymer interactions) can be taken as a measure of /~h, Eq. (3.168) can be rewritten in the form fo

= goo~(s2}lo/2

(3.169)

where Ko is a constant for a given system. Since ($2)~o/2 is proportional to n1/~ and for highly expanded coils o~ is approximately proportional to n1/1° (see Problem 3.14), Eq, (3.159) predicts n’~ (3.170) fo °= K’o where /~o is another constant and 0.5 _< ao <_ 0.6. Comparison of Eqs. (3.167) and (3.170) shows that while for both the free-draining polymer nondraining polymer the frictional coefficient increases with polymer chain size, the effect however is less pronounced for the nondraining polymer molecule. The frictional behavior of real polymer molecules comprises contributions of both free-draining and non-draining polymer molecules represented by Eqs. (3.167) and (3.170), respectively. The free-draining contribution is dominant for very short chains and for elongated rodlike molecules, but for flexible (i.e., coiled) chain molecules it decreases rapidly as the chain length increases. Since most polymers consist of long flexible chain molecules, their frictional properties approximate to those of nondraining polymer molecules. In the following section, the nondraining behavior is therefore considerd further, specifically in relation to the viscosity of dilute polymer solutions. Viscosity

of Dilute

Polymer Solutions

Einstein derived the following equation for the viscosity of suspensions of rigid, uncharged, spherical particles which do not interact with the suspension medium:

212

Chapter 3

where r/ and r/o are the viscosities of the suspension and the suspension medium,respectively, and ¢2 is the volumefraction of the spherical particles. Equation (3.171) can be rewritten r/sp

-- (5/2)¢2

(3.172) where ~Tsp -- (~/ - ~7o)/r/o is ~ownas the specific viscosiq. It may be assumed that under the action of a shear stress the pol~er coil with the solvent enclosed by it behaves like an Einsteinian sphere, and hence, the viscosiT of polymer solution should obey Eq..(3.172) for noninteracting spherical particles. Noninteraction of the polymer coils requires infinite dilution and this is achieved mathematically by defining a quanti~ called the in~nsic v~cosiq, [~], according to equation [q]

= lira

(q,v

/ c)

(3.173)

If Vh is the hydrodynamic volume of each polymer molecule (it is assumed that all polymer molecules are of the same molecular weight) then ¢2 = (c/M)NA,~ (3.174) where c is the polymer concentration (mass per unit volume), ~f is the molar mass of the polymer (mass per mol), and NAvis Avogadro’s number. Substituting Eq.(3.174) into Eq. (3.172) gives

= A more satisfacto~ form of this equation to satis~ the condition of noninteraction of polymer coils is given in te~s of intrinsic ~scosi~, [~], defined by Eq. (3.173). Thus, IV]

= (5/2)(g~v/M)(V~)~,o

(3.176)

This equation can be rea~anged to give an equation for the hydrodynamic volume of an impermeable (nondraining) polymer molecule in infinitely dilute solution :

=

( .17z)

Thus, the quantity [~]M is proportional to (Yh)c~0.

Problem 3.23 The relative viscosity of a benzene solution of natural rubber of concentration 0.05 g/100 cm3 is measuredto be 1.18 at 25°C. The molecular weight of the rubber is 200,000. Calculate an estimate of the hydrodynamic volume of the rubber molecule. State the assumptionsmadefor the calculation. Answer: In order to use Eq. (3.175) it is necessary to assumethat the polymercoils are impermeableand noninteracting. Since the natural rubber is a high-molecularweight flexible polymer and the solution is very dilute, these conditions may

213

Polymersin Solution

be assumed to be closely approximated. Further it should be assumed, though incorrectly, that all moleculesare of the samemolecularweight(i.e., homodisperse). r~s p --

-1 = 1.18- 1 = 0.18 r/o c = 0.0005 g -3 cm From Eq. (3.175): 2r/~M vh 5CNAv 2(0.18)(2.0 x 105 g mo1-1) 5(0.0005 g cm-3)(6.02 × 1023 tool-’) = 4.8 x 10-~7 3cm

In dealinz with linear polymers, the root-mean-square end-to-end distance (r2)l/~’can be taken as a measure of the size. Therefore, assuming that (Vh)c-~O is proportional to @2)3/2, we obtain

[r/]= ~((r~)~--/-------~)

(3.178)

where ~ is a constant. If (r 2) is determined from dissymmetry measurements on the light scattered by dilute solutions of a polymer fraction, the results being extrapolated to infinite dilution, and if the intrinsic viscosity is determined in the same solvent and at the same temperature, then it is possible to calculate ~ from Eq. (3.178). It is important to use wellfractionated samples for these measurements, since Eq. (3.178) is actually derived for monodisperse samples. So when measurements are performed with a heterodisperse polymer sample, that is, with a sample which includes species covering a considerable range in molecular weight, then a number average of @2)3/2 over the molecular weight distribution should be used in conjunction with number average molecular weight Mnin Eq. (3.178). Analysis of experimental data for [r]] and @-~)1/2 by light scattering of solution led Flory to the conclusion that the parameter ~ should have a universal value for all linear polymers in all solvents. For nondraining polymer molecules, if2, knownas Flory constant, is independent of chain structure and chain length, and is dependent only on the spatial distribution of segments about the center of gravity in the molecular coil. For randomlycoiled linear polymers this distribution is approximately Gaussian, and even moderately branched polymers may present a similar distribution of segments. For nondraining polymer coils with Gaussian distribution of segments the best value of ~5 determined on the basis of theoretical calculations and experimental measurements is 2.5x1023 (cm3 g-1)(g mol-1 cm-), or simplymo1-1,whenIt/] is expressedin cm3/gand _ _(~’))1/2 in cm. The value is 2.5x1021 (dL g-])(g mo1-1-a) when [r /] is expressed in

214

Chapter 3

dL/g and (r2), 1/2 in cm. This is often quoted as 62 for all polymer-solvent systems. In reality, 62 decreases [26] from its theoretical value of 2.87×1023 valid for unperturbed gaussian coils (2.87x1021 if [r]] is expressed in dL/g) to about 2.1x1023 for polyr0ers in good solvents. According to Eq. (3.178), the contribution of a high polymer molecule to the viscosity should be that of an equivalent sphere having a volume proportional to (r2) 3/2. Since for a randomly coiled linear chain the mean square radius of gyration (S2) is related to meansquare end-to-end distance (r 2) by (S2) = (r2)/6 [cf. Eq. (2.4)], Eq. (3.178) can be rewritten [r/I= 62’((S-~/-)3/2)

(3.179)

where ~ is another Flory constant given by 62~ = 63/262 and hence has the value of 3.6×1024 mo1-1 when [r/] is expressed in cm3/g and (~q2)1/2 in cm. To get at the underlying factors influencing viscosity, it is desirable to separate (r 2) 1/2 into its componentfactors, viz., the unperturbed dimension (r -2) 01/2 ai~d ~ linear expansio~nfactor o%for hydrodynamicchain dimensions [cf. Eq. (3.123)]. Equation (3.178) may then be recast as follows:

[7] = 62 Since {r2}o is directly proportional to the number, n, of chain segments, it is also directly proportional to M. Therefore, (r2)o/M is a constant and Eq. (3.180) is more commonlywritten in the form of the Flory-Fox equation [27] : a M1/2 (3.181)

[7] = Ko

where

No= 62

(3.1S2)

If the preceding analysis of hydrodynamiceffects of the polymer molecule is valid, Ko should be a constant independent both of the polymer molecular weight and of the solvent. It may, however, vary somewhat with the temperature inasmuch as the unperturbed dimension (r2)01/2 may change with temperature, because this dimension is modified by hindrances to free rotation the effects of which will, in general, be temperature-dependent. The Flory-Fox equation predicts that the intrinsic viscosity is dependent on (i) the stiffness of the polymer chain (through the term (r2)o/M),(ii) the molecular weight of the polymer (through M~/~), and (iii) the solventpolymer-temperature combination (through o%). Under theta conditions, c%is unity and Eq. (3.180)predicts a proportionality of [r/] on the square root of the molecular weight (M°’S). This behavior has been confirmed experimentally for several polymer-solvent systems. The exponent 0.5 is a

Polymers in Solution

215

virtual lower limit, inasmuchappreciably poorer solvents will not dissolve the polymer. In very good solvents, wh.ere there is large expansion of polymer coil, oe,, is proportional to M1/10 (see Problem 3.14) and hence the intrinsic viscosity varies as M(a/~+3/1°),i.e., as M°’s. This is an upper limit. For other solvents and non-theta conditions, the behavior is intermediate between these two limits. The Flory-Fox equation (3.180) thus suggests a general relationship of the form [r/] = KMa (3.183) where K and a are empirical constants for a given polymer-solventtemperature combination, and a increases in the range 0.5 < a < 0.8 with the degree of expansion of the molecular coils from their unperturbed dimensions. Under theta conditions where the molecular coils have unperturbed dimensions, a = 0.5. This important relationship between [r/] and M is commonly known as the Mark-Houwink equation or Mark-HouwinkSakurada equation and was first proposed on the basis of experimental data. The Mark-Houwinkconstants K and a are evaluated from [r/] and Mdata of calibration samples with narrow molecular weight distributions. Generally, a plot of log[r/] against log Mis fitted to a straight line from which K and a are obtained. In an "ideal" solvent (i.e., a poor solvent at the theta temperature for which the second virial coefficient vanishes), also knownas theta soh, ent, o~ = 1 and Eq. (3.181) reduces (3.184) [r/]0 = 1/2 KoM It has been established that Ko normally is independent of the solvent and the molecular weight of the polymer, though often dependent to some extent on the temperature. It is therefore possible to deduce values for the expansion factor in good solvents from intrinsic viscosities measuredin them. From Eqs. (3.181) and (3.184) the linear expansion factor c~e, which is a measure of long range interactions and pertains to hydrodynamicchain dimensions, is thus given by 1/3

(3.185) where [r/] is the intrinsic viscosity in the given solvent and [U]0 is the intrinsic viscosity of the same polymer in a theta solvent. The factor by which the molecule is enlarged maythus be calculated from the ratio of the intrinsic viscosity of the polymerin a good solvent to its intrinsic viscosity in an "ideal" solvent at (about) the same temperature. It has been suggested that o~ thus obtained indirectly from viscosity measurements is related to the more direct measurement of o~ of Eqs. (3.123) and (3.124) c~ = c~°’81 (3.186) Considerable experimental evidence exists to support this conclusion. O~r~ : ([r/]

/ [7-]]0)

216

Chapter 3

Determination from Viscosity

of

Polymer

Molecular

Dimensions

The molecular dimensions of a polymer chain in any solvent can be calculated directly from light scattering measurements,if the polymercoil is large enough to scatter in an asymmetric manner (see Light Scattering Methodin Chapter 4). However, when the chain is too short to be measured accurately by light scattering method an alternative technique has to be used. If the universal constancyof ff is accepted, it is possible to calculate the average dimensions of polymer molecules in solution merely from knowledge of their intrinsic viscosities and molecular weights. Moreparticularly, it is possible to calculate the natural, or unperturbed, dimensions of the polymer chain from the knowledgeof intrinsic viscosity in a theta solvent [28,29].

Problem3.24 The intrinsic viscosity of polystyrene of molecularweight 3.2 x 105 in toluene at 30°Cwasdeterminedto be 0.846 dL/g. In a theta solvent (cyclohexane at 34°C) the samepolymerhad an intrinsic viscosity of 0.464 dL/g. Calculate (a) unperturbed end-to-end distance of the polymermolecule, (b) end-to-end distance of the polymer in toluene solution at 30°C, and (c) volumeexpansion factor toluene solution. (~ = 2.5x 1023 mo1-1) Answer: [r/]0 = 0.464 dL g-1 = 46.4 cm3 g-1 From Eq. (3.184): Ko = [r/]o / 1/2 M = (46.4 cma g-l) / (3.2 × 105 = 0.082 cm3 mol 1/2 g-3/2 FromEq. (3.182):

1/ g 2mol-1)

(r~)~o/2 = KoMZ/~I (0.082 cm3 tool ’/2 3/2 g-a/~)(3.2 × 105 g mol-1) (2.5 × 1023 mo1-1) = 5.94 × 10 -17 3cm (r2)1o/~ = 3.90 × 10-6 cm = 390 ~4 (b) FromEq. (3.178): -(84.6 cm3 -I) g-1)(3.2 × 105 g mol -1) (2.5 × 1023tool = 108.28 × 10-ls 3cm (r2) V~ = 4.76x10 -6 cm = 476

217

Polymersin Solution

(c) Neglectingthe effect on Ko of small temperaturedifference between[r/] and [r/]0 measurements,the hydrodynamicchain dimension expansion factor, c%, can be obtained from Eq. (3.185). Thus,

=

[(0.846 dL g-l)]1/3 = 1.22

dLg-1)j

From Eq. (3.186): Average linear expansion factor, a = (a,) 1/°sl = 1.28 Volumeexpansion factor , 53 = (1.28) 3 = 2.1 Problem3.25 For a fractionated sample of c/s-l,4-polybutadiene of molecular weight1.23 x 105intrinsic viscosities weremeasured[30] in three different solvents at respective theta temperatures. From the results given below determine the variation of the unperturbeddimensionsof the polymermoleculewith temperature. Solvent n-Heptane Isobutyl acetate n-Propyl acetate

0-Temperature~°C) - 1.0 20.5 35.5

[r/]0, dug 0.670 0.656 0.645

The Ko values for the temperatures -1, 20.5 and 35.5°C are calculated from Eq. (3.184) using [r/]0 values and M= 1.23×105g tool -1. These are plotted as lnKo vs. T°K. From the slope, dln Ko/dT -3. = 1.09 × 10 Differentiation of Eq. (3.182) with respect to T yields dba(r2)o 2dlnKo -- 0.73 .3 x 10 dT 3 dT Note: The parameter dln{v2)o/dT appears in the theoretical thermoelasticity of macromolecularsubstances.

treatments of

It is not always possible to find a suitable theta solvent for a polymer and methods have been developed which allow unperturbed dimensions to be estimated in nonideal (good) solvents. For small expansions of Gaussian polymer chains the expansion parameter, c%, for the hydrodynamic chain dimensions is given by a closed expression of the form a~3 = 1 + b.z (3.187) where z is the excluded volume parameter defined by Eq. (3.116) and b a constant. (The value of b is uncertain, e.g., values of 1.55 and 1.05 have been obtained from different theories.) Combination of this relation with the Flory-Fox equation (3.181) gives [r/]

= KoM1/2

1/ + 2bKozM

(3.188)

Chap~e~ 3

O.OOZ,

0.003 0

E ._1 "o 0.002 ~ 0.001 0

0

I

I

I

I I z, O0

I

I

I I I 800 M1/2 1]2) (91/2 moC

I

I I 1200

I

I

I 1600

Figure 3.14 Plot of [r/]M-1/~ against M1/2 for polystyrene solutions in benzene at 20°C (Problem 3.26).

which upon rearrangement, recognizing that according to Eq. (3.116) z proportional to M1/2, leads to [film -:/2

:/ = ~ Ko + B’M

(3.189) t where B depends upon the chain structure and polymer-solvent interaction, and is a constant for a gdven polymer-solvent pair at a given temperature. Thus [q] and Mdata of calibration samples with molecular weight distributions can be plotted as [r/]M -1/2 against M1/2 to give Ko as the intercept. The value of Ko can then be used to evaluate unperturbed chain dimensions and the expansion factor for the chain dimensions.

Problem 3.26 Viscosity measurementson polystyrene fractions of different molecular weights in benzeneat 20°C yielded the following values for intrinsic viscosities. Molecular weight M [r/], dL/g 44,500 0.268 65,500 0.356 1.07 262,000 694,000 2.07 5.54 2,550,000 Evaluate for polystyrene of molecular weight l0 s (a) the unperturbed end-to-end

Polymers in Solution

219

chain lengths and (b) the volume expansion factor in benzene at 20°C. Assume if9 = 2.5×10 z3 mo1-1. Answer: In Fig. 3.14, [~]M-V~ is plotted against M~/2 and a straight lin~ is fitted. The intercept at M1/~ --~ O, yields Ko = 0.00117 dL tool 1/2 g-3/~__ 30.117 cm moll/2 g-3/2. (a) From Eq. (3.182):

__

(r~)~o/z=[(0.117 cmZm°lVZ g-Ve)(106(2.5 × 10~z mot-~)g m°I-1)3/211/3 7.76 x 10-6 cm (776 _~)

=

(b) From Eqs. (3.184)-(3.186): ’/°’ 0:~ 8’ = (.[~]M- V~.’~ From Fig. (3.14), mo1-1. Therefore,

[~]_/~/./-1/2

=

0.00275

dL molI/2 g-312

for

M = 106 g

1/0.81

0:3 _~_ I(0.275 cm3 mo11/2 g-3/2)] [.(0.117 cm~ molV2 g-Ve)j

= 2.87

The interpretation of [r/] for branched polymers and copolymers is considerably more complicated. The effect of branching is to increase the segment density within the molecular coil leading to a smaller size for a molecule with the same molecular weight. Thus, a branched polymer molecule has a smaller hydrodynamic volume, and hence a lower intrinsic viscosity, than a similar linear polymer of the same molecular weight.

REFERENCES 1. P. J. Flory, J. Chem.Phys., i0, 51 (1942). 2. M. L. Huggins, (a)Ann. N. Y. Acad. $eL, 43, 1 (1942); (b)J. Phys. Chem., 46, 151 (1942); (c)J. Am. Chem. Sot., 64, 1712 (1942). 3. C. E. Bawn, R. E J. Freeman, and A. R. Kamaliddin, Trans. Farad. Soc., 46, 677 (1950). 4. S. Glasstone, Thermodynamicsfor Chemists, p. 373, Van Nostrand, Princeton (1947). 5. T. G. Fox and E J. Flory, J. Am. Chem. Soc., 73, 1909 (1951); 73, 1915 (1951).

220 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30.

Chapter 3 P. J. Flory and W. E Krigbaum, J. Am. Chem. Soc., 75, 1775 (1953). R. E Smith, J. Chem. Phys., 38, 1463 (1963). M. Kurata, W. Stockmayer, and A.,Roig, J. Chem. Phys., 33, 151 (1960). E J. Flory, Principles of Polymer Chemistry, Cornell Univ. Press, NewYork (1953). A. R. Schultz and P. J. Flory, J. Am. Chem. Soc., 74, 4760 (1952). J. H. Hildebrand, J. M. Prausnitz and R. L. Scott, Regular and Related Solutions, Van Nostrand Reinhold, NewYork (1970). A. Rudin, The Elements of Polymer Science and Engineering, p. 430, Academic Press, Orlando (1982). H. Burrell, Official Digest, 27, 726 (1955). R. C. Nelson, R. W. Hemwall, and G. D. Edwards, J. Paint Technol., 42, 636 (1970). P. A. Small, J. Appl. Chem., 3, 71 (1953). K. L. Hoy, Z Paint Technol., 42, 76 (1970). S. Krause, J. Macromol.Sci. - Macromol. Rev., C7, 251 (1972). H. Burrell, Polymer Handbook (J. Brhndup and E. Immergut, eds.), Wiley, New York (1975). C. M. Hansen, J. Paint Technol., 39, 104, 511 (1967). A. E M. Barton, Handbook of Solubility Parameters and Other Cohesion Parameters, CRCPress, Boca Raton, Florida (1983). Union Carbide Corporation, Tables of Solubility Parameters, 3rd ed., Chemicals and PlaStics Research and Development Dept., Terry Town, N.Y. (1975). C. M. Hansen and A. Beerbower, Kirk Othmer Encycl. Chem. Technol. (A. Standen, ed.), 2nd ed., Suppl. 889, 910, Interscience, NewYork (1971). J. E Teas, J. Paint Technol., 40, 519 (1968). E. B. Bagley, T. E Nielson, and J. M. Scigliano, J. Paint Technol., 43, 35 (1971). E J. Flory and T. G. Fox, Jr., J. Polymer Sci., 5, 745 (1950). W. R. Krigbaum and D. K. Carpenter, J. Phys. Chem., 59, 1166 (1955). E J. Flory and T. G. Fox, Jr., J. Am. Chem. Soc., 73, 1904 (1951). M. Kurata and W. H. Stockmayer, Fortschr Hochpolym. Forsch. (Advances in Polymer Science), 3, 196 (1963). W. H. Stockmayer and M. Fixman, J. Polymer Sci., C1, 137 (1963). G. Moraglio, Europ. Polym. J., 1, 103 (1965).

EXERCISES 3.1.

3.2.

The heat of mixing bf two components is not zero if it is not an ideal solution. What should be the maximumpositive value of AHmixat 25°C so that the mixing is possible ? [Ans. -1] 410.43 cal tool Using the simple lattice theory evaluate the number of distinguishable arrangements, ~/, for 100 cma solution of styrene in xylene in which the concentration of styrene is 10-2 mol/L at 20°C. What is the entropy of mixing in the preparation of the solution. (Density of xylene at 20°C = 0.86 g/cma.) [Ans. ~ = 102x102~; A,..q’mix = 6.4xi0 -2 J °K-l]

221

Polymers in Solution

3.3. Suppose that the dissolved styrene monomerin Exercise 3.2 is completely converted to dissolved polymer molecules each of DP = 1000. Assuming a coordination number of 10, calculate the number of distinguishable arrangements of solute and solvent in 100 cm3 of this solution and compare with the result of Exercise 3.2. Calculate the entropy of mixing in the preparation of this solution. ~0 [Ans. ~ = 103×10 ; ASmix = 8.3x10 -3 J °K-I] 3.4. Calculate (a) the enthalpy of mixing and (b) Gibbs free ener~,y change dissolving 10-5 mol of poly(methyl methacrylate) of ~ = 105 and of p 1.20 g/cm3 in a 150 g of chloroform (p = 1.49 g/cm3) at 25°C. The value of X is 0.377. Assumethat the volumes are additive. [Arts. (a) 25.5 J; (b) --16.1 3.5. Using the lattice theory and assuming a coordination number of 12 calculate the entropy change for the conversion of 10-6 mol of polymer of DP = 1000 from a perfectly ordered state to a randomly ordered state. [Ans. 13.0×10-3 J °K-l] 3.6. Use the Flory-Huggins equation for A#I to calculate an estimate of X for solutions of natural rubber (~, = 2.5x105) in benzene, given that vapor pressure measurements show that the activity of the solvent in a solution with ~b2 = 0.250 is 0.989. [Arts. 0.426] 3.7. Experimental data of vapor pressure measurements of toluene-polyheptene-1 system at 30°C [P. J. Tait and P. J. Livesey, Polymer, 11, 389 (1970)] are recorded beldw. The amount of polymer in each case is 0.1378 g while the amount of solvent (toluene) varies and so the vapor pressure (Pl) of solvent over the solution.

3.8.

Wt. of toluene Vapor pressure, p Wt. of toluene Vapor pressure, p (g) (cm Hg) (g) (cm Hg) 0.0012 0.124 0..0279 2.085 0.0048 0.0579 0.506 2.893 0.0077 0.716 0.1239 3.400 0.0133 1.308 0.3153 3.640 Determine the interaction parameter X for the polymer-solvent system, assuming that no volume changes occur on mixing. [Data: Polymer molecular weight = 2.24×105; polymer density at 30°C = 0.863 g/cmZ; vapor pressure of pure toluene at 30°C is 3.686 cm Hg.] [Ans. 0.49] The freezing points of acetic acid solutions of cellulose acetate of average molecular weight 65,000 were determined from their cooling curves. The freezing points obtained for solutions of different polymer weight fraction (w2) concentrations are given below: 0.0506 0.1020 0.1545 0.2082 0.2627

T, °K 289.64 289.15 288.35 287.36 286.15

222

Chapter 3 Calculate the polymer-solvent interaction parameter X corresponding to different concentrations of the polymer. Commenton the results obtained in comparison with the X value of 0.21 obtained by the osmotic pressure measurements at 25°C. [Data: Freezing point of acetic acid = 289.85 °K; molar heat of fusion of acetic acid = 2803 cal/mol; density of acetic acid at 25°C = 1.049 g/cm3; density of polymer at 25°C.= 1.30 g/cm3.] [Ans. X = -1.56, -1.146, -0.990, -0.813, -0.676 at w2 = 0.0516, 0.1020, 0.1545, 0.2082, and 0.2627,respectively.]

3.9.

Osmotic measurements on polyisobutylene (mol. wt. 1.46×106) in toluene at various temperatures yielded the following values for the second virial coefficient (A2) Temperature, °C A2×104, cm3 tool g-2 -5 0 5

0.41 0.68 0.95

Determine for the polymer-solvent system (a) the temperature at which theta conditions are attained, (b) the entropy of dilution parameter ¢, and (c) the heat of dilution parameter n at 0°C. [Data: Specific volume polymer at 0°C = 1.0 cma/g; molar volume of toluene at 0°C = 106.1 cm3/mol.] [Ans. (a) --12°C; (b) 0.16; (c) 3.10. Measurements of osmotic pressure were carried out [E Danusso, G. Moraglio, and G. Gianotti, J. Polym. Sci., 51, 475 (1961)] at different concentrations of c/s-l,4-polybutadiene in isobutyl acetate at various temperatures, using different molecular weight fractions of the polymer. (a) From the results presented below as II’/c2 vs. c2, where H’ is the osmotic pressure in g/cm2 a, and c2 is the solution concentration in g/cm determine the 0-temperature for polyisobutylene in isobutyl acetate. C° ) From the 0-value obtained, evaluate the entropy of dilution parameter ~p and the heat of dilution parameter n at 25°C. Fraction I

Tempera- c2 × 102 I-[’/c2 ’ c2 × 102 II’/c2 C2 × 102 ture (°C) 119.8 0.298 115.2 30 1.021 125.5 0.705 25 1.027 113.9 0.710 111.8 0.300 111.5 23 1.029 110.4 0.711 108.4 0.301 105.7 II 21 1.030 150~5 0.783 149.1 0.522 149.8 144.5 19 1.033 144.7 0.785 143.4 0.523 0.926 169.9 0.708 169.7 0.504 169.4 III 20 [Specific volume of ~olymer = 1.1 cma/g; molar volume ofisobutyl acetate at 25°C = 133.8 cma/mol.] [Arts. (a) 20.5°C; (b) ~b = 0.260, t~ = 0.256] 3.11. The intrinsic viscosity of different fractions of c/s-l,4-polybutadiene (PB) was measured in isobutyl acetate at 20.5°C. Based on the viscometric results IF. Danusso, G. Moraglio, and G. Gianotti, J. Polym. Sci., 51, 475 (1961)] given below, determine whether theta conditions have been attained in the solution.

223

Polymers in Solution Mol. wt. (M) PB fraction 227,000 188,000 148,000 109,000 86,000 68,000 53,000

[r/I, dL/g 0.88 0.80 0.73 0.61 0.56 0.51 0.41

[Ans. The data fit to an equation: [r/] = 1.91x10 -3 M0"49s. The value of the exponent (-,~0.5) indicates that theta conditions are attained.[ 3.12. Values of second virial coefficient (A2) of a polydimethylsiloxane gumof molar mass (M) 4.43x105 g/mol in several solvents are given below along with data of molar volumes (V~) at room temperature. Solvent [(C}Ia)2SiO]4 n-Heptane Toluene

V1, cma/mol 307.8 147.5 106.9

.42,

mol cm3 g-2 -4 6.35 x 10 .4 6.47 x 10 .4 4.06 x 10

It has been found that mean square end-to-end distance of polymer molecule and molar mass are related by (
Solvent [(CH3)2SIO]4 r~-Heptane Toluene

X 0.312 0.412 0.458

o~ 1.44 1.43 1.35

~3 3.02 2.92 2.46

3.13. The second virial coefficient (I’2) of a test sample of polystyrene (M~ = 4.8x10 n) in toluene (p = 0.845 g/cm~ at 25°C) was found to be 219 cmZ/g at 25°C. The partial specific volume of the polymer in the solution at this temperature was found to be 0.91 cm3/g. Evaluate the interaction parameter X. [Ans. 0.4941 A 3.14. solution of poly(methyl methacrylate) (p = 1.20 g/cm~, ~i = 3-5x10~) in chloroform (p = 1.49 g/cm3 at 20°C) has been prepared by dissolving 100 mg of the polymer in 200 mLof the solvent. Estimate the osmotic pressure of the resulting solution. IX = 0.377] [Ans. 4 X 10.2 atm]

224

C1aap~er3

3.15. The second virial coefficient F2 of a sample of polyisobutylene (Mn 428,000) in chlorobenzene at 25°C is 94.5 cm3/g. Calculate the osmotic pressure in g/cm2 of a 7.0× 10-8 mol/L solution of this polymer in chlorobenzene at 25°C and compare with the value calculated for an ideal solution. [Density of chlorobenzene at 25°C = 1.11 g/cm3.] [Ans. 0.23 gcm-2 -2] (ideal 0.18 gcm 3.16. What is the "excluded volume effect" ? Show the dependence Of the expansion factor c~ on the molecular weight of the polymer. Howdoes the mean square end-to-end distance (r 2) vary with the molecular weight for theta and better solvents ? 3.17. Osmotic pressure determinations were made at 27°C on a series of solutions of a globular protein in water. The following results were obtained: Protein concentration, amg/cm 1 2 3 4 5

Osmotic pressure, mm H~_O 1.64 3.29 4.95 6.62 8.31

(a) Calculate the molecular weight of the protein and its second virial coefficient. (b) Calculate the interaction parameter and the excluded volume and compare your result with the partial molar volume assuming that the partial specific volume is 0.75 cm3/g. Commenton the result of this comparison. [Density of water at 27°C = 0.996 g/cm3.] [Ans. (a) 1,56,200; 2.2x10-5 mol cm3 g-2 (b) 0.499; 1.I 3 mo1-1 ( cf. 0 .12 m3 mol-1).] 3.18. Polyisobutylene of molecular weight 1.46×106 was dissolved in toluene at 65°C and the solution was slowly cooled till it became turbid. Calculate an estimate of the volume fraction of polymer in the separated phase. [Polymer density = 0.92 g/cm3; molar volume of toluene = 106.9 cm3/mol] [Arts. 0.0082] 3.19. Make suitable derivations from the Flory-Huggins theory to show that when phase separation takes place in a polymer solution, the proportion of x-mer in the polymer-rich phase increases as x increases. 3.20. A series of solutions of polyisobutylene fractions of different molecular weights (M) in ethylbenzene were slowly cooled and the temperature at which the stirred solution became turbid on slow cooling was recorded for each concentration. The values of Tc together with polymer molecular weights are given below. M To, °C 24.4 8.2 x 106 23.8 1.8 × 106 17.0 1.8 × 105 10.7 5.4 × 104 Calculate from the data an estimate of the theta temperature /9 and the

225

Polymers in Solution entropic dilution parameter ¢. [Polymer density = 0.92 g/cm3; volume of ethylbenzene = 122 cm3/mol] [Ans. 26°C; 0.849]

molar

3.21. The critical temperature (Tc) at which the solutions of cis-l,4-polybutadiene in n-heptane underwent separation into two phases were measured for three carefully fractionated polymer samples having different molecular weights (M). From the data, given below, determine the theta temperature (/9) the polymer in n-heptane. -3 M x 10 295 940 1550

To, °C -14.0 -8.5 -6.8

3.22. Calculate an estimate of the solubility parameter for acetone at 20°C from its heat of vaporization at the boiling point (56°C), (AH~)~oc = cal/mol, and heat capacities in liquid and gaseous states, given by Cp(liquid) = 17.251 + 44.31 × 10-3T cal mo1-1 °K-1 Cp(gas) = 5.371 + 49.227 x 10-3T - 15.182 x 10-~T 2 cal mo1-1 z. Density of acetone at 20°C = 0.791 g/cm [Ans. 9.82 (cal cm-Z)1/2 1/2 or ]20.1 MPa 3.23. Using the values of molar attraction constants from Table 3.2, calculate the solubility parameter values for (a) polystyrene (p = 1.05 g/cmZ), polyacrylonitrile (p = 1.18 g/cm3), and (c) poly(ethylene terephthalate) = 1.38 g/cmZ). ~ns. (a) 8.9, (b) 12.7, (c) 11.2, all in (cal 1/2] 3.24. Derive the following expression

x = V ~ (~ - 6~)~/RT which relates parameters

61

the Flory-Huggins interaction parameter X to the solubility and ~52 of polymer and solvent. What are its limitations ?

3.25. Calculate an estimate of the interaction parameter X for Buna-N rubber Coutadiene-acrylonitrile copolymer) and n-hexane at 25°C from the solubility parameter of the solvent (~51) and the polymer (~52). [Data: ~5~ = 14.7x103 (J m-3)1/2; ~52 = 18.7×103 (J m-3)1/~; molar volume of n-hexane at = 131.6 cm3/mol.] [Arts. 1.1] 3.26. Dioctyl sebacate (~5 = 8.7) which is used as a plasticizer for poly(vinyl chloride) is to be substituted by a mixture of tritolyl phosphate (6 = 9.8) and aromatic oils (6 = 8.0) on the basis of equal ~ value. Calculate the composition of the mixture. [Ans. Volumefractions: tritolyl phosphate 0.39, aromatic oils 0.61] 3.27. Using Hansen parameters (Tables 3.5 and 3.6) determine if poly(methyl methacrylate) (PMMA)is expected to dissolve in a solvent mixture of (by volume) chloroform/benzene.

226

Chapter 3

3.28. The intrinsic viscosities of polyisobutylene of molecular weight 5.58x l0 s in cyclohexane at 30°C and in benzene at 24°C (theta temperature) are 2.48 dL/g and 0.799 dL/g, respectively. Calculate (a) unperturbed end-to-end chain length of the polymer, (b) end-to-end chain length of the polymer in cyclohexane at 30°C, and (c) volume expansion factor in cyclohexane 30°C. Take ~ = 2.5x1023. [Ans. (a) 562 ~; (b) 821 .~; (c) 3.29. Polyisobutylene fraction of molecular weight 540,000 was used for viscosity measurements in cyclohexane and benzene. The intrinsic viscosity values obtained were 2.48 dL/g in cyclohexane at 30°C and 0.80 dL/g in benzene at the theta temperature (24°C). The observed relation between molecular weight and unperturbed end-to-end distance is given by (r2)10/2 0. 76 x 10-a M1/2 cm Evaluate the total thermodynamic interaction (¢-~) for the polyisobutylenecyclohexane system at 30°C. [Data: Polymer density = 0.92 g/cma; molar volume of cyclohexane = 109 cma/mol at 30°C.] [Ans. 0.129] 3.30. Viscosity measurements were made on solutions of fractionated cis-l,4polybutadiene samples in toluene at 30°C and in n-heptane at -I°C (theta temperature), yielding the following values of intrinsic viscosities (in alL/g): Fraction 1 2 3

[r/] in toluene at 30°C 2.267 1.493 1.056

The viscosity-molecular 30°C is given by

[r/] 0 in n-heptane at -I°C 0.890 0.672 0.508

weight relationship

for the polymer in toluene at

[r/] = 2.27 x 10-~ 0’7~ M Derive a relationship between the unperturbed root-mean-square end-to-end distance of the polymer molecule and the molecular weight. The value of the universal parameter, @, in tolnene is 2.5 x 1021 dL tool -1 -a. cm 2 -s ~/2 [Ans. (r2)~ol = 0.91x 10 M cm]

227

Polymers in Solution APPENDIX

3.1

REGULAR SOLUTIONS:

SOLUBILITY

PARAMETER

Whenan ideal solution is formed from its components, there occurs no enthalpy change (AHmix = 0) and the entropy change (ASmix) is ideal, the latter being characterized by completely random mixing of all species in a mixture where the components have similar sizes and shapes. A so-called "regular solution" is, however, obtained when AHmi x is nonideal (i.e., non-zero, either positive or negative) but AS’mi is still ideal. Onthe molecular level, an ideal mixture is thus x one in which the different types of molecules, A and B, for example, behave exactly as if they are surrounded by molecules of their ownkind, that is, all intermolecular interactions are equivalent. A regular solution, on the other hand, can form only if the random distribution of molecules persists even in the presence of A-B interactions which differ from the purely A-Aand B-Binteractions of the original components A and B. The concept has proved valuable in the development of an understanding of miscibility criteria and of deviations from ideality. The requirement that a regular solution has ideal ASmix, and hence a random molecular distribution, despite the existence of A-B interactions that lead to a nonideal AHmix,effectively restricts regular solutions to those systems in whichonly dispersion forces are important, because the orientation effects of polar molecules cause nonrandommolecular distributions. Consider a regular mixture containing NA molecules of component A, each of which has molecular volume vA and can, make zA contacts with other molecules. The corresponding values for component B are NB, VB, and zB, respectively. Each A-Acontact contributes an interaction energy WAA,and the corresponding energies for (B-B) and (A-B) contacts WBBand WAB.Assume that only first-neighbor contacts need to be taken into consideration and that the mixing is random. If a molecule is selected at random, one may then assume that the probability that it makescontact with a molecule of a particular componentis proportional to the volume fraction of that component. If this randomly selected molecule were of componentA its energy of interaction with its neighbors would therefore be ZAWAANAVA/Vmi x + ZAWABNBVB/Vmix,where the total volume of the mixture Vmix is equal to NAy A + NBV B. The energy of the interaction of NA molecules of component A with the rest of the system is then obtained by multiplying both the terms in the previous sum by NAand dividing the first term by 2, i.e. ZAWAAN~VA/2Vmi x + ZAWABNANBVB/Vmi x. (The division of the first term by 2 is necessitated by the fact that it takes two A molecules to make an A-A contact.) Similarly, the interaction energy of B molecules o f c omponent B with the rest of the system is ZBWBBN~3VB/2Vmi x + ZBWABNANBvA/Vmi x. The total contact energy of the system E, is given by the sumof the interaction energies of A-A, B-B, and A-B contacts in the system. Interaction

energy of A-A contacts

= ZAWAAN~VA/2Vmi x

(A3.1)

228

Chapter 3

Interaction

energy of B-B contacts

= ZBWBBN~3vB/2Vmi x

Interaction

energy of A-B contacts = 1 ZAWABNANBVB + ZBCaABNANBV A

(A3.2)

Therefore,

E~ = [zA~o~g~vA + ~o~NAN~(zavB + zB~) + z~gg~J/2Y~ (A3.4) Substituting

--

Vmix by NAVA+ NBVB, Eq. (A3.4) can be manipulated

÷

x [~(za,~

1

÷

2(NAVA + NB~)

+ z~va) - ~zav~ - ~]

(A3.5)

To eliminate WAB it ~s assumed that (A3.6) In effect, this takes WABto be equal to the geometric mean of WAAand WBB.The geometric mean rule, based partly on theoretical principles and partly on observation, is expected to hold, however, only in situations where dispersion forces provide the only significant interaction energy. With the help of Eq. (A3.6), Eq. (A3.5) can be written in the Et = NA ZAWAA+ NB ZB°’~BB 2 2 NAVA + NBVB \ 2VA ] \ 2-~--B ] J The first ~o terms on the right hand side of Eq. (A3.7) represent the interaction energies of isolated ~mponentsA and B, while the last term represents the change in internal ener~ AEm~of the system when the species are robed. If the contact energies can be assumed to be independent of temperature, the eathalpy change on m~ing, AHm~is then AHm~

NANBVAVB

AEm~

2I/zAnY5

1/2

--

/~B~BB

~ 1/2]

(A3.8)

~e te~s (zAw~/2VA) ~/2 and (ZBWBB/2vB) V2 represent solubili~ parameters SA and ~B, respectively. It is’convenient to recast Eq. (A3.8) in the fo~

(~- ~)~ \ g~,.---~¥YB----~] NAVA

NBVB

= ~A ~ u~ (~a - ~)~ where CA and CB are volume fractions

(A3.~ of components A and B in the mixture.

Chapter

4

Polymer

Molecular

Weights

INTRODUCTION The general problem of the size and shape of polymer molecules stands at the very heart of polymer science and engineering. If the molecular weight and molecular weight distribution for a polymer are known along with a good understanding of its chain conformation, many properties of the polymer can be predicted. While the question of molecular weights, in general, pervades the entire area of chemistry, polymers have classically presented several special problems. Twoof them are the high molecular weight and the polydispersity. Polymer molecular weights are very much larger than those found for simple substances and typically range from a few thousand to a million or more. Moreover, the molecular weight of synthetic polymers within any sample is not uniform, unlike those of conventional chemicals. The mechanism of polymer formation involving growth and termination of polymer chains, which are subject to variations during manufacture, results in the production of a mixture of macromolecules that are chemically identical but different in sizes. The nature of distribution of these molecular sizes in the product greatly influences both the mechanical properties of solid thermoplastics and their processing behavior at elevated temperatures. This is one reason that the plastics market contains different grades of each polymer. All varieties are often chemically identical, but someof their molecular weight-dependent properties may differ enough that the polymers cannot be interchanged in a given application. Owingto the heterogeneity of polymers with respect to molecular weight, the numerical value assigned to the molecular weight of a polymer depends on the way in which the heterogeneity is averaged. Thus, if the molecular weight is computed by dividing the total mass by the total number of molecules, which is equivalent to weighting the molecular weight of each 229

230

Chapter4

species by its mole fraction, then we obtain the number-averagemolecular weight Mn. Alternatively, the molecular weight of each species may be we___ightedby its weight fractio___n to give the weight-averagemolecularweight Mw. The relation between Mn and Mwdepends on the form of molecular weight distribution. Only for a homogeneoussample of a polymer are the two averages equal, and otherwise Mw> Mn. The principal m___ethods for the measurement of the number-average molecular weight Mn make use of the well-known properties of dilute solution, such as osmotic pressure, elevation of the boiling point, and depression of the freezing point (i.e., the colligative properties of solutions), since these are all proportional to the number of dissolved solute molecules. Measurementof the turbidity of a dilute polymer solution, i.e., the intensity of light scattered relative t_~o the intensity of the incident beam, is the standard method for obtaining Mw. Gel permeation chromatography is essentially a process for the separation of p_9]ymermo___leculesaccording to their size and affords determination of both Mn and Mw. The most widely used characterization procedure is, however, viscosity because it is the easiest of the various methods and requires no complicated instrument. All these methods are described in this chapter. One of the characteristics of molecular size or weight distributions of synthetic polymers is their central tendency or average. Wetherefore review first the fundamentalsof small particle statistics as these apply to synthetic polymers and determine from them the various molecular weight averages that are current in polymer science. MOLECULAR Arithmetic

WEIGHT

AVERAGES

Mean

The distribution of molecular weights in a polymer sample is commonly expressed as the proportions of the sample with particular molecular weights. The mass of data contained in the distribution can be analyzed more easily by condensing the information into parameters that present a concise picture of the distribution and describe its various aspects. One such summarizing parameter is the arithmetic mean that is often used with synthetic polymers. The various molecular weight averages used for polymers can be shown to be simply arithmetic means of molecular weight distributions. Let us assume that unit volumeof a polymer sample consists of a total of A molecules comprising al molecules with molecular weight IV_fl, c~2 molecules with molecular weight M2, . .... aj mole__~culeswith molecular weight Mj. The arithmetic mean molecular weight Mis given as usual by the total measuredquantity divided by the total numberof elements. That is,

231

PolymerMolecular Weights a~M~ T a2M2 + "" + aiMi al + a2 + .’’ + aj

--

aiM~ + a2M2 + "" -tA

ajMj

al A Mt

aJMj

+ M2 + ...

+ -~

(4.1)

The ratio ai/A is the proportion of molecules with molecular weight Mi. If we call this proportion fi, the arithmetic mean molecular weight is given by -~ = flM~ + f2M~ + ... + fjMj : ~ f~M~ (4.2) i Equation (4.2) defines the arithmetic mean of the distribution of molecular weights. ~most ~1 molecular weight averages can be de~ed from this equation. Number-Average Molecular Weight Degree of Polymerization ( D P~)

( M~) and

The distribution we have assumed above is a number d~tribu~on, since the record consists of numbers of molecules of specified sizes. If we substitute the proportion of species fi, which have molecular weight M~, by the corresponding mole fraction ni in~Eq. (4.2), we obtain the definition number-average molecular weight Mnrepresenting the number distribution : M = ~-~niMi = M,~ (4.3) i The differential numberfunction is simply the mole fraction hi, and a plot of ni versus Mi yields a differential number distribution curve, as in Fig. 4.1(a). If the distribution is normalized, the area under the n/-- ~/i curve in Fig. 4.1(a) will be unity. In mathematical terms, the cumulative number (or mole) fraction defined as M ~(M~)

= ~

(4.4) i where n~ is the mole fraction of molecules with molecular weight Mi and ~(Mi) is the cumulative mole fraction with molecular weight ~_ Mi. plot of ~(Mi) against the corresponding Mi yields an integral number distribution curve, as in Fig. 4.1(b). The units of ordinate are mole fractions and extend from 0 to 1; the distribution is therefore said to be normalized. While Eq. (4.~)__ gives a simple definition of the number average molecular weight Mn, we can drive other equivalent definitions following

232

Chapter 4

blol.ecu(arweight

Motecu[arweight, number distribution curve. (b) A norcurve.

Figure4.1 (a) A normalized differential malized integral number distribution

a simple arithmetic. For this let us define which have already been used above.

the following

terms, some of

ni : mole fraction of species i (that is, molecules of same size with molecular weight Mi) in a sample moles of species i total of all Ni’s .weight fraction of species i weight of species i sum of all Wi’s It now follows that

nl

= Ni/E

(4.4a)

Ni = Ni/N

(4.4b)

Wi = NiMi

,a = yc,/ E w, = N,M,/Z N, M,

(4.4c)

and

(4.5)

"~,~ = ~_, n~M~ - E NiM~

_ zw~ E N,

The number-average

=

(4.6)

E(W,/M,)

degree of polymerization,

DP.

1

_ w DP,,, is

defined

as

Number-average molecular weight Met weight

M,~

E niMi _ ~, nixi

(4.7)

233

PolymerMolecular Weights which may also be written as DP,~ = ~_,

(4.7a)

n~x

where nx is the number fraction repeating units. Alternatively, from Eq. (4.5)

of molecules containing

]9P,~-M,~

_ 1 Mo MoE(w,/M~) which may also be written as

_ E(wl/x,)

of molecules containing

Weight-A verage Molecular Weight Degree of Polymerization ( D P~

1.

(4.8)

(4.8a)

DR,, = 1 / ~_,(w=/x) where wx is the weight fraction repeating units.

x number of

x number of

( Mw) and

The situation for weight distribution corresponds to that for a number distribution described in the previous section. If we had recorded the weight of each species in the sample, rather than the number of molecules of each size, the array of data wouldconstitute a weight distribution. The differential weight fraction is simply the weight fraction wl, while the integral (cumulative) weight fraction ~(Mi) is given by ~b(M{,) = ~ (4.9) i . and is equal to the weight fraction of all specms with molecular weight not greater than Mi. A plot of wi against Mi yields a differential weight distribution curve, as in Fig. 4.2(a). If ff~(Mi) is normalized, the scale of the ordinate in this figure goes from 0 to 1 and the area under the curve equals unity. Figure 4.2(b) depicts a simple integral weight distribution, normalized by recording fractions of the total weight rather than actual weights of differential species. The proportion of sample with size Mi is expressed in the present case as the corresponding weight fraction. Substituting wi for fi in Eq. (4.1) produces the following expression for the arithmetic mean of the weight distribution : M = ~,w~Mi = Mw (4.10) Mwis the weight-average moledular weight which from Eqs. (4.10) and (4.4c) can also be expressed -~

-

ENiMi2 E N~ M~

(4.11)

234

Chapter 4

(b)

(a)

Motecu[arweight

Mo~ecu{ar weight ~ NIi

Figure4.2 (a) A normalized differential malized integral wei~t distribution

The weight-average

weight distribution

degree of polymerization,

DPw =

D.[iw, is defined

Weight-average molecular Mer weight

M~ = Mo which may also be written

curve. (b) A

curve.

weight

E w~M~ Mo - ~-" w~x~

(4.12)

as

DR,. = ~_, w~x where wz is the weight repeating units.

as

fraction

(4.12a) of molecules

containing

x number of

Problem 4.1 Pol,vmer samples A and B are monodisperse polyisobutylenes, while sample C is a polydisperse polyisobutylene. Sample A is known to have a molecular weight twice that of B and ~o for sample C is given as 1.8×105. Deduce the M,~ for sample C from the following two measurements on a mixture of all three samples. The mixture contains 30 g of A, 40 g of B, and 30 g of C. Light scattering measurements give a molecular weight of 94,000, while the measurement of osmotic pressure gives a molecular weight of 50,000. Answer: Noting that light scattering measurements give weight-average molecular weight, use of Eq. (4.10) for the polymer mixture gives (P4.1.1) 0,.30(~)A q- 0.40(~w)B + 0.30(~w)C = Since polymer samples A and B are monodisperse,

(~.)A = (~7~)A and

(~,~)B

= (~/~)~

235

PolymerMolecular Weights

Also, it is given that MA= 2MB.Thus, Eq. (P4.1.1) becomes 0.60MB+ 0.40MB+ 0.30(1.8 x 105) = 94,000 MB = 40,000 Since ~,, is given by average mass per mole in the mixture and the osmotic measurementgives Mn,it follows from Eq. (4.6) that 100

30 40

30 2--~

30 = 50,000

40 30 + ~ + (~)c

--

1 500

(M~)c = 48,000

The molecular weight averages discussed above express the central tendency of the molecular weight distribution. However, the distribution of sizes in a polymer sample is not completely defined by its central tendency. The breadth and shape of the distribution curve must also be knownas they reflect the dispersion of the measured quantities about their mean. If all species in a polymer sample have the same__molecula_j_r weight (that is, the polymer is monodisperse) then M,~ = M~= Mz. Such monodispersity is, however, unknownin synthetic polymers and it is always true that Mz> M~o> Mn. The ratio M~/M,, or (~--]~,/~,)-1, commonlytaken to be a measure of the polydispersity of the sample. This ratio, called the polydispersity index (PDI), is not a soundstatistical measure, however, and it is easy to make incorrect inferences from the magnitude of the Mw/Mnratio. The breadth and shape of the distribution curve are characterized most efficiently with parameters derived from the moments of distribution. MOLECULAR

WEIGHTS

IN

TERMS

OF

MOMENTS

In previous sections we have seen that average molecular weights are arithmetic means of distributions of molecular weights. An alternative and generally more useful definition can be given in terms of moments of distribution that facilitate generalizations beyond the two averages we have considered to this point. This approach also clarifies the estimation of parameters related to the breadth and symmetryof the distribution. A momentin mechanics is generally defined as U.~ j= Fd (4.13) where Uj is the jth moment,abOout a s~ecified line or plane a of a vector or scalar quantity _b-’ (e.g., force, weight, mass, area), d is the distance from F to the reference line or plane, and j is a number indicating the power to which d is raised. (For example, the first momentof a force or

236

Chapter 4

weight about an axis is defined as the product of the force and the distance of the line of action of the force from the axis. It is commonlyknownas the torque. The second momentof the force about the same axis, i.e. j = 2, is the moment of inertia.) If F is composed of elements ~/, each located a distance d/ from the same reference, the momentis the sum of the individual momentsof each element:

u? = ZFd?

(4.14)

i In polymer science the mathematical formulation for moments corresponds to that in Eq. (4.14). The ordinate at M= 0 in the graph of the molecular weight distribution [Figs. 4.1(a) and 4.2(a)] is commonly chosen as the reference though it is usually not mentioned explicitly. The distance d/from the reference line is measured along the abscissa in terms of the molecular weight Mi, the quantity Fi is replaced by the quantity or proportion of polymer with molecular weight Mi, and 3" assumes a wider range of values (0, 1, 2, 3 .... ) than in mechanics. A general definition a statistical momentof a molecular weight distribution taken about zero is then

Uj = Z qiMi j

(4.15)

where qi is the quantity of polymer in the sample with molecular weight Mi. The prime superscript used in the symbol indicates that the moment is taken about the /~/_/ = 0 axis. It requires to define qi appropriately and assign numerical values to j to obtain equations for different momentsof distribution. To distinguish between numberdistribution and weight distribution we shall henceforth use the notation nU to refer to a momentof the number distribution and wUto denote a momentof the weight distribution.

Problem 4.2 Write general equations for statistical momentsfor (a) number distribution and (b) weight distribution of molecularweights. Answer: (a) Numberdistribution Unnormalized : ~U~ = ~ ~ NiMi j Normalized: ,~Uj = E niMi

(P4.2.1) (P4.2.2)

(b) Weightdistribution Unnormalized: wU’ = ~WiM/ Normalized : ~oUj = ~ wiM/

(P4.2.3) (P4.2.4)

Weightdistributions are usually encounteredduring analysis of polymersamples, while numlSerdistributions are moreuseful in considerationof polymerization kinetics.

237

PolymerMolecular Weights

Arithmetic Mean as a Ratio of Moments In general terms, the ratio of the first momentto the zeroth momentof any distribution defines the arithmetic mean. For an unnormalized number distribution, the zeroth (j = 0) and first (j = 1) momentsof the distribution about zero are given, respectively, by [cf. Eq. (P4.2.1)]:

,~U~ -= EN~(M~)°= ~-~Ni

(4.16)

= N

i

nU~ = Z N~(M~) 1 = Z N~M~

(4.17)

- ~N~M~/~N~

(4.18)

i is the arithmetic The ratio of these moments mean of the number distribution and by comparing with Eq.(4.5) we can write

M - U,’

= M~,

n 0 The arithmetic mean of an unnormalized weight distribution given by [cf. Eq. (4.10)]:

is likewise

1 M - wU~’ _ ~W~(M~) _ Z:w~M~= ° _ 2~M~ ~U0’ ~ W~(M~) ~W~ In tions. ential equals

(4.19)

the above two examples, we have chosen unnormalized distribuFor normalized distributions, the area under the curve for the differnumberdistribution [Fig. 4.1(a)] or weight distribution [Fig. 4.2(a)] unity. That is, ~U~ = En~(M~)°

= Z:~

=

~u; = Zw~(M~)°=Z~ = 1

(4.20)

(4.21)

It is now seen from Eqs. (4.18) and (4.19) that the arithmetic mean numerically equal to the first momentof the normalized distribution.

Extension to Other Molecular Weight Averages Wemay define an average in general as the ratio of successive moments of the distribution. Wehave seen above that Mn, the number average molecular weight, is equal to the ratio of the first to the zeroth moment of the number distribution. If we now take ratios of successively higher moments of the number distribution, other average molecular weights are described : ! 2

~2 U,

~ N~M~

(4.22) (4.23)

n U4’ ,! U

’l E N~M~

(4.24)

238

Chapter4

This process of taking ratios of successive momentsto obtain higher averages can continue without limit. In practice, however, the averages usually quoted are limited to M,m_Mw, M_~and the viscosity average molecular weight My. While Mn, Mwand My may be measured directly (see later), it usually necessary__ko measure the detailed distribution to estimate M=and other averages. M. is given by 1/a -_- |~ NiMia+l" M. l-~ N-----~/

(4.25)

where a is a constant. A derivation of Eq. (4.25) is given in a later section (see p. 279). Combination of Eq. (4.25) with Eq. (4.4c) -M~ = L[~

1/a

r~ M. ~] anx/a ~WiMia] J = tLw~

(4.26)

In terms of moments, My is given by

My

=

[~j

(4.27)

and

’11/ -~=’~,~j t~orrr

(4.28)

corresponding to numberdistribution and weight distribution,

respectively.

Problem 4.3 A sample of poly(vinyl chloride) is composedaccording to the followingfractional distribution: Weightfraction, wi 0.04 0.23 0.31 0.25 0.13 0.04 -3 Meanmol. wt., Mi x 10 7 11 16 23 31 39 (a) Compute M,~, M~, and M=. (b) Howmany molecules per gram are there in the polymer? Answer: The wi vs. Mi data are used to makethe following table: 0.04 0.23 0.31 0.25 0.13 0.04

7,000 11,000 16,000 23,000 31,000 39,000

280 2,530 4,960 5,750 4,030 1,560 19,110

-5 0.57 x 10 -5 2.09 x 10 -s 1.94 x 10 -5 1.09 x 10 -5 0.42x 10 -5 0.10 x 10 -5 6.21 x 10

1.96 x 106 T 2.78 x 10 T 7.94 x l0 1.32 x 108 s1.25x10 T 6.08 x l0 s4.27 x l0

(a) FromEqs. (4.6), (4.10), and (4.23): 1 1 = 16, 100 6.21 x lO-S E(wi/Ivli)

239

PolymerMolecular Weights

~ E NiMi 2~ NiMi

2E wiMi Y~ wiMi

4.27 x s 10 = 22,344 19, 110

(b) Numberof molecules per gram = (~’~ ~i)(Avogadro’s number)

=(6.21×10-5)(6.02 × = 3.74 x 1019 molecules/g

MOLECULAR

WEIGHT

DETERMINATION

Two fundamentally different approaches are used for the measurement of polymer molecular weights: absolute methods and secondary methods. Absolute methods give values that provide a direct estimate of the molecular weight. Secondary methods, on the other hand, yield comparisons between the molecular weights of different polymers and must be calibrated with a reference molecular weight that has been studied by one of the absolute approaches. Studies of colligative properties, light scattering, and sedimentation under ultracentrifugation of polymer solutions are several methods for the determination of absolute molecular weights. The measurement of any colligative property yields the number-average molecular weight. An absolute method for weight-average molecular weight is provided by light scattering measurements. The ratio Mw/Mn, determined by light scattering measurements and from osmotic pressure measurements on the same solutions, is a measure of the width of the molecular weight distribution. Ultracentrifugation experiments yield Mz values. They are used primarily for biological polymers. The random coil conformation of most synthetic polymers in solutions makes it difficult to interpret these data and this method is little used with such materials. Amongthe absolute methods of molecular weight determination, only the osmotic method and the light scattering methodare given primary consideration in this chapter. Thoughthe absolute methods for the determination of molecular weights are well established, both theoretically and experimentally, the absolute measurements are difficult to carry out, are time-consuming, and often require expensive apparatus. For these reasons, for routine determinations of molecular weight, the much faster secondary methods, such as solution viscosity and gel permeation chromatography, are commonlyused. These methods require prior establishment of empirical relationships that relate the molecular weight to the viscosity of the polymer solution or to the retention times in a gel-permeation column. Once such calibration has been done, the secondary methods provide a fast, simple, and accurate

240

Chapter 4

way to obtain molecular weights. The solution viscosity and gel permeation chromatography methods are described in a later part of this Chapter. __Most of the methods for measuring number-average molecular weights (Mn) rely on the colligative properties of solutions. These include boiling point elevation (ebulliometry), freezing point depression (cryoscopy), vapor pressure lowering, and osmotic pressure (membraneosmometry), which are all proportional to the number of dissolved solute molecules and so can be used to obtain Mn. Membraneosmometryis, however, the only colli_g_ative property measurement that is practical for direct measurement of Mn of high molecular weights, because the other colligative properties give rather small effects unless the molecular weight of the polymer is less than about 20,000 (see Table 4.1). Twoother techniques are also used to measure Mn of relatively lowmolecular-weight polymers. These are end-group analysis and vapor phase osmometry. End-Group

Analysis

End-group analysis can be used to determine Mn of polymer samples if the substance contains detectable end groups, and the number of such end groups per molecule is known beforehand. Whenone knows the exact number of end groups per molecule, the measurement of the concentration of end groups gives the number of moles and hence the molecular weight. Since the concentration of endgroups varies inversely with molecular weight, end-group methods tend to become unreliable at higher molecular weights. In fact, most methods of chemical analysis becomevery inaccurate for molecular weights substantially greater than 15,000. End-groupanalysis is restricted to low-molecular-weight polymers with well-defined structures and__0_distinguishable end groups. The upper limit for accurate measurement of ~’/n is dependent upon the sensitivity of the technique used to measure the end group concentration, but typicall3i is 10,000 to 15,000. End-group analysis has been applied mainly to condensation polymers, since these polymers by their very nature have reactive functional end groups. The end groups are often acidic or basic in nature, as exemplified by the carboxylic groups of polyesters or the amine groups of polyamides; such groups are conveniently estimated by titration. Colorimetry, radioisotope, and spectroscopic analyses are amonganalytical techniques that are of value for special types of end groups. From the experimental data Mnis derived according to -~n

--

f.w.e

(4.29)

where f is the functionality or number of reactive groups per molecule in the polymer sample, w is the weight of the polymer, g is the amount of reagent used in the titration, and e is the equivalent weight of the reagent.

Polymer Molecular Weights

241

Table 4.1 A Comparison of Colligative Properties of a 1% (w/v) Polymer Solution with Molecular Weight 20,000 Property Vapor pressure lowering Boiling point elevation Freezing point depression Osmotic pressure aFrom Ref. 1.

a Value 0.004 mm Hg 0.0013°C 0.0025°C 15 cm solvent

Problem4.4 A sample (3.0 g) of carboxyl terminated polybutadiene required titration with 20 mL 0.1 N KOHto reach a phenolphthalein point. Calculate Mnof the polymer.

(CTPB) end

Answer: Here f = 2 eq mol-I, e = 56 g eq-I, w = 3.0 g (20 mL)(0.1 eq L-I)(56 -1 ) a = = 0.112 -1) (1000 roLL

g

From Eq. (4.29), ~ = (2 eq mo1-1)(3.0 g)(56 (0.112 g)

-~ ) = 3, 00 0 g to ol -1

End-group analysis yields the equivalent weight of the polymer, Me, which is the mass of the polymer per mole of end groups. If for a polydisperse polymer sample, Ni is the number of moles of polymer molecules of molecular weight Mi and f is the number of analyzable end groups per polymer molecule, then

M~ = Z NiMi/ Since f is rewritten as

a constant

E INi

and Mn = ~ NiMi/~

M¢ = --~,~/:

Ni,

(4.30) Eq. (4.30)

can

(4.31)

The functionality f of low-molecular-weight functionalized prepolymers (e.g., polyether polyols used in the preparation of polyurethanes) is often determined from Eq. (4.31) by combining functional group analysis with another suitable method of molecular weight determination such as vapor phase osmometry. Note that in this respect the functional groups do not have to be end groups.

Problem 4.5 A sample (2.0 g) of polyether polyol prepolymer (~,~ = 2048) dissolved in chlorohydrocarbon solvent was treated with excess succinic anhydride

242

Chapter4

to convert each hydroxylgroup in the polyol to a carboxyl group by formation of succinic half-ester. A sample(1.0 g) of this treated polymerrecovered from the solution by precipitation (in excess of ethanol) required 12.8 mLof N/10 KOH for carboxyl titration. Determinethe hydroxylfunctionality of the polyol. Answer: One molecule of succinic anhydride is added to each hydroxyl group, CHACO\ DOt’I

+ I - /O ~ ~OCOCt-I2CH2COOH CH.~CO Carboxyl end group in 1 g of succinic anhydride-reacted polymer(SAP) = (12.8 mL)(0.1xl0-3 -1) mol mL = 1.28x10 -s mol Equivalent weight (mass per mol of carboxyl group) of SAP = (1 g)/(1.28x10 -3 tool) = 781 gmo1-1 Molar mass of succinic anhydride = 100 g mo1-1 Equivalent weight (mass per mol of hydroxyl group) of polyol, M~= -1 (781 - 100) or 681 g tool From Eq. (4.31), -~) (2048 g mol -~) g tool (681

The end-group analysis method for Mndetermination cannot be used in manycases of practical interest because f in Eq. (4.29) is not known.This particularly true for branched or cross-linked polymers with variable number of end groups per molecule. There is also the problem of solvent selection because the choice of solvents is obviously limited to those that dissolve the polymer. This usually rules out aqueous solutions, in which manyanalytical reactions are normally carried out. The solvent selected must not react with the reagent used in the analytical method and must not react with the polymer during the process of solution in any way that may interfere with the subsequent end group estimation. As it is usually necessary to dissolve the polymer in organic solvents, the techniques of nonaqueoustitration are of particular importance in the estimation of acidic and basic groups. Colligafive Ebulliometry

Property

Measurement

(Boiling-Point

Elevation)

In this method, the boiling point of a solution of knownconcentration is compared to that of the solvent at the same pressure. For ideally dilute

243

PolymerMolecular Weights

solutions, the elevation of the boiling point, T- Tb, is related to the normal boiling temperature of the solvent Tb, its molar latent heat of evaporation Le, and molecular weight M1, and also to the molecular weight of the solute Mg. and relative weights of solvent and solute W1and W2,respectively, by .RTb 2 W2 M1 A% = T-Tb = Lo " 11"M2 (4.32) For convenience, Eq. (4.32) is rewritten

/XTb = RT M1 IO00W2

1000L~ W1M2 (4.33) = kern2 where the solvent’s molal boiling-point elevation constant lee is defined by ke-

RTb2M1 - 1000L, and the solute molality mg~(in units of moles per kilogram) 1000W2 m2 -WIM2 For benzene, (1.987 cal mo1-1 °K-1)(353 °K)2(78 g -1) k e .~ (1000 g kg-1)(7, 497 cal mo1-1)

(4.34)

(4.35)

= 2.6 °K kg mo1-1 Someother values of ke defined by Eq. (4.34) are: water 0.51, acetic acid 3.0, benzene 2.5, acetone 1.7, methyl alcohol 0.8, chloroform 3.8, carbon tetrachloride 5.0. To determine a molecular weight, one measures ATbfor a dilute solution of solute in solvent and calculates m2 from Eq. (4.33). The molecular weight M~of the solute is then calculated from Eq. (4.35). Since Eq. (4.33) applies only for an ideally dilute solution, a precise determination of molecular weight requires that ATbbe found for a few different solute concentrations c2; one then plots M2[as calculated from Eq. (4.35)] vs. c2 and extrapolates to zero Ebulliometry, like end-group analysis, is limited to low-molecular-weight polymers.-By use of thermistors sensitive to 0.0001°C, it is possible to measure molecular weight values up to 50,000, although typical limits (using Beckmanthermometer with temperature difference measurement accuracy of 0.001°C) are 5,000. Cryoscopy

(Freezing-Point

Depression)

This is a classical method for measurementof molecular weights of organic compounds. Calculation of the freezing-point depression proceeds exactly

244

Ch~p~er 4

the same way as for the boiling-point elevation. Equations that correspond to Eqs. (4.33) and (4.34) for ideally dilute solutions AT.f

= - k.fm2

(4.36)

k.f

- RT}M1 (4.37) 1000L] where AT] =_ T-TI is the freezing-point depression, TI is the freezing point of pure solvent, and LI is the molar latent heat of fusion. The solvent’s molal freezing-point depression constant ]~,f is calculated from Eq. (4.37) the same way as }e is calculated from Eq. (4.34). For water, for example, (1.987 cal mo1-1 °K-1)(273 °K)2(18.01 -~ ) }.f = (1000 g kg-1)(1,436 cal mo1-1) -~ = 1.86 °K kg mol Some/~f values so obtained are: water 1.8, acetic acid 3.8, benzene 5.1, succinonitrile 20.3, camphor 40. The large /~I of camphor makes it especially useful in molecular weight determinations. A comparison with ]% values given earlier shows that the cryoscopic effect is somewhatlarger than the elevation in boiling point achieved with typical solvents (cf Table 4.1). This increased sensitivity, together with the simplicity of the apparatus is one of its most attractive features. Thoughthe sensitivity can be increased considerably by using camphoras solvent, as it has a high cryoscopic constant, its use is limited to polymers of good thermal stability because of its rather high melting point (179°C). Succinonitrile, in comparison, has a melting point below 60°C and also a fairly high cryoscopic constant. Like ebulliometry, the cryoscopic method is also limited to relatively low-molecular-weight polymers with Mn up to 50,000. A newer variation of this methodinvolves the solute-induced depression of the nematic-isotropic transition temperature for a liquid crystal solvent such as p-azoxyanisole [2]. This technique appears to be particularly convenient for high-molecularweight polymers and may be applicable for Mn values as high as 106. Membrane

Osmometry

Osmotic pressure is the most important amongall colligative properties so far as synthetic polymers are concerned. To explain osmotic pressure, let us imagine a box (Fig. 4.3) divided into two chambers by a semipermeable membranethat allows a solvent A to pass through it but does not allow the passage of solute B. Suppose that the left chamber is filled with pure solvent A and the right chamber with a solution of B in A such that initially the heights of the liquids in the two capillary tubes are equal. The chambers are thus initially at equal pressures, that is, . PL = PR, where the subscripts stand for left and right. Let us assume that the membrane

245

PolymerMolecular Weights

t!

Pure

Osmotic head(h)

I

~t 3 ,-, t2 ,.t I

Solution of BinA

Semipermeable membrane Figure 4.3 Schematic diagram showing the development of osmotic head as a function of time, wheretl represents the initial measuringtube levels, ~2 the levels after an elapsed time, and t3 the levels whenstatic equilibrium occurs.

is thermally conducting and thermal equilibrium is always maintained, that is, TL = TR = T. The chemical potential of A on the left lZA,L is #~4, representing chemical potential of pure solvent A. If the solution on the right is dilute enough to be considered ideally dilute, then /~A,R= /~. + RTIn ZA, which is less than lZA,L = /z~4, since XAbeing mole fraction of A in solution is less than 1. Since lZA,L > lZA,R, solvent A will flow through the membranefrom left to right and the liquid height in the right tube will rise, thereby increasing the pressure in th__e right chamber. For an ideally dilute solution, the p_~_r.tial molar volumeVA for the solvent is the same as for the pure solvent VA ~- W~, and since, in general, (O#i/OP) = Vi, one can write (OttA/OP )T = VA---V~ (4.38) Since Y.~ is positive, the increase in pressure increases ttA,R until eventually equilibrium is reached with ]ZA,R = ~A,L = ~°A. (It may be noted that since the membraneis impermeable to B, there is no equilibrium relation for Let the equilibrium pressures in the left and right chambers be /9 and P q- l-I, respectively. Wecall 1-I the osmoticpressure. It is the extra pressure that must be applied to make ~A in the solution equal to t~, so as to achieve membraneequilibrium for species A between the solution and pure A. If the solution in the right chamber is dilute enough to be considered

246

Chapter 4

ideally dilute, then at equilibrium or #°A(P , T ) = #°A(P + l-I, T ) + RTlnxA (4.39) wherethe right side of the equation follows from the general relation: #~ = #~ + RTlnx~. Since (O#i/OT)p = - Si and (O#i/OP)T Vi, it follows that dt~°A = --S~4dT + V~dP (4.40) Thus, at constant T, dl~°A = V~dP. Integration from P to P-{-II gives lZ°A(P + H, T) - #°A(P,T ) = f e+nV~dP JR Substitution of Eq. (4.41) into Eq. (4.39) gives RT In

XA :

--

f P+n V~dP

dP

(4.41)

(4.42)

Liquids being rather incompressible, V.~ varies little with pressure and we can take V~as essentially constant. The integral in Eq. (4.42) then becomesV~II and Eq. (4.42) thus gives II lnx A = -(RT/V2) with XA = 1-- XB, we have lnXA = --X, B -- X~B/2 ....

(4.43)

II = (RT/V~)x, Since the solution is quite dilute, wehave nB nA q- nB and RT n~ FI -V~ nA where nA and nB are the number of moles of solution that is in membrane equilibrium with

(4.44)

"~ where, since xB << 1, we can neglect x~ and higher powers. Therefore,

nB nA (4.45) solvent and solute in the pure solvent A.

Problem4.6 A solution containing 1.018 g of a protein per 100 g of water is found to have an osmotic pressure of 10.5 torr at 25°C. Estimate the molecular weight of the protein.

Answer: Theosmoticpressure (in atm) 10.5 torr H -= 0.0138 atm 760 torr/atm

247

PolymerMolecular Weights

For water: density at 25°C and 1 atm = 0.997 g cm-a; molar mass = 18.0 mo1-1. So, V.~ = 18.054 cm3 -1) mo1-1 and nA = (100 g)/(18.0 g tool 5.555 mol. From Eq. (4.45): IIV~nA (0.0138 atm)(18.054 3 mo1-1)(5.555 mo (82.06 cm3 atm mo1-1 °K-1)(298 °K) = 5.659 × 10-5 mol Hence, 1.018 g M = = 18,000 gmo1-1 5.659 x 10-5 mol Since the solution is quite dilute, the solution volumeV is approximately equal to that of the solvent nAV~ and Eq. (4.45) becomes HV

=

nBRT

(4.46)

or H = CBRT (4.47) where the concentration CB in moles/volume equals, nB/V. Equation (4.46), which has formal resemblance to the equation of state for ideal gas, PV ---- nI~’, where n is the number of moles of gas in volumeV, is called van’t Hoff’s law, valid in the limit of infinite dilution. For solutions that are not ideally dilute, Eq. (4.43) should be written as

II (4.48) A = -(RT/V~) In ~i’AX to include the activity coefficient 7A that measures deviations from ideally dilute-solution behavior. However,a different (but equivalent) expression for II in nonideally dilute solutions is often more convenient than Eq. (4.48). A result that can be derived from the statistical-mechanical theory for nonelectrolyte solutions developed by McMillan and Mayer [3] in 1945 showed that the osmotic pressure in a nonideally dilute two-component solution is given by YI = RT (M~lc, + A2c~ + Azc~ +...) (4.49) which has formal resemblanceto the virial equation for gases. In Eq. (4.56), MBis the solute molecular weight and CB is the solute mass concentration defined as CB = mB/V (4.50) where mB is the mass of solute B in the solution of Volume V. The quantities A2, A3 .... are related to the solute-solvent intermolecular forces in the given solvent and are functions of temperature. In the limit of infinite dilution, cB goes to zero, and Eq. (4.49) becomes rI

= RTcB/MB

= RTmB/MBV

= RTNs/V

= CBRT

(4.51)

248

Chapter 4

which is the van’t Hoff law [Eq. (4.47)]. CBin Eq. (4.51) is the concentration of solute B in moles per unit volume. Problem4.7 Find the osmotic pressure of a solution of 1.0 g glucose (C~H1206) in 1000 cm3 of water at 1 atm and 25°C. Answer: -1 Molar mass of glucose = 180.15 g tool NB= (1.0 g)/(180.16 g mo1-1) = 0.00555 Substitution in Eq. (4.45) gives (0.00555 mol)(82.06 cm3 atm mo1-1 °K-1)(298 °K) 3) (1000 cm = 0.135 atm = 102.6 torr

H =

Note the large value of l"i for the very dilute 0.0055 mol/L glucose solution in Problem 4.7. Since the density of water is 1/13.6 times that of mercury, an osmotic pressure of 102 torr (102 mmHg) corresponds to a height of 10.2 cm x 13.6 = 139 cm = 4.5 ft of liquid in the right hand tube in Fig. 4.3. The large value of 1-i results from the fact that the chemical potential of a component of a condensed phase is rather insensitive to pressure. Hence it needs a large value of 1-I to change the chemical potential of A in the solution so that it equals the chemical potential of pure A at pressure P. The large values of osmotic pressure given by dilute solutions make osmotic pressure measurements valuable in determining molecular weights of substances with high molecular weights such as polymers. Though the osmotic pressure according to Eq. (4.51) is inversely proportional to solute molecular weight, it has a relatively large, measurable value even for highmolecular-weight polymers in dilute solutions. Vlrlal

Equations

In polymer science, the measuredsolute concentrations are taken as accurate without substituting them by fictitious concentrations called "activities" and deviations from ideality are expressed in the coefficients of the concentration terms. Thus, the osmotic data of a real solution are expressed, according to Eq. (4.49) and omitting the subscript for solute,

where A2and Aaare called the second and third virial coefficients. Unfortunately, there is no uniformity in the exact form of the virial equations used

PolymerMolecular Wei~ht~ in polymer science. Alternative forms of Eq. (4.52) include

RT [1 + r2c + r3c2 + ..- ] M

c

(4.53)

and

H RT -+ Bc + Cc2 + ... c M

(4.54)

The three forms are equivalent if

B = RTA2 = (RT/M)F2

(4.55)

The virial coefficients are often reported without specifying the equation to which they apply, and this can usually be deduced only by inspecting the units of the virial coefficient. The value of the third virial coefficient P3 in Eq. (4.53) is often assumed to be equal to (P2/2) 2, so that Eq. (4.53) can be rewritten (~)1/2

= (__~_)1/2

(1 q_

iv c)

(4.56)

This form is sometimesconvenient to use for extrapolation of I-i/c data to zero concentration because, for good solvents, plots given by expressions (4.52)-(4.54) are not linear. At low concentrations, the effects of the c2 terms in any of the above virial equations, Eqs. (4.52) to (4.54), will be very small, and the data 1-I/c versus e are expected to be linear with intercepts at c = 0 yietding values of M-1 and slopes that are measures of the second virial coefficient of the polymer solution.

Problem4.8 Showthat the molecular weight determined from osmotic pressure measurementsis the numberaverage molecular weight. Answer: Consider a whole polymer to be made up of a series of monodisperse macromolecules i with concentration (weight/volume) ci and,molecular weight Mi. For each monodisperse species i one can write from Eq. (4.52) for very dilute solutions, RTc~

~ =

M~

For the polydisperse polymer H = ~I-I4 = RT~-~(ci/Mi) and c = wherewi is the weight fraction of species i in volumeV. For the osmotic molecularweight one can then write :

250

Chapter4 RTc II RT~ RT ~(c~/M~) 1

~(w~/M~)

[From Eq. (4.6)]

The nonideality of polymer solutions is incorporated in the virial coefficients. Predicting nonideality of polymer solutions means, in reality, predictions of the second virial coefficient, because this is the coefficient which can be measured most accurately. Better solvents generally produce greater swelling of macromoleculesand result .in higher virial coefficients. A number of factors influence the magnitude of the second virial coefficient. These include the nature of the polymer and solvent, the molecular weight distribution of the polymer and its mean molecular weight, concentration and temperature of the solution, and the extent of branching in the polymer chain. The second virial coefficient, A2or B, decreases with increasing molecular weight and increased branching of the polymer as both factors tend to result in more compactstructures that are less swollen by the solvent. Most polymers become more soluble in their solvents as the temperature is increased. This is reflected in an increase of the virial coefficient with temperature and in a reduction of its value as the temperature is reduced. At a sufficiently low temperature, the second virial coefficient mayactually be zero. This is the Flory theta temperature, which has been defined in Chapter 3 as that temperature at which a given polymer of infinitely high molecular weight would be insoluble at great dilution in a given solvent. At the theta temperature a polymer in solution is on the threshold of precipitation. A solvent, or a mixture of solvents, for which the theta temperature is close to room temperature (or whatever temperature is to be employed) is a theta solvent for the particular polymer. Such a solvent neither expands nor contracts the macromolecule, which is then said to be in its "unperturbed" state. The diameter of the polymer chain random coil in solution is then equal to the diameter it would have in the amorphous bulk polymer at the same temperature. Use of theta solvents enables the experimenter to obtain polymer molecules that are unperturbed by solvent but separated from each other far enough not to be entangled. However, theta solvents or solutions under theta conditions are not normally used for polymer molecular weight measurements because the solute is on the verge of precipitation. Practical

Aspects

of Osmornetry

Osmometersconsist basically of a solvent compartment and a solution compartment separated by a semipermeable membrane and equipped w]th a

Polymer Molecular Weigh~

251

method for measuring the equilibrium hydrostatic pressures on the two compartments. In static osmometers,the heights of liquid in capillary tubes attached to the solvent and solution compartments (Fig. 4.3) are measured. At equilibrium, the hydrostatic pressure corresponding to the difference in liquid heights is the osmotic pressure. The principal disadvantage of this static equilibrium procedure is the length of time required for attainment of equilibrium. After the first contact of the solvent and solution with the membrane, periods of a few hours or more may be required for the pressure difference to stabilize, and this equilibrium process must be repeated for each of the several concentrations of polymer solution in a molecular weight measurement. However, this problem has disappeared with the advent of high-speed dynamic osmometers. The osmotic pressure, it may be recalled, is defined as the pressure that must be applied to the solution to maintain equilibrium when solution and pure solvent are separated by a semipermeable membrane. Thus, by attaching a counter pressure device to the solution tube (Fig. 4.3), the magnitude of the applied external pressure required to prevent change in the liquid levels in the solvent and solution tubes gives the osmotic pressure of the solution. This method of determining the osmotic pressure is conveniently referred to as the ’dynamic equilibrium’ technique. It is especially useful whenrapid determinations of osmotic pressure are required. Dynamic osmometers reach equilibrium pressures in 10 to 30 minutes and indicate osmotic pressure automatically. Several types are available. Some commonlyused models employ sensors to measure solvent flow through the membraneand adjust a counteracting pressure to maintain zero net flow. A commercially available automatic osmometeroperates on the null-point principle. In this high-speed membraneosmometer schematically represented in Fig. 4.4, the movementof an air bubble inside the capillary, immediately below the solvent cell indicates the solvent flow to the solution cell. Such movementis immediately detected by a photocell, which in turn is coupled to a servomechanism. If any movementof the air bubble is detected by a photocell, the servomechanismis stimulated to movethe solvent reservoir upward or downwardin order to adjust the hydrostatic pressure such that the solvent flow is completely arrested. The pressure head of the reservoir gives the osmotic head. Someosmometersalso use strain gauges on flexible diaphragms to measure the osmotic pressure directly. Preparation of the semipermeable membrane is tricky. The membrane must not be attacked by the solvent and its impermeability to solute molecules must be reconciled with a reasonably high permeability to solvent so that osmotic equilibrium is achieved in a reasonable time. If the membrane is too permeable, however, large leakage of low-molecular-weight polymer species will occur resulting in lower observed osmotic pressure and hence higher calculated molecular weight. Extremely careful control of membranestructure is required in order to obtain the highest possible degree of solute rejection. Cellulose and cellulose acetate membranesare the most widely used types with synthetic polymer solutions. The raw data obtained in osmotic pressure experiments are pressures (osmotic heads) in terms of heights of solvent columns at various polymer

252

Chapter 4

Servomotor

~

Amplifier

~Sampte

= Elv"’°rlI=i

~tecto~

Figure 4.4 Schematic diagram of essential

components of a high-speed

mem-

brane osmometer.

concentrations. The pressure values are usually in centimeters of solvent (h) and the concentrations (c) maybe in grams per cubic centimeter, per deciliter (100 cm3), or per liter, and so on. In direct application the data, hie is plotted against c and extrapolated to (h/c)o at zero concentration. The columnheight (osmotic head) h is then converted osmoticpressure 1-i by H = hpg (4.57) wherep is t___hedensityof the solventand g is the gravitational acceleration constant. Mnfollows from Eqs. (4.52)-(4.54), whichin the limit of c -~+ reduce to

It is necessary to rememberthat the units of/~ must correspondto those of (H/C)o. Thus with h (cm), p (g/cm3), and g (cm/s2), R should ergs mo1-1°K-1. On the other hand, if h, p, and g are in SI units, /~ should be in J mo1-1°K-1.

Polymer Molecular Weights

253

The second virial coefficient straight line portion of the (1-I/c) c2 terms in Eqs. (4.52)-(4.54).

can be obtained from the slope of the versus c plot essentially by dropping the

Problem 4.9 At 20°C, the osmotic pressure of a polycarbonate in chlorobenzene solution with the following results: Concentration (g/L) Osmotic pressure (cm chlorobenzene)

1.95 0.20

2.93 0.36

3.91 0.53

The density of chlorobenzene is 1.10" g/cm3. Calculate molecular weight and (b) the second virial coefficient.

(a)

was measured 5.86 0.98

for the polymer: (a)

From the osmotic data we obtain the following table: c

(100~cma) 1.95 2.93 3.91 5.86

h (cmC6HsC0 0.20 0.36 0.53 0.98

h (cmC6H~Cl)(1000cma) c g 0.102 0.123 0.135 0.167

The next step is to plot (h/c) against c. Whenthis is done, we get a straight line (Fig. 4.5) of slope 0.016 and intercept 0.072 on the ordinate, i.e., -1) (h/c) Ic-~0 = 0.072 (cm C6H5C1)(1000 cm3)(g -a = )72.0 (cm C~H~Cl)(cma)(g Pressure exerted by a column of C6H~CI, h cm long, lI = hpg

Therefore,

= (h cm)(1.10 g cm-3)(980 cm -z) = 1078h gcm-1 -2 s 72.0 × 1078 (g cm-1 -’) s-2)(cmZ)(g

Hence,

77,616 cm2 -~ s

77,616 cm~ -~ s Mn Noting that _R should be in ergs mo1-1 °K-1 to yield ~ in g mo1-1 we then obtain r ~ (8.314 x 10 ergs mo1-1 °~K-1)(293 °K) M n (77,616 cm2 -2) s = 3.1 x 105 g mo1-1

254

Chapter4

(b) Slope of h/c versus c linear plot (Fig. 4.5)

= -2) 0.016 (cm C~HBCI)(106cm6)(g s-2)(cm6)(g = 0.016 x 1078 × 10~ (g cm-1 -2) = 1.725 x 10 Neglectingc2 and higher powersof c in Eq. (4.52) for dilute solutions, this slope can be equated to A~_/Tfl. Hence A2RT= 1.725 x l0 T crn ~ g-1 s-2 (1.725 x l0T g-1 cm5 s-2) or A2 = (8.314 x 10v ergs tool -z °K-1)(293 °K) = 7.0 x 10-4 tool cm3 g-~

It is to be expected from Eq. (4.52), that measurementsof the osmotic pressures of the solutions of the same polymer in different solvents should yield plots with a commonintercept (at c = 0) but different slopes (see Fig. 4.6), since the second virial coefficient, which reflects polymer-solvent interactions, will be different in solvents of differing solvent power. For example, the second virial coefficient can be related to the Flory-Huggins interaction parameter X (see p. 162) A2 = (½ - X )v221V~ °

(4.59)

where v2 is the specific volume of the polymer, V1 o is the molar volume of the solvent, and X is an interaction energy per tool ofthe solvent divided by RT. For a theta solvent, X = 0.5 and A2 = 0. Better solvents have lower X values and higher second virial coefficients. Equation (4.59) can be used to determine X from osmotic measurements (cf. Problem 3.6 in Chapter 3). 0.20

0.10

/ 0

? t

I 1

I t 2 3 t. 3) ¢ (g/lO00 crn

I 5

Figure 4.5 Molecular weight determination from osmotic head (h) and concentration (c) data (Problem4.9).

255

PolymerMolecular Weights

~

lncreasing solvent power

Theta solvent

Figure 4.6 Reducedosmotic pressure (H/c) versus concentration (c) plots the samepolymersamplein different solvents of increasing solvent power.

Problem 4.10 Measurementof osmotic pressure for solutions of two polymers A and B in the samesolvent (benzene) at 27°C yielded the following data: Conc., cA Osmoticpressure, II (g/dL) (cm of solvent) 0.400 1.60 0.900 4.44 1.200 6.91 1.400 8.95 1.800 13.01 3] [Solvent density = 0.87 g/cm3; polymerdensity = 1.05 g/cm (a) EstimateM,, and the secondvirial coefficient I’2 for the polymer. (b) Determinethe polymer-solvent interaction parameter Answer: (a) Using Eq. (4.56), (II/c) 1/2 is plotted versus c in Fig. 4.7. The value of 1/2 is obtained by extrapolating to c = 0. This gives (II/c)¢=_ 0 1/~ = 1.80 cml/~ 1/2 (II/c)~__ dL g-V2 o

256

Chapter 4

Dimensions: H is in cm (solvent), R -

riM cT

-1. and M is in g t ool c is in g -1,

Therefore,

cm dL mo1-1 °K-1

Note that 1 atmosphere

= = =

pressure exerted by a column of mercury 76 cm long (76 cm)(13.6 g -3) -2 1033 g cm

R = ( 0"082 Latm~mol°K]( 1033 gatm cm-z~] (0.-~g)(10]~___.~L) = 0.974 x 10z cm dL mo1-1 °K-1 RT = (0.974 x 103 cm dL mo1-1 °K-l)

(300 °K)=2.92

x 105 cm dL mo1-1

From Eq. (4.56) at e =0: RT

Mn --

(ri/C)~--0 (2.92 x 105 cm dL mo1-1) (1.80 cml/2 dL1/2 g-1/2)2

= 90,100

g mo1-1

From Fig. 4.7, slope = 0.50 cml/2 dL3/2 g-3/2 From Eq. (4.56),

[’2 --

2 x slope (RT/M)l/2

2 (0.50 cmV2-dL3/2 s [(2.92 x 10 cm dL mo1-1) / (90, 100 g mol-l)]l/2 = 0.555 dLg-1 = 55.5 cm3 g-1 (b) Consider Eq. (4.59): v2 = (1.05 g era-3) -1 V~° = (78 gmo1-1)/(0.87 Since A2 = [’a/M, X --

I

Eq. (4.59)

=

0.952 cm3 g-1 -3) = 8 9. 65 cm 3 mo1- 1 becomes

°FzV,

For Polymer A/benzene :

X --

-1 1 (55.5 cm3 g-l)(89.65 ctn 3 tool 2 (90,100 g mo1-1) (0.952 3 g- 1)2 0.44

PolymerMolecular Weights

t

1

0

I

1

0.4

I

0.8

257

1

i

1.2

I

I

1.6

I

I

2.0

!

2.4

-1 c~g dL Figure 4.7 Plot

of

(II/c) 1/2 versus e (Problem 4.10).

The practical range of molecular weights that can be measured by membrane osmometry is approximately 30,000 to one million. The upper limit is set by the smallest osmotic pressure that can be measured at the concentrations that can be used with polymer solutions. The lower limit, on the other hand, depends on the permeability of the membranetoward low-molecular-weight polymers. For measurements of Mn less than 30,000 another technique knownas vapor-phase osmometrydescribed next is more suitable. Vapor-Phase

Osmometry

Vapor-phase osmometry is based on vapor pressure lowering, which is a colligative property. The method therefore gives Mn. CombiningR.aoutt’s law and Dalton’s law we have P = P1 + P’2 = xl.P1 ° q- xzP2 (4.60) where /9 is the vapor pressure of the solution; P1 and P~ are the vapor pressures, respectively, of the solvent and solute in solution; P1° and /2.20 are the pure component vapor pressures of the solvent and solute, respectively; zl and z2 are the mole fractions, respectively, of the solvent and solute in solution. Whenthe solute is nonvolatile, as is the case with high-molecular-weight polymers, the vapor phase consists only of solvent. Therefore Eq. (4.60)

258

Chapter 4

reduces to

P = xlP~ °= (1-x2)P~

(4.61)

Rearranging,

x2 = 1 p_ p~o_p

p1 o P1o

_ -AP op1

(4.62)

Here Ap is the vapor pressure lowering given by difference between the vapor pressure of the solvent above the solution and the vapor pressure of the pure solvent at the same temperature. Rewriting Eq. (4.52)

Ap °P1

- -x~ =

W~/M~ (W1/M1)(W2,/M2)

(4.63)

where W and Mare weight and molecular weight, respectively, with the subscript 1 used for solvent and 2 for solute, we obtain for dilute solutions

, P~ .~o = -(W~/M1) - W~M~- (a .6a) in which ~o is the molar volumeof the solvent and c2 is the concentration (massNolume) of the solute. The expanded virial form of Eq. (4.54) vapor pressure lowe~ng is

°P1 [cf. Eq. (4.54)] where B and ~ are virial coe~cients. As ~P is inversely proportional to molecular weight of the solute, its magnitude for pol~er solutions is quite small and diNcult to measure direr. For example, at 10 ~ concentration in benzene, a polymer with Mn equal to 20,000 produces a vapor pressure lowering of about 2x10-a mm Hg at room temperature. The limits of a~uracy of pressure measuremen~are about half this value. It is therefore more accurate and convenient to conve~this vapor pressure difference into a temperatur e difference. This is accomp~shed in the method called vapor-phase osmomet~. The method is also ~lled vapor-pressure osmome~or more accurately the~oelec~c differen~al vapor pres~re lowe~ng. Ideally, the temperature difference ~T, co~esponding to the vapor pressure difference ~P in Eq. (4.65), ~n be deduced from the ClausiusClapeyron equation

~P RT ~

°P1 ¯ where ~H~is the latent heat of vapo~zation of the soNent at temperature ~ Combining Eq. (4.66) with Eq. (4.65) we obtain

~ _ ~° + ~ + c4 + ...

(4.~)

PolymerMolecuMrWei~ht~

25~

Thus, in theory, the molecular weight of the solute can be determined by measuring AT/c2 and extrapolating this ratio to zero c2. Practical

Aspects

In the vapor-pressure osmometer, there is no membrane. Two matched thermistors are located in a thermostated chamber that is saturated with solvent vapor (Fig. 4.8). With a hypodermicsyringe a drop of solution placed on one thermistor and a drop of solvent on the other thermistor. The solution has a lower vapor pressure than the solvent at the same temperature, and so the solvent vapor condenses on the solution droplet. The solution droplet, therefore, starts getting diluted as well as heated up by the latent heat of condensation of solvent condensing on it. Owingto the temperature rise and increased dilution, the vapor pressure of the solution droplet increases steadily. The process of condensation and the resultant temperature rise continues till the vapor pressure of the solution droplet at the new elevated temperature becomesequal to that of the pure solvent at the original temperature. In a steady state, the total rise in temperature AT can be related by an analog of Eq. (4.67): AT _c2 ks [~-----~

+ Bc~ + Cc~ q- - - . ] (4.68)

where ks is an instrument constant. It cannot be calculated a priori. In practice, the apparatus in use is calibrated for a given solvent, temperature, and thermistor pair, by using solutes of knownmolecular weight. The two thermistors on which the solution droplet and the solvent droplet are placed are arranged in an Wheatstone bridge circuit in such a way that the temperature rise can be measuredvery accurately as a function of the bridge imbalance output voltage, AV. The operating equation is AV K -+ KBc (4.69) c M [cf. Eq. (4.67)] where K is the calibration constant. A plot of AV/c versus c (where c is the solution concentration) is made and extrapolated to zero concentration to obtain the ordinate intercept (AV/c)c~o. The calibration constant f can be computed using the equation K = M(AV/c)c_o (4.70) where m is the molecular weight of the known standard sample. For determining the molecular weight of an unknownsample, solutions of the sample ,are made in different concentrations in the same solvent used for the standard sample and the whole procedure is repeated to obtain the ordinate intercept (AV/c)c__O. The molecular weight of the unknown sample is then given by ~,~ = K/(AV/c)c_+o (4.71)

260

Chapter 4

Gatvanometer

~ ~ ~ .... /’~ ~",-,~

II ~

~- Synchronous rectifying

Syringe guide Chamber Thermal. block

_...--~ Thermistor. beads

~

Figure 4.8 Schematic diagram of a vapor phase osmometer.

Problem 4.11 Following are the vapor phase osmometry data for a standard polystyrene of known molecular weight and an experimental sample of hydroxyl terminated polybutadiene (HTPB) in toluene solutions at 70°C. Calculate the molecular weight of HTPB. Polymer Standard polystyrene of M,, = 1800

HTPB of unknown molecular weight

Concentration, c (WL) 6 9 12 15 6 9 12 15

Bridge output, (#V) 107 164 224 287

AV

85 129 176 ,225

Answer: The data in columns 2 and 3 are plotted as AV/c versus c both for the polystyrene standard (Fig. 4.9a) and for HTPB(Fig. 4.9b).

261

Polymer Molecular Weights Polystyrene standard (Fig. 4.9a): M,~ = 1800

(zxy/c)o_ o = 16.95 K = 1800×16.95 = 30,510 HTPBsample (Fig. 4.9b): (AV/c)¢_~o = 13.45 ~,~

= K __ 30,5113 = 2268 13.45 ( A V/c) c-,o

Since there is some question as to whether the calibration constant K is independent of the molecular weight of the calibration standards, the safest procedure is to use calibration standards that are in the same molecular weight range, more or less, as the unknown materials that are to be determined. [Molecular weights of polymeric species that are based on calibrations involving low-molecular-weight compounds, like benzil and hydrazobenzene, may be erroneously low in some vapor-phase osmometers. Fortunately, the low-molecular-weight anionic polystyrenes, which are usually used as gel permeation chromatography standards (see later), are also suitable for vapor-phase osmometry standards.] The upper limit of molecular weights for vapor-phase osmometry is considered to be 20,000. Development of more sensitive machines has extended this limit to 50,000. The method is convenient and relatively rapid, although multiple concentrations and extrapolation to infinite dilution are still required.

2o _ (a) Potystyrene standard

(b) HTPBsampte

17 16 0

~ ~.

~ 8

I

I

I

12 16 0 h Concentration, c (g -1)

Figure 4.9 Plots of AV/c versus c (vapor-pressure 4.11.)

I

8

I

I

12 16

13

osmometry data of Problem

262

Chapter4

Light-Scattering

Method

Introduction The measurement of tight scattering by polymer molecules in solution is a widely used technique for the determination of absolute values of weight-average molecular weight, Mw. This technique, which is based on the optical hetero~neity of polymer solutions, was developed by Debye in 1944. Besides Mw,light-scattering measurements can give information about the size and shape of polymer molecules in solution and also about parameters that characterize the interaction between solvent and polymer molecules. The experimental technique of light scattering is, however, exacting. This is mainly because of the considerable difference in magnitude between the intensity of the incident beam and that of light scattered by the polymer solution. Intensity

of Scattered

Light

: Rayleigh

1Latio

In order to understand the basic phenomenaof light scattering by polymer solutions one must start by considering the scattering of light by a gas. A theoretical treatment of the scattering of light by the molecules of a gas was given by Lord Rayleigh in 1871. A plane-polarized beam of light can be regarded as consisting of a periodic electric field and a periodic magnetic field, perpendicular to each other and to the direction of propagation, each at a given point varying sinusoidally with time. For example, if the light wavetravels in the z-direction, then the electric and magnetic vectors would vibrate in y and z directions, respectively (Fig. 4.10).

--Instantaneous electric field vector

In.cident

--Magnetic field vector Figure 4.10 Light shownas a transverse wave.

PolymerMolecular Weights

Focussed monochromatic light beam

I

263

Samptecet[ [

\

\/A’I ° / \\ --x ~

/ ..... ~ ~.

r o e:90 ~=~U

Transmitted beam

I e--oo ~ ia / = Movabte ~ / photocetl connected to /~.~ ~~ gatvanometer

Figure 4.11 Main features of the apparatus required to measure the light scattered from a polymersolution.

According to classical theory, the electrons and nuclei in the particle oscillate about their equilibrium positions in synchronywith the electric field of the incident radiation. If the incident light waveis being transmitted along the x-direction and the electric field vector is in the y-direction, then a fluctuating dipole will be induced in the particle along the y-direction. This dipole is the source of the scattered radiation and will act as a secondary source, radiating energy in all directions. Since the dipoles oscillate with the same frequency as the incident light, the scattered radiation from these dipoles also has the same frequency (and hence wavelength) and is referred to as elastic (or Rayleigh) scattering (i.e., with zero energy change). Almost all of the scattered radiation is of this type. A small amountof the scattered radiation, however, has a higher, or lower, wavelength than the incident radiation and arises from inelastic (or Raman)scattering (i.e., with nonzero energy change). Such inelastically scattered light carries information relating to bond vibrations and is the basis of Ramanscattering. However, in this section it is the elastically scattered light from moleculesthat is of interest, since it enables the molecular weight to be determined. A’schematic diagram that shows the basic features of the apparatus required for a light scattering experiment is given in Fig. 4.11. For a dilute gas, the intensity of scattered light as a function of scattering angle 0 is given by the Rayleigh equation 2ior 27r 2 (~-l)2M (4.72) = P,4 -/k4NAv c Io(1 + cos2 0)

Chapter 4 In this expression, Io is the intensity of light of wavelength A incident on the system; iO is the intensity of scattered light per unit volume of the system that is detected at angle 0 to the incident beam direction and at a distance r from the center of the system; Mis the molecular weight of the gas; ~ is the refractive index; c is the density or concentration of the gas in mass per unit volume, and -]VAv is Avogadro’s number. The term /~0 is called the Rayleigh ratio. It is the reduced relative scattering intensity defined by 2io r (4.73) Ro = Io(1+cos20). and is independent of both r and 0 on the basis of the Rayleigh equation (4.72). The Rayleigh ratio is sometimes defined as the quantity i90 r2/Io, which is the reduced scattering intensity at 0 = 90°, that is, Rg0. Equation (4.72) shows that the scattered .intensity is inversely proportional to the fourth power of the wavelength of light used (A). Thus, the shorter wavelengths are scattered more intensely than longer wavelength light. (This is whythe sky is blue.) More intense signals can therefore obtained by using light of shorter wavelength. Serious difficulties arise in extending the Rayleigh theory_ of light scattering in dilute gases to liquids. The main reason for these difficulties is that, unlike in gases, strong intermolecular forces are present in liquids. However,these difficulties could be avoided by a completely different approach to light scattering in liquids, workedout by Einstein [41, in which the scattering is considered to arise from local fluctuations in the density due to the thermal motions of the molecules. The density fluctuations lead directly to local fluctuations in the refractive index and hence to scattering of the incident light. Einstein’s expression for the Rayleigh ratio of a pure liquid is given by

Ro = io(l

2 ~ dfi ior RT ’’ / ~ --/7~ +cos~O) - 27r 14N,~v /3 \ dp]

(4.74)

where p is the hydrodynamic pressure on the liquid and /3 is the compressibility, that is, /3 = -(1/W )(OV/Op)T. The other terms are as described earlier. High-molecular-weight polymers do not usually exist as liquids but they can often be dissolved in liquid solvents. The presence of the dissolved solute causes additional scattering of light by a solution (over and above that of the pure solvent). However,the lack of a theoretical treatment of this additional scattering in liquid solutions held up the use of light scattering measurements to determine the molecular weights and sizes of polymers until in 1944 when Peter Debye [5] put forward such a treatment. Debye pictured the additional scattering of light by a solution to result from local fluctuations in the concentration of the solute. In a treatment analogous to that of Einstein for the density fluctuations in pure liquids, Debye considered the local fluctuations in solute concentration, due to random thermal motion, to be opposed by the osmotic pressure of the solution. With

PolymerMolecular Weights

265

this modelhe derived the expression for the Rayleigh ratio of the scattering due to solute as R 0 ~-io(1 -Jl-cos20) -- ,~4NA v ~,n°~c (c~l-i/c~c.)T r2 ~ RTc , i~o 2~r 2 [~ ~ dh~ (4.75) Here the prime denotes the excess scattering from the liquid due to the solute; h and ~? are the refractive indices of the solution and solvent, respectively; d~/dc is the specific refractive index increment with concentration; c is the concentration of solute in mass per unit volume;l-I is the osmotic pressure of the solution, and the other terms are as described previously. R~, commonlyknownas the excess Rayleigh ratio, is the difference between the Rayleigh ratios of the solution and the pure solvent: Rj = R0(solution)R0(solvent) (4.76) The excess Rayleig_h ratio R~ can be derived from the "raw" galvanometer readings Ig and lgs when the sample container contains the solution and the solvent, respectively, with the photocell of Fig. 4.11 positioned at an angle 0 in both cases. The equation used is

RJ = k ( Ig - ) sin0

(4.77) 1 + cos 9 0 The value of k can be obtained from the

where k is the conversion factor. equation k Ig8 sinO Ro e0 - 1 + cos

(4.78)

using Igs measured at an angle 0 and knownvalue of the Rayleigh ratio ~ for the solvent. [Note that ./g and Igs are multiplied by sin 0 to correct for the unavoidable variation of effective scattering volumeas 0 is varied. Multiplication by sin 0 actually computesthe value the galvanometer would have read, had the "viewed" volume remained constant. This is made clear in Fig. 4.12. Turbidity As a result of the interaction of light and a scattering particle someof the energy which was associated with the incident ray is radiated in directions away,from the direction of incident radiation. Thusthe intensity of incident light is diminished by the amount radiated in all other directions by the dipoles in the particle. The total amount of scattered light can thus be deduced from the decrease in intensity of the incident beam. If Io and I are the intensities of the beambefore and after passing through a length/~ of the medium,they can be related in terms of Beer’s law for the absorption

266

Chapter 4

Figure 4.12 The variation of effective scattering volume(shaded) with angle. A, incident beam; B, polymersolution or solvent; C, scattering cell; D, light trap; E, scattered light beam; F, movablephotocell connected to galvanometer.

of light as follows:

I _ e_r~

(4.79)

Io

where 7- is the measure of the decrease of the incident beam intensity per unit length of the mediumand is called the turbidity. It has the dimensions of reciprocal length. Wemay write Eq. (4.79) in the form

e_r, _ Io- L_Io- I’/ Io Io

- 1 I’/ Io

(4.80)

where Is is the total intensity of light that is scattered by the solution and Is ~ is the total intensity scattered per unit path length. The fraction of light scattered is generally very small and to a good approximation the exponential in Eq. (4.80) can be expressed

... 1- e (4.81) A combination of Eqs. (4.,80) and (4.81), with neglect of higher powers ~-£, leads to

r = I’~/Io

(4.82)

267

PolymerMolecular Weights

as the relationship between the total intensity of scattered light per unit path and the turbidity. The total intensity of light scattered per unit path length through the solution is the intensity scattered through all angles of polar coordinates, or

s: = fo" fo sin 0 d0

Since r 2 i~ = IoR~ (1 + cos 2 0 ) [of. Eq. (4.75)],

Io

(4.83) Eq. (4.83) becomes

- v = fo’fo2"R~(l+cos20)sinadOd¢

(4.84)

Substituting for R~ from Eq. (4.75), Eq. (4.84) can be written vA4NA v (_

de

d") 2 RTc

(4.8~) Thevalueoftheproduct ofthedefinite integrals is16~’/3, sothatweobtain therelationship between theturbidity andth~totalscattered intensity as (_ 2dh~

~-- So- \3~-z~-~)(~rI/oo)r

(4.s~)

This expression provides us with a wayto determine molecular weights of polymersfrom light scattering of polymersolutions as shownbelow. A comparisonof Eqs, (4,86) and (4,75) showsthat the relationship betweenthe turbidity and the Rayleighratio is

~- ~0 =

Zo(~ +~o~0)

This expression enables us to determine the turbidity the intensity of light scattered at given angles. Turbidity

and Molecular

Weight

of

by measurement of

Polymer

The connection between turbidity and the molecular weight of a solute may be obtained by substitution of (OI~/OC)Tinto Eq. (4.86). Representing the osmotic pressure for a monodisperse (i.e., single molecular weight) solute by a virial equation [cf. Eq. (4.52)] in the form

where -42 and Aa are the second and third virial coefficients, differentiation

we obtain by

268

Chapter 4

Substituting in Eq. (4.84) we obtain 7-

= k~]

C ~no--~c)

/

(~--+2A2c+3Aac

Rearranging we have

2+...

) (4.90)

He 1 -+ 2A2c + 3A3c ~ + ... ",M where the function H is a lumped constant given by H = k,3A--U-NN~v / ~no-~-~c )

(4.91)

(4.92)

The procedure for determination of molecular weight according to Eqs. (4.91) and (4.92) is therefore as follows. Take a series of polymer solutions (dust free) of different concentrations. Choose a particular angle to the incident beam (usually 90°) and determine excess Rayleigh ratio R~ (or excess scattering intensity i~), and hence turbidity from Eq. (4.87), each solution. Calculate H from Eq. (4.92) using measured values of and ~o. Evaluate Hc/’~ for each concentration and plot this against c~ By extrapolation of the linear portion to zero concentration the intercept on the Hc/~- axis gives 1/Mdirectly, while the initial slope yields the second virial coefficient. Problem 4.12 Given below are typical light scattering data for solutions of polystyrene in benzene(fi, o = 1.5130) with ,k = 4360~ at 25°C: Conc. (g/lO0 cm~) (~ - ~o) ×102 r×lO ~ -t) (cm 0.1778 0.020 0.091 0.3708 0.041 0.142 0.6244 0.069 0.187 0.7736 0.206 0.086 1.0230 0.113 0.228 Determinethe molecular weight of the polymer. Answer: The value of the refractive index incrementdfz/dc is neededat infinite dilution, but there is little concentration dependencein the normal concentration range used for light scattering of polymersolutions. The required value can therefore be obtained from ~ - ~o c

Aft c

d~ dc

Polymer Molecular Weights

269

Therefore Eq. (4.92) can be written H = \3A4NAv] / 32"n’3 ~o-~

(P4.12.1)

[The value of (d~/dc) can also be obtained from the plot of (~ - ~o) against and Eq. (4.92) can be directly used.] For c = 0.1778 g/100 cm3, using Eq. (P4.12.1): H

32(3"143)3 3(4360 x 10-s cm)~ -1) (6.02 x 1023 tool -6 = 4.41 x 10 tool cm2 g-2

(0.02 ×z) -210gcm (1"513)2 I [(0.17-~

=

So~

(4.41 x 10-8 tool cm2 g-2) (0.1778 x 10-~ -3) g cm

Hc

(0.091

~-

X 10 -2

-1) cm

= 8.61 x 10 Similarly, Holt values are calculated for other c values and are plotted against c in Fig. 4.13. From the intercept at c = 0, 1 M

(H..~__~) = 6.7110 -6 mol g-1 c--0 M = 1.49 × 10 ~ g mo1-1

24

2O

(T)¢-o 0

0.2

0. t"

0.6

0.8

1.0

1.2

cx 10 2(g 3) cn~

Figure4.13 Light-scattering in benzene (Problem 4.12).

determination of the molecular weight of polystyrene

270

Chapter 4

It is also customary to define another optical Eq. (4.92)1

K-

constant K such that [cf.

3H I 16~r - \MNav] ~n°-~c)

It is easy to see from Eqs. (4.87),

Kc 3Hc R~ - 167rR~

(4.91),

(4.93)

and (4.93)

Hc - T --

1 M +2A2c+3A3c 2+ --(4.94) Hence M and A2 ~n be dete~ined from a plot of Kc/R~ agMnst c in the same way as from the plot of Hc/v versus

c using Eq. (4.91).

Problem4.13 A solution

of polystyrene in benzene at 25°C used for light scattering experiments with light of wavelength 4358 )1 has the following values: h = 1.5130, dh/dc = 0.111 cm3 g-1. Determine the optical

constant K for the system.

Answer: From Eq. (4.93) 2 (3.143) ~ (1.5130) ~ (0.111 cm3 (6.023 × 10z3 mo1-1) (4358 × 4-s = 2.564 × 10 -T cm2 g-2 mol (It may be noted that K is a constant for a independent of the concentration and molecular K

=

g-1)2 cm) polymer-solvent weight.)

Lystem and is

Problem4.14 Measurements of excess scattering

for a solution of polystyrene in benzene at 25°C with light of wavelength 4358 ~. 6elded the following data 3C, gflO0 cm -1 ~ × 105, cm

0.132 3.88I

[ 0.197 15"44

] 0.296 7.63

0:445 10.1

0.667 13.3

Determine (a) the molecular weight of the polymer and (b) the second virial coefficient for this particular polymer-solvent system. (The value of K for the polymer-solvent system is 2.564x10 -T cm2 g-2 mol.) Answer: cm3 = 1.32×10 -3 -3, gcm -’ ~ (2.564 × 10 ~m g-~ tool)(1.32 -~) (3.88 x 10-~ cm -~ 8.72 × 10 tool g-~

For c = 0.132 g/100 gc P~0

× ~0-~ -3 g )~m

= Similarly, Kc/R~ values for other concentrations (c) are calculated and these are plotted against c in Fig. 4.14.

271

PolymerMolecular Weights (a) Eq. (4.94) : Fromthe intercept of the graph Kc/I~o axis (Fi g. 4.1 4), M

~-0 -1 M = 1.31 x 105 g tool

(b) Eq. (4.94): Fromthe slope of the graph (Fig. 4.14), 2A2 = 8.0x10 -4 3molg-2cm Therefore, A2 = 4.0 x 10-4 atool g-2 cm Eq. (4.55): ~other fo~ of the second vidal ~efficient F2 = A2M = (4.0x10 -4 molg-ecm a)(1.31x10~

-~) gmol

= 52 ~3 g-1 The molecular weight obtained by application of Eq. (4.94) to a polydisperse polymer will be some average over the molecular weight distribution characteristic of the polymer. It can be easily shown that the molecular weight determined from light scattering measurement is Mw.

15 13

Intercept =7.6x1(~6mol g-1

I

0

I

0.2

I

I

I

I

O.Z. 0.6 -3) c x 102(gcrn

I

I

0.8

I

1.0

Figure 4.14 Plot of Kc/l~o versus c in the absence of interference effects (Problem4.14).

27"2

Chapter 4

Problem4.15 Show that the molecular determined by the light scattering

weight of a polydisperse polymer method is a weight-average molecular weight.

Answer: To see what kind of average molecular weight we measure by light scattering of a polydisperse polymer, we examine the contributions made by individual species to the turbidity ~-. For a solution in the limit of infinite dilution, Eq. (4.91) becomes "c = HeM If the solute molecules are independent agents and contribute additively the observed turbidity, one can write

and

to

c = ~ ci

where ~’i, ca, and Mi refer to the turbidity, weight concentration, and molecular weight of monodisperse species i, which is one of the components of the mixture that makes up the polymer sample.__ The average molecular weight Mis obtained from the overall turbidity ~- and concentration e: --~

= r

Hc

_ HZc~M,

H E ca

_ Zc,

M,

(P4.15.1)

E c~

Since ca = N~Mi/V, where Ni is the number of moles of spedes i having molecular weight Mi in volume V of solution, it is seen on substitution in Eq. (P4.15.1) that

EN, M~

Polymer Dimensions of Scattering Intrapartlcle

and Corrections

for

Dissymmetry

Interference

Equation (4.94) is derived on the assumption that each solute molecule small enough compared to the wavelength of the incident light to act as a point source of secondary radiation, so that the intensity of the scattered light is symmetrically distributed as shown in Fig. 4.15. This condition is satisfied by vinyl polymers having a degree of polymerization less than about 500. If any linear dimension of the solute particle is as great as about A~/20 or greater, as for large size polymer molecules, then the secondary radiations from dipoles in various regions of the scattering molecule may vary in phase at a given viewing point. (Note that t i s t he wavelength o f the light in solution. For dilute solutions it is equal to A/~o, where ~o is the refractive index of the solvent.) The resulting interference will depend on the size and shape of the molecule and on the observation angle.

273

Polymer Molecular Weights

Problem 4.16 Using light of wavelength 5461 ~ and benzene as solvent (refractive index 1.5014), what is the limiting size (root mean square end-to-end distance) of a polymer coil above which the molecule can no longer be "regarded as a point source in refraction. Answer:

(~/~o) < r 2 >1/2 = "__ = 20 20 5461 ~4 -= 182.74 20 x 1.5014 This represents the critical value of the diameter or length of the scattering molecule. Limiting

\

~_, t-- Locus Scattering iden~ica[ dipo[e

(b) ° 90

\

\\ \ \

135~

Figure 4.15 (a) Scattering envelope for point scatterer

with unpolarized incident light. (b) Scattering by a random coil which is comparable in size with the wavelength of the incident radiation. (c) A scale diagram showing the effect of interference. Curve A shows a two-dimensional scattering envelope from a polymer solution and curve B shows the scattering envelope which would have resulted from the same solution in the absence of interference. Distance from the scattering particle to the boundaries of the envelope represents the magnitude of scattered light as a function of angle. The scattering envelopes are cylindrically symmetrical about the direction of the incident light.

274

C1aapter4

The general effect of intraparticle interference can be illustrated with reference to Fig. 4.15(b), in which scattering by a random coil which is comparable in size with the wavelength of the incident radiation is shown. The scattered light received at the plane AB(zero angle) is in phase no matter what part of the polymer molecule is acting as the source of the secondary radiation. However, as 0 increases, there occurs an increasing path difference between the light received from the different parts of the molecule, and so the intensity of the scattered radiation is reduced because of destructive interference due to phase differences between the scattered rays. This interference effect will moreoverbe greater the larger the scattering angle 0, and consequently the radiation envelope will not be symmetrical. Both the~e effects are illustrated in Fig. 4.15(c). The envelopes are cylindrically symmetrical about the incident ray in both cases of interference and noninterference but the envelope in the presence of interference effects is no longer symmetrical about a plane through the scatterer and normal to the incident radiation. Thus the scattering is less in the direction for which 0 = 90°-t-fl than for 0 = 90°-/~. This effect is called dissymmetry. The observedratio of these intensities is usually referred to as the dissymmetrycoefficient [6] designated by z~, i.e. z;~ =

.~ Z90o+~

--

.~ ZTr--0

--

/~

7r--0

The root-mean-square distance. (r2) 1/2 between the ends of the polymer chain is a convenient measure of the diameter of the randomly coiled polymer molecule. The dissymmetry Coefficient z~ will be unity for (T2}1/2 < ,~t/20 and will increase as (7"211/2 increases. In general, macromolecules with a linear structure in good solvents and with molecular weights between 105 and 107, have diameters between 200 and 3000 N. In light scattering measurements using mercury arcs (,~ = 4,000-5,000 ,~) the source of light, the particle diameter is therefore larger than/V/20. Intraparticle interference effects, described above, diminish as the viewing angle 0 approaches zero degrees to the incident light and at zero angle the scattering is unperturbed so that Eq. (4.94) can be used for determination of molecular weight. Laser light scattering photometers are now commercially available in which scattering can be measured accurately at angles at least as low as 3° off the incident beam path. The optics of older commercial instruments that are still in wide use are restricted to angles greater than about 30° to the incident beam, and for this reason it is often preferred to apply appropriate corrections to scattering intensities measured at such large angles. It is seen from Fig. 4.15(c) that (i) the forward scatter exceeds backwardscatter, that is, the dissymmetrycoefficient, z, exceeds 1, and (ii) the forward scatter at zero angle (0 = °) i s t he s ame w hether i nterference effects are present or not. It is therefore easy to decide whether interference effects must be considered or not from experimentally observed values of /~ and ~r-0, if the two quantities _~ and R~t 0 differ significantly from one another, interference effects are present. These effects may be taken

275

PolymerMolecular Weights

into account by defining a parameter P(O ), called the particle scattering factor. It is simply the ratio of the scattering intensity to the intensity in the absence of interference, measured at the same angle 0: "; (~0)observed

( .~!O)obsewed

P (0)

i.,¢reorence =

io,or or.ce

(4.95)

The R~ in our previous equation (4.94) is (R~)no interference, since their derivation the scattering molecules have been considered as point sources. Thus Eq. (4.94) may be rearranged in terms of the observed and P(O) as Kc Kc !

P (0) (R~)nointerference + 2A~.c + ~A~c~ + ...

= --~ 1-’o

(4.96)

According to Zimm, the following approximation may be written Kc

1 -+ 2A2c (4.97) R~ "r M.P(O) (omitting the subscript ’no interference’ for simplicity) Since, as noted above, the forward scatter at zero angle is the same whether interference effects are present or not, t 9 (0) = 1 at 0 0% Therefore, if Hc/~- or Kc/R~measured at different values of 0 and c is extrapolated to both 0 = 0 and c = 0, Eq. (4.97) reduces --

Hc

--

e-~0, 0-~0

c-~0, 0-~o

M

The molecular weight of the solute can thus be obtained without making any assumptions about the shape of the polymer molecule. Scattering

Factor

Analytical expressions for the particle scattering factor P (0) are required so that an appropriate extrapolation to 0 = 0 can be performed in order to eliminate the effects of interference. Debyeshowed that the functional form of 19(0) depends on the shape of the scatterers. The following scattering functions have been derived for three models in terms of a polymer dimension, the wavelength of the light, and the refractive index of the solvent [7]. I.

Randomcoil polymer P(O) --

2 u2 [e-"

- (1-

u)]

(4.99)

276

Chapter4 where 2) u = -~

sin~

(s

(4.100)

and @2) is the mean square radius of gyration of the polymer molecule (see p. 61). It will be recalled that the radius of gyration is related to the root-mean-square end-to-end distance (r 2) by (82) 1/2

-

~

(4.101)

@2}1/2

Substituting this to replace @2) by @2) in Eq. (4.100) leads u 2.

=

2

(~)(~)(27rsin~)

(4.102)

Spherical polymer molecules P(O)

= ~-5

(sinv

- vcosv)

(4.103)

where (2~rd~ v = \ ~, ] sin~ 3.

(4.104)

Rigid-rod polymer

P(O)= -~ s(2~)- sinx where

(4.1o~)

z sin Y dy

fo

S(x)

Y

x

=

sin ~

L

= length of rod

In Fig. 4.16 the particle scattering function P(O) is plotted for random coils, spheres, and rods as a function of respective parameters. Nowconsidering the important case of random coils, when u << 1, one can approximate Eq. (4.99)

P(O)= 1 - u/3

(4.107)

Since 1/(1 - z) ~ 1 + x when x << 1, then from Eq. (4.107)

VP(o)= 1 + ~/~

(~.~0s)

277

PolymerMolecular Weights

1.0

0.8 Rods 0.6 )heres

P(e) 0.4

F[exib[e Coil.s

0.2

0

1.0

2.0

3.0

211"d ¯ e /’2"ILL . e\ -- },, sm’~")spheres ,.~’~F" sin ~’)rods (V’~)crao~tdsom’( Fi.~__re 4.16 Particle scattering factor P(O) as a function of V’-~, that is, V/2/S((r2)1/2 / A’)[27rsin(0/2)] [cf. Eq. (4.102)], (27rd/A’)sin(/9/2) for spheres of diameterd, or (2~rL/A’)sin(/9/9.) for rods of length L. (After Ref.

Substituting for u from Eq. (4.102) gives O (8~r2 1/P(O) = 1 + ~,9X,~] (r2> sin2~

(4.109)

Substituting this in Eq. (4.97) one then obtains

R~ - "r

- M+

~9A ’~)

(r2)sin~(0/2)

+ 2A2c (4.110)

o~,i~ o~~u~stit~t~ ~o~t~¢Cx~s~ion [(VM)(S~/9~’~)(~)] t~¢ S, one obtains:

Chapter4

278

Kc/R~

= HC/T = (l/M)

This equation contains concentration c (for a (if the concentration is ments have demonstrated

Zimm

Ss in2(O/2) +

2A 2c (4 .111)

a linear dependence of the factor Kc/R~ ~n the constant angle of observation) and on sin (0/2) held constant). Numerouslight-scattering measurethe applicability of Eq. (4.110).

Plots

The double extrapolation to zero 0 and zero c required for the use of Eq. (4.104) is effectively done on the same plot by the Zimm method [9]. The rationale for this method follows from the functional form of _P(0), described above, which shows that the ratio of the observed scattering intensity at an angle 0 to the intensity which would be observed if there were no intraparticle destructive interference is a function of the parameter sin2(O/2). Zimm~lots consist of graphs in which Kc/_R~ (or He~r)is plotted against sin (0/2) + bc, where b is a constant arbitrarily chosen to give an open display of the experimental data, and to enable the two extrapolations to c = 0 and 0 = 0 to be carried out with comparable accuracy. (It is often convenient to take b -- 100.) In practice, intensities of scattered light are measured at a series of concentrations and at several angles for each concentration. The Kc/R~ (or He/r) values are plotted, as shown in Fig. 4.17. The extrapolated points on the 0 = 0 line, for example, are the intensities of the lines through the Kc/R~ values for a fixed c and various 0 values with the ordinates at the corresponding bc values. Similarly, the c = 0 line is drawn through the intersections of the lines through the Kc/R~ values for a fixed 0 and various c values with the corresponding sin2(0/2) ordinates. The 0 = 0 and c = 0 lines intersect on the ordinate and the intercept equals 1/m.

Problem 4.17 Scattering from benzene and a series of polystyrene solutions was measuredwith a light-scattering photometer. The resulting experimental data [9], given in Table P4.17.1, consist of a series of galvanometerreadings ./’g and lgs, for solution and solvent, respectively, with a photomultipliersituated at the various angles shownto an incident beamof unpolarized, monochromaticlight of wavelength 5461 ~. Determinethe molecular weight of the polymer by constructing a Zimmplot. Use the values: ~.o = 1.5014, d~/dc = 0.106 cma g-l,. /~0o of benzene = 16.3 x 10-8 -x. cm

Polymer Molecular Weights

279

Table P4.17.1 Values of Ig and Ias at various c and 0 c 0 30 ° 45 ° 60 ° 75 ° 90 ° 105 ° 120 ° 135 ° °150 3 g/cm -3 440 2.0 × 10 1542 917 607 461 408 540 755 1235 -3 x 10 1.5 1383 820 550 413 363 384 475 660 1080 -3 x 10 1.0 1158 682 455 343 301 319 392 540 880 -3 339 464 755 0.75 x 10 998 590 396 297 263 275 -3 607 0.50 × 10 803 477 319 241 214 224 275 376 Pure benzene 282 170 128 105 100 105 127 170 285 Data from Ref. 10. AllswQl" :

From Eq. (4.78): R90o = k (fgs)9oo k - /~oo

_ 16.3

× 10 -6 cm-1 = 16.3

100

× 10 -8 -t cm

The values of ex~ss Rayleigh ratio R~ calculated from Eq. (4.77) using the above value of k and data of Table P4.17.1 are re~rded in Table P4.17.2. Table P4.17.2 Values of _,~xl06 in cm-1 calculated from data in Table P4.17.1 30 ° 45 ° 60 ° 75 ° 90 ° 105 ° 120 ° 135 ° °150 -3 2.0 x 10 -3 1.5 × 10 -3 1.0 × 10 -3 0.75 x 10 -3 0.50 × 10

58.7 51.3 40.8 33.3 24.3

57.4 49.9 39.3 32.3 23.6

54.1 47.6 36.9 30.2 21.6

52.5 45.4 35.1 28.3 20.1

50.2 42.9 32.8 26.6 18.6

49.4 41.2 31.6 25.1 17.5

46.6 39.3 29.9 23.9 16.7

44.9 37.6 28.4 22.6 15.8

44.2 37.0 27.7 21.9 15.0

From Eq. (4.93): K

2 (3.143) 2 (1.5014) 2 (0.106 cm3 g-l)2 4(6.023 × 1023 mo1-1) (5.461 × 10-5) = 9.34 × 10 -8 cm2 g-2 mol

=

The values of Kc/.R~ calculated using the aforesaid K and data in Table P4.17.2 are recorded in Table P4.17.3. A Zimmplot obtained from these data is shown in Fig. 4.17.

280

Chapter 4

Table P4.17.3 Values of (Kc/R~) × 6 in too l g-1 calc ulated from dam in Table P4.1?.2 ° c 0 30 ° 45 ° 60 ° 75 ° 90 ° 105 ° 120 ° 135 ° 150 3g/cm -3 2.0 x 10 3.18 3.25 3.45 3.56 3.72 3.78 4.01 4.16 4.22 -3 1.5 x 10 2.73 2.81 2.94 3.08 3.26 3.40 3.56 3.72 3.78 -3 1.0 × 10 2.29 2.37 2.53 2.66 2.85 2.95 3.12 3.29 3.37 .3 0.75 x 10 2.10 2.17 2.32 2.47 2.63 2.79 2.93 3.10 3.20 -3 0.50 × 10 1.92 1.98 2.16 2.32 2.51 2.67 2.79 2.95 3.11 From the intercept

of the e = 0 and 0 = 0 lines on the Kc/R~ axis (Fig. 4.17): = 1.37x

~" e--~0,0-*0 (Kc)

10-6 tool g-1

From Eq. (4.98): M -

1 1.37,x 10-6 g m°l-1 = 730,000 g mo1-1

Problem4.18 For the polystyrene

sample in Problem 4.17 calculate (a) the second virial coefficient, (b) the root mean square end-to-end distance, and (c) the root-mean-square radius of gyration. Answer: (a) According to Eq. (4.111), the limiting slope of the/9 = 0 line Kc/R~ versus c plot is 2A2. From the Zimmplot in Fig. 4.17, limiting slope = 8.75 × 10-4 mol g-2 cm3. Therefore, A2 = 4.375 x 10-4 3. mol g-2 cm From Eq. (4.62): Another form of the second virial

coefficient

x 10-4 mol g-2 cm3) -a) (7.3 = 3.2x 102 cm3 g-1

r2 = A2M = (4.375

x 105 g tool

(b) The root-mean-square end-to-end distance of the polymer chains is found from the limiting slope of the c = 0 line. From Eq. (4.111): Limiting slope of the c = 0 line =

~9--~,]

<

>

where From the Zimmplot in Fig. 4.17: slope = 1.26 x 10-6 mol g-~ <

t2

>

x -6 to ol g- ’)(7.3 8~ -~° ~ = 1.386 x 10 cm (9)(1.26

x 105 gmol -~) (5 461 x 10-8 cm~

281

PolymerMolecular Weights 2>1/2
=

1.18x10 -5 cm = 1180

(c) FromEq. (4.101): < r2 >1/2

2>I/2 <s

Dissymmetry

--

1180:4

= 482:4

Method

An alternativd method of determining molecular weight by light scattering measurements is the dissymmetry method, in which the value of P (0) for a particular angle-usually 90°-is obtained from measurements of the dissymmetry of the scattered envelope. Thoughless reliable than the double extrapolation (Zimm method) described above, the dissymmetry methodhas the greatest merit of simplicity and rapidity. A series of dustfree polymer solutions of varying concentration are prepared and for each solution, the values of R45, /r~90 and /~135 are obtained from which are subtracted the corresponding solvent values to give the excess Rayleigh ¯ I ! ratzos R45, Rgo, and R135. At each concentration, two quantities are calculated, namely, (i) the dissymmetry coefficient z45 = R45//~135, and (ii) Kc/R~o. By extrapolation to zero concentration, the limiting values of z45 and K-~//~o at zero concentration are deteimined which then enable the particle scattering factor P(O), root-mean-square end-to-end distance < r 2 >1/2 and the molecular weight M to be found from available correlations for z45 (Fig. 4.18) and P (0) (Fig. 4.16).

Problem 4.19 Using the light-scattering data given in Problem 4.17 for a polystyrene sample with benzene as the solvent determine by the dissymmetry method(a) the molecularweight of the polymer,(b) size of the polymermolecule in solution, and (c) secondvirial coefficient for the polymer/solventsystem. Answer: Takingthe /~ data of TableP4.17.1 the various values of z45 at the five different concentrations are calculated as R~5/R~3 ~. .These are shownin Table P4.19.1. Table P4.19.1 Dissymmetry coefficient versus concentration for polystyrene sample in Problem4.17 -3 c × 10 3 gcm 2.0 1.50 1.0 0.75 0.50 Z45 = R~5/R~3 1.278 1.327 1.384 1.429 1.494 ~ Theplot of z4s against c is usually curved, but a graphof 1/(z,s - 1) against e is usually linear and hence moresuitable for extrapolation. Both these curves are shownin Fig. 4.19. By extrapolation,

(~,

1

~-~0

=

orIz45-o = 1.64.

282

Chapter 4

Correspondingto this value of dissymmetrycoefficient one obtains from Fig. 4.18 the ratio < r’z>l/2/A’ = 0.30. Having found < r 2 >1/2/~,, u may be computed angle is (4.100) then obtained from Eq.0° and (4.99) Fig. 4.16. It is found from Eq. for any angle the or particle-scattering factor inP this (0) atmanner that u(90°) = 1.185 and P (90) = 0.70. (a) The values of Kc/l~o from TableP4.17.1 are plotted against c in Fig. 4.20. On extrapolation of the straight line to c = 0 gives c--~0

_-

FromEq. (4.97) 1 ----- 2.04 x 10 MP(O) -1 Substituting P (90) = 0.70 obtained above then gives M= 700,000 g tool

’(b) Since’ = 0.30, (r2) ~/~ = 0.30,V

1.09x 10Scm = 1090 (c) FromFig. 4.20: slope = 2A2 = 8.33 x 10 -4 mol cm3 g-2 A2 = 4.16 X 10 -4 mol cm3 g-2 The agreement between the values of M, < r 2 >1/2 and A~ obtained by the dissymmetrymethodand the Zimmmethodare seen to be quite satisfactory.

Light-Scattering

Instrumentation

Light-scattering photometers essentially contain a light source, means for providing a collimated light beam incident on the sample, and means for detecting the intensity of scattered light at various angles from the incident beam path. Scattered light is detected by means of a photomultiplier. The sample cell is positioned on a precision goniometer, which enables the photomultiplier detector to be movedaccurately to a wide range of scattering angles. Cylindrical cells are the most advantageous since measurementscan be made at any angle. The ceil is correctly positioned in a holder and accurately located on the axis of the instrument. At any angle, solvent scattering is subtracted from solution scattering to ensure that the scattering that is taken into account in the molecular weight calculation is due to solute alone. Since io ~ -/o A-4 [see Eq. (4.74)], greatest sensitivity is achieved by using high-intensity visible radiation ideally of short wavelength. Modern instruments use monochromaticlaser radiation, e.g., He-Ne(A = 632.8 nm) and Ar ion (A = 488.0 nm or 514.5 nm).

Polymer Molecular

283

Weights

c = 0.002 o- ~c=0.002 c=0.0020=135

c=o.oo2 e=soo "S~---~/ o / /f¢--o.oo1~ ~/ Z.-~7"--~8=150 / c=o.oo2 ~=~so ~ / ~ / //¢=o.oo~o c=0.002 0=75 o ~/*/ :=0.002 e=60° "~:~’~-/

/ /

E 3.0

e=o~ ,’ ~

/

~/X

," / U~:o.o~s c~O

~J’~2.o

~

Constant

.~-’’~

tne

c=O

1.0

I

0.2

0

~

I

0.4

I

I

I

0.6 100c + ~in2e

I

0.8

I

I

I

~ .0

Figure 4.17 Zimm plot centration

units

for the polystyrene sample of Problem 4.17¯ The conemployed are g/cmz. The symbols o represent extrapolated points.

Randomcoils

Spheres

3 Z/.5o 2

1

o

I

1.2

0.2

o.~

o.6

o.~

~.o

L
of polymer size.

284

Chapter 4

1-7

5.0

1.6

1.5 3.0 z 1./-* 2.0 1.3

1.2

I 0.5

0

I I 1.0 1.5 c-3 x) 103(g cm

I 2.0

1.0 2.5

Figure 4.19 Two plots which may be made to obtain the limiting value of the dissymmetry coefficient

at zero concentration (Problem 4.19).

4.0

3.5

3.0

2.5

2-0

Intercept = 2.0/. x l(~6mot

1.5 0

f 0.5

I I I 1.0 1.5 2.0 3 c x 10 (g cm-3i

2.5

Figure4.20 Concentration dependence of Kc/t?Jao for the sample of polystyrene of Problem 4.19.

PolymerMolecular Weights

285

Since the square of the refractive index increment (dfi/dc) appears in the light scattering equations, e.g., Eq. (4.92), this value__ must accurately determined in order to perform calculations of /l’/w, A2, and (S2)z. The value of dh/dc is needed at infinite dilution, but there is very little concentration dependence in the normal concentration range used for light scattel-ing of polymer solutions. The required value can therefore be obtained from measurements at finite concentrations (see Problem 4.12). Since the sensitivity of the measurements depends strongly on the refractive index increment, the value of dh/dc should be as large as possible. Thussolvents that have refractive indices substantially different from that of the polymer should be chosen (typically, to give dh/dc of about 0.1 cm3 g-l). A direct measurementof Ah is preferable to individual measurements of fi, and fi, o and so the preferred method of measuring d~/dc is differential refractometry, which measures the refraction of a light-beam passing through a divided cell composedof solvent and solution compartments that are separated by a transparent partition. The most dreaded enemyof light-scattering practioners is dust. A single large dust particle mayscatter more light than the liquid itself. Elimination of dust is not easy; it is accomplished by either centrifugation or filtration. Of course, the cell itself and all t.he filtering equipment must be dust-free to begin with.

Dilute Solution Viscometry The viscosity of even dilute polymersolutions is considerably higher than that of the solvent. The increase in viscosity dependson several factors, such as, temperature, solvent-polymer interaction, polymer concentration, and the sizes of the polymer molecules. These have been discussed in Chapter 3. The last mentioned dependence on size permits estimation of an average molecular weight from solution viscosi__~. The average molecular weight th~at is measured is the viscosity average My, which differs from -~-[n and Mw described so far in this chapter. The relation betweenviscosity increase and molecular weight that is used for the calculation of Myis given by the MarkHouwink-Sakurada (MHS)equation [Eq. (3.183)], described in Chapter 3. Before viscosity increase data are used to calculate Myof the polymer, it is necessary to eliminate the effects of polymer concentration. The methods whereby this is achieved are described in a later section. These methods, ho__wever,do not removethe effects of polymer-solvent interactions, and so My of a given polymer sample will depend to some extent on the solvent used in the solution viscosity measurements. The viscosity method of molecular weight determination requires very little investment in apparatus and can be carried out quite rapidly by flow time measurements with a simple glass viscometer (see p. 291) set up in constant temperature bath. As a result, this is.the most widely used method for measuring a polymer molecular weight average.

286

Ch~pCer4

Wefirst consider below the commonlyused nomenclature for solution viscosity and then describe in later sections the definition and significance of viscosity average molecular weight (My) and the method of its determination from solution viscosity. Solution Viscosity Terminology There are two systems of nomenclature in the field of solution viscosity: the trivial nomenclature and the IUPACnomenclature [11]. The trivial nomenclature is vague and misleading but is widely used, while the IUPAC nomenclature is clear and precise but is rarely used. Table 4.2 lists the commonlyused names with their symbols and definitions. It maybe noted that the viscosity terms listed in Table 4.2 are not viscosities at all. Thus~Tr and rlsp are actually unitless ratios of viscosities. The intrinsic viscosity [r/] is a ratio of viscosities divided by concentration and so has the units of reciprocal concentration, commonlyquoted in cm37gor dL/g. VlscoMty-A

verage

Molecula~

Weight

Consider a polymer sample to be made up of a series of i monodisperse macromolecules each with concentration (weight/volume) 14 and molecular weight Mi. Then, c~ = NiMi (4.112) where gi is the concentration in terms of moles/volume. From the definition of intrinsic viscosity [r/] given earlier [cf. Eq. (3.173)] rl-r] (4.113) [~7] °) = c--*0 liml(C where ~7 is the viscosity of a solution of concentration c and ~o is that of the pure solvent, one can write for the species i in very dilute solution (~i/r]o)-

1 = c/[r/~]

(4.114)

Also, according to Mark-Houwink-Sakurada (MHS) equation (3.183) the assumption that the constants K and a are independent of molecular weight, one can write [rill = KMi ’~ (4.115) Substituting Eqs. (4.112) and (4.115) in Eq. (4.114) ~+1 ~r ~ ~/r

I = .....

(4.116)

Wemay regard the viscosity of the solution of a whole polymer as the sum of the contributions of the i monodisperse species that make up the

287

Polymer Molecular Weights

Table 4.2 Solution Viscosity Nomenclature Name Relative viscosity Specific viscosity Reduced viscosity

Symbol r/~ r/sp r/sp / c

Inherent viscosity

r/i~h

bIntrinsic viscosity

[r/]

aDefinition

a r/ = solution viscosity; r/o = solvent viscosity bAlsocalled the limiting viscosity number.

polymer.That is, -

=

i

whole From Eqs. (4.113) and (4.117),

)

_ c-~O C

(4.117)

’~+1 = K x-" N,.M,.

"

(4.118)

AT. M, a÷l c--~O C

whole

However, and so

(4.119)

[r/] = lira ( ~(" ~~" ~N’M’a+I ~ ~ ,EN~M~ c~0 i Th~ MH8 ~uation

[cf.

Eq.

(~.173)] =

K~:

(4.120)

where M~is the ~scosiW-average molecular weight of the whole polymer. Substituting Eq. (4.119) in Eq. (4.120) gives

K~ lim

K

So in the limit of infinite

~Mi

/

i

i

dilution,

~ = ~/~ [EN~M~a+~/E

N~M~]

(4.122)

~ alternative deflation in te~s of weight fraction polymer ~n be derived by noting that

composition

where Wi and W are the concentrations then converted to the fo~

Equation (4.122)

~ = [E

in weight/vol. wiMia]

Ua

of the

(4.123)

288

Chapter 4

where wi is the weight fraction of species i. In terms of moments, --

] a[~-~,]

:

[wU

(4.124)

Note that My, like Mr~ or i~o, is a function of the molecular weight distribution of the polymer but unlike the latter it is also a function of the solvent (through the exponent a). T.__hus, a ~g~en polymer sample can characterize__d by a single value of Mnor Mw,but it may have different values of Mydepending on the solvent in which [r/] is measur__ed.In general, the broader the molecular weight distribution, the more My may vary in different solvents. For a monodisperse polymer, however, My = M,~= iw. Note that according to Eq. (4.123), My = Mn for a = -1 and My = Mwfor a = 1. For polymers that assume random coil shapes in solution, it is found that 0.5 < a < 0,8. Since a is thus closer to 1 than to -1, ~ will be much closer to Mwthan__to Mn. Once K and a are known,-~v__ is mucheasier to measure than Mw.__,and it is thus convenient to take My as an approximate measure of Mw. This approximation is often useful but not always very reliable for broad distribution polymers. Problem4.20 Showthat the intrinsic viscosity of a mixture of polymers is the weight average value of the intrinsic viscosities of the components of the mixture in the given solvent. Answer: Let [~] be the intrinsic viscosity of species i (monodispersemacromolecules)in particular solvent. The MHS equation [Eq. (4.120)] for this species [rh] = KM~" (P4.20.1) FromEq. (4.123), for the whole polymer, (P4.20.2) (~.)" = ~ w, Mi" Substituting Eq. (P4.20.2) in Eq. (4.120) and then combiningwith Eq. (P4.20.1) gives

Calibration

of

the

Mark-Houwink-Sakurada

Equation

The classical method for determining K and a values of the Mark-HouwinkSakurada (MHS)equation involves fractionation of a whole polymer into subspecies, or fractions, with narrow molecular weight distributions. An average molecular__ weight can be dete___rmined on each such fraction, by osmometry(Mn) or light scattering (Mw), and, if the fractions are narrow

289

PolymerMolecular Weights

enough, the measured average can be approximated to Myof monodisperse polymer. The intrinsic viscosities meas~uredat a constant temperature for a number of such fractions of knownMy are fitted to the equation ln[r~] = InK + aln(~,~) (4.125) to yield the MHSconstants K and a for the particular polymer/solvent systemat the temperature of viscosi~ty measurement__=_. (Since actual fractions are not really monodisperse and My is closer to Mwthan to Mn , it is a better practice to determine the molecular weight by light scattering than by osmometry.) The two constants K and a are derived from the intercept and slope of a linear least squares fit to [r/] -- Mvalues for a series of fractionated polymers. The method assumes that K and a are fixed for a given polymer type and solvent and do not vary with polymer molecular weight. This is not strictly true, however, and the MHSconstants determined for higher-molecular-weight species may depend on the molecular weight range. Tabulations of such constants therefore usually list the molecular weights of fractions for which the particular K and a values were determined. Such a list for some commonsystems of more general interest is presented in Table 4.3. Measurement

of

IntrinMc

Viscosity

This determination is performed very easily with simple glass viscometers. Since the viscosity of a liquid depends markedly on temperature, viscosity measurements must be made at a carefully controlled temperature (within :t=O.l°C). Before a measurement, the viscometer is therefore equilibrated in a carefully controlled thermostatic bath at the required temperature. The most commonlyused glass viscometers are the Ostwald and Ubbelohde types shown in Fig. 4.21. In the Ostwald viscometer, a given volume of liquid is introduced into bulb B through stem A and is drawn up by suction into bulb C and above the fiducial mark a. The suction is released and the time taken by the liquid meniscus to pass between the fiducial marks a and b in bulb C is measured. The average driving force during the flow of this volumeof liquid through the capillary tube is proportional to the average difference in the levels of the liquids in the two tubes (i.e., proportional to h, as shownin Fig. 4.21a). In order that this driving force ~s the samein all cases, it is clearly essential that the sameamountof liquid should always be taken in bulb B. The requirement that the same amount ’of liquid should always be used does not apply in the case of the Ubbelohdesuspended level viscometer [13], shown in Fig. 4.21(b). A modified design of the Ubbelohde viscometer shown in Fig. 4.21(c). For measurement with the Ubbelohde viscometer measured volume of polymer solution with knownconcentration is pipetted into bulb B through stem A. This solution is transformed into bulb C by applying a pressure on A with compressed air while column D is kept

290

Chapter4

aTable 4.3 Mark-Houwink-SakuradaConstants (K and a) Polymer Polystyrene

Solvent Benzene bMEK Toluene

Temp. Mol. wt. range K × 102 -4) (°C) (M × 10 (cm3 g-l) 20 0.12-14 1.23 20-40 0.8-400 3.82 20-30 2-200 1.05

a 0.72 0.58 0.72

Polyisobutylene Cyclohexane Toluene

30 25

0.05-320 14-34

2.88 8.70

0.69 0.56

Poly(methyl methacrylate)

Acetone Chloroform bMEK

25 25 25

8-140 8-140 8-140

0.75 0.48 0.68

0.70 0.80 0.72

Poly(vinyl acetate)

Acetone Methanol

30 30

2.7-130 2.7-130

1.02 3.14

0.72 0.60

Poly(vinyl alcohol)

water

25

0.85-17

30.0

0.50

Poly(acrylonitrile)

e DMF

25

3-37

2.33

0.75

Natural rubber

Toluene

25

4-150

5.0

0.67

Cellulose Acetone 2-14 20 0.238 triacetate aData from Ref. 12; t~Methylethyl ketone; CDimethylformamide

1.0

closed. Whenthe pressure is released, the solution in bulb E and column D drains back into bulb B and the end of the capillary remains free of liquid. The solution flows from bulb C through the capillary and around the sides of the bulb E into bulb B. The volume of liquid in B has no effect on the rate of flow through the capillary because there is no back pressure on the liquid emerging from the capillary as the bulb E is open to atmosphere. The flow time t for the solution meniscus to pass between the fiducial marks a and b in bulb C above the capillary is noted. Since the volume of solution in B has no effect on the flow time t, the solution in B can be diluted in situ by adding a measured amount of solvent through A. The diluted solution, whose concentration is easily calculated from the solvent added, is then raised up into C, as before, and the new flow time is measured. In this way, the concentration of the solution in B can be

291

Polymer Molecular Weights

A

D

A

D

/ Capill.ary

~ Capillary

api!.[ary

B

(a)

(c)

(b)

Figure4.21 Glass viscometers. (a) Ostwald viscometer; (b) Ubbelhode suspendedlevel viscometer; (c) A modified suspended-level viscometer (see text for description).

changed by successive dilution with measured volumes of solvent and the corresponding flow times can be determined. With the assumption of a constant flow rate, the flow time t can be related to the viscosity r/ of the liquid by the Hagen-Poiseuille equation

[14] ~]

= 71"/’4

mp ~ /

8V ~

(4.126)

where Ap is the pressure difference between the ends of a capillary tube of length ~ and radius r, and V is the volume of liquid that flows through the capillary tube in time t. For measurements carried out in a viscometer, in which the capillary is in a vertical position, the driving force is simply the weight of the liquid itself. Therefore, the pressure difference, /kp, which is the driving force per unit area, is given by

Ap = hpg

(4.127)

whereh is

the average height of the liquid during measurement, p the density of the solution and g the gravitational acceleration constant. Substitution of Eq. (4.127) in Eq. (4.126) yields

~rr4 hg pt ~/ -- 8 V t The applicability

of Eq. (4.128)

demands that

(4.128) the flow be Newtonian or

292

Chapter4

viscous. This will be true providedthat a dimensionlessquantity, called the Reynoldsnumber,is less than 1000. In terms of the variables of Eq. (4.128) this condition is given by 2Vp < 1000 (4.129) This condition is readily satisfied for the apparatus and liquids usually used for measurements of the viscosities of polymersolutions. However,in addition to the requirementsof viscous flow, the derivation of Eq. (4.128) relies on the followingassumptions. (a) It is assumedthat all of the potential energyof the driving force expendedin overcoming the frictional resistance. This is not strictly true, however,since someenergymustbe expendedto accelerate the liquid in the tube. Whenthis "kinetic energy correction" is made, Eq. (4.128) becomes pV (4.130) 8~r~t For measurementsof absolute viscosity ri, the correction term can amount to 10 to 15%but, for measurements of relative viscosities ri/rio (see Table 4.2), whichare of interest in polymerchemistry, the error introduced by the use of Eq. (4.128) instead of Eq. (4.130) is usually less than 2% Problem4.21). (b) It is assumedthat the velocity of the liquid at the walls of the capillary is zero, i.e. there is no slippageof the liquid alongthe walls. This assumptionis usually valid for liquids that wet the capillary walls. If the kinetic energy corrections and slippage are neglected, a very simple relation can be obtainedfor the relative viscosity (Table4.2). Thus, denotingthe termsrelated to solventwith subscriptzero, a ratio of viscosities of solution and solvent in terms of respective flow times for the sameQ through the samecapillary is obtained from Eq. (4.128) ri pt -(4.131) rio Poto For dilute solutions p is very close to Po and Eq. (4.131) simplifies ri t -(4.132) to rio Thus, the ratio of viscosities neededin Eq. (4.113) can be obtained from flow times without measuringabsolute viscosities. ri

-

~rr4hgpt 8V~

Problem4.21 Supposethat the Ostwaldviscometer shownin Fig. 4.21(a) has r = 0.02 cm, ~ = 10.0cm, V = 5.0cm3, and h = 15 cm. Whatpercentage error will be introducedbyneglectof the kinetic energycorrectionin Eq.(4.,130) in determining : 3) whichhas a flowtime (a) Theabsolute viscosity of chloroform(p = 1.49 g/cm

293

PolymerMolecular Weights

of 206 s at 20°C. (b) The relative viscosity of a solution of poly(methylmethacrylate)in chloroform whoseflow time in the sameviscometer is 280 s. Answer: (a) Eq. (4.130) can be written ’1 = A.t- B/t whereA = 7rr 4hgp/8V£

and B = pV/87rL

A = (3.14) (0.02 4 (15cm) (980.63)cm s -2) (1.49 g cm-3) 8 (5.0 cm (10.0 cm) 2.75 × 10-5 g cm-1 -2 s B = (1.49

gcm-3)(5.0 8(3.14) (10.0

cm3) = 2.97x 10 -2 -1 gcm

Therefore, ~7 = (2.75 x 10-5 g cm-~ s -2) (206 s) - (2.97 x -2 g cm-1) / (206 s) = 5.52 x 10-3 g cm-1 s -1 or Poise (2.97 x 10-2 g cm-~)/(206 × 100 = 2.6% (5.52 x 10-3 -~) gcm-’ s (b) Let t = flow time of solution; to = flow time of solvent; For dilute solution, Psoln. -- Psolvent; Hence(rl/rlo)withoutcorrection-~l;/to; (r//~o)withcorrection : (~) °- B B--) % error

=

%error

= 1 -

2At

x100 = 1.2%

The intrinsic viscosity [r/] is defined in Eq. (4.113) as a limit at zero concentration. Since the r//r/o ratios are obtained from Eq. (4.132) with measurements made at finite concentrations of the solution, it becomes necessary to extrapolate the data to zero concentration in order satisfy Eq. (4.113). There are a variety of ways to carry out this extrapolation. The variation in solution viscosity (r/) with increasing concentration (c) be expressed as a power series in c. The equations usually used are the Huggins equation [15]:

294

Chapter

and the Kraemer equation

[16]:

_ [7] - iv]2 c - [v]3c2....

vioh-

It is easy to show that both equations should extrapolate to a common intercept equal to. [].Ther/ usual calculation procedure thus involves a double extrapolation of Eqs. (4.133) and (4.134) on the same plot Problem 4.22) to determine [r]] and hence My from the MHSequation.

Problem4.22 The following are data [17] from viscosity measurements with an Ostwald viscometer (r = 1.5 x 10-2 cm, ~ = 11 cm) on a solution of poly(methyl methacrylate) in chloroform at 20°C. Concentration (g/cma) x l02 Flow time (s) 0.0000 170.1 0.03535 178.1 0.05152 182.0 0.06484 185.2 0.100 194.3 0.200 219.8 0.400 275.6 (a) Determine[r/] by plotting r~sp/c and r/i~ against e. (b) Find M~for this polymer, for which the MHSequation 10- 3 --0.80 M~ cma/g.

is [r/]

= 3.4

AIISV¢~ ;

Neglecting kinetic energy correction since the error is small, r~p/c

--

t - to toC ’

and r~h =

In(t/to) c

where to = flow time for solvent = 170.1 s and t = flow time for solution. Both rlsp/c and r~mu are plotted on the same graph in Fig. 4.22. The common intercept of the plots on the ordinate at c = 0 gives [r/]

= (~0/c)~0

= (r~)~=o

= 1.325

From Eq. (4.120): -(1.325 × 102~ 1/0.8 M. = ~ ~-~iO_= 5.47x10 5]

A useful initial concentration for solution viscometry of most synthetic polymers is about 1 g/100 cm3. High-molecular-weight polymers may require lower concentrations to produce a linear plot (Fig. 4.22) of (rl/rlo -- 1)/c

295

PolymerMolecular Weights

1.4

1.2

1.0

I

I

1

0.1

0

1

I

0.2

I

0.3

I

I

0.4

Concentration,c x 3) 102 (g c~ Figure 4.22 Plot of rl~p/c and ~inh against c (Problem4.22).

against c; which does not curve away from the c-axis at higher concentrations. At very low concentrations such plots may curve upwards, which is attributed to adsorption of polymer on the capillary walls. Gel Permeation

Chromatography

Gel permeation chromatography (GPC)is essentially a process for the separation of polymer molecules according to their size. The separation occurs as the solute molecules in a flowing liquid movethrough a stationary bed of porous particles. The method has been used extensively in biochemistry to separate biological polymer molecules from small molecule contaminants (with the use of Sephadex column). Application of the method to synthetic polymer chemistry in the 1970s has revolutionized the procedures for polymer characterization and molecular weight determination. The principle that underlies the method can be explained as follows: Assumethat a dilute solution containing a broad molecular weight distribution of polymer chains and oligomers is allowed to flow through a column packed with finely divided solid particles, which have pores (tunnels) of diameter, say, 1000 24. As the dissolved solute passes each particle, molecules with dimensions smaller than 1000 24 will enter the pores and will permeate the pore space under the influence of the usual thermal

Chapter4

296

motions (Fig. 4.23). Thus, the smaller molecules will have longer effective paths than larger molecules and will hence be "delayed" in their passage (elution) through the column. On the other hand, larger polymer molecules with coil size greater than 1000 24 will be unable to enter the pores and will thus be swept along with the solvent front to appear in the exit from the GPCcolumn ahead of the smaller molecules (Fig.4.23). In reality, the columnpacking itself has a distribution of pore sizes and this improves the effectiveness of the fractionation process for the whole molecular weight distribution. Thus, the GPCmethod is essentially a. process for the fractionation of polymers according to their size, and hence according to their molecular weight. However, the molecular weight cannot be determined directly. Molecular weight can be calculated from the GPCdata only after calibration of the GPCsystem in terms of retention time or elution volume with polymer standards of known molecular weights (see p. 298). J~xperimen~al

Arrangement

A schematic layout of a typical GPCunit is shown in Fig. 4.24. Since the elution rates by gravity flow through a vertical column, as in conventional chromatography, will be slow and nonreproducible, a mechanical pumpis usually employed to force the sample and the elution solvent through the column at pressures of up to 1000 to 4000 psi and at a rate of 2 to 3 ml/min. The sample is injected into the column entry from a graduated hypodermicsyringe (typically, 0.5 to 3 ml of a 0.005 to 0.1%solution of the polymer) by means of a mechanical inlet device. The pulse of the polymer solution injected into the column entry becomes diluted and attenuated as the different species are separated on the column packings. The GPCcolumns are usually 3/8-inch diameter stainless steel tubes with a combined length of 3 to 10 ft, or more, depending on the type of packing (stationary phase) used. The most commontype and the first be widely used for analysis of synthetic polymers consists of polystyrene gels, called styragel particles (hence the nameof gel permeation). These are highly porous polystyrene beads and also highly cross-linked so that they can be packed firmly without clogging the columns when the solvent flows through them under pressure. A combination of high degree of porosity and controlled pore sizes is achieved by copolymerization of styrene and divinylbenzene in mixed solvents that are good solvents for the monomers but have marginal affinity for polystyrene [18]. (For GPCin aqueous systems the most commonpackings are cross-linked dextran or acrylamide polymers and porous glass.) The particle size of the stationary phase has a profound effect on the resolution of the separation. Gel permeation being a diffusioncontrolled phenomenon, the speed and efficiency of the separation will be increased as the stationary particle size is decreased and the relative surface area is increased. Thus, small particles (,~-, 10/~mdiameter), called microstyragel particles, allow faster separations with smaller samples and shorter columns than.do larger particles (37 to 74 #m).

297

Polymer Molecular Weights

" ’~" ~ e "x ..~.. ,’ i ~

Substrate particle with pores (tunnel.s)

X ~X

~X x

~X

’,x

Xx ~ x x

X X

X

Out

Fibre 4.23 Schematic of the pNnciple of separation

by gel pe~ea~on c~omatography. Black circles represent molecules of rail sizes ~ pore diameter, while crosses represent molecules of ~ii sizes > pore diameter. ~ a dis~bution of d~erent molecular sizes enters the column at the same ~me, the molecule~ will emerge from the ~lumn in sequence, distributed a~ording to molecular size, ~om larger to smaller.

Figure 4.24 Schematic diagram of a gel permeation chromatography apparatus.

Chapter4

298

After passage through the column system, the flowing eluant passes through a detector, which responds to the weight concentration of polymer in the eluant. The most commondetector is a differential refractometer. It measures the difference in refractive index between the eluted solution and the pure solvent. This difference is proportional to the amount of polymer in solution. Spectrophotometers are also used as alternative or auxiliary detectors. Whenultraviolet detection is used, the spectrophotometer is usually set to a particular wavelength (e.g., to the aromatic absorption region of a polymer that contains phenyl rings) and the abs0rbance is monitored as a function of elution volume. The elution volume (also called the retention volume), that is, the volume of solvent that has passed through the GPCcolumn set from the time of injection of the sample, is conveniently monitored by means of a small siphon, which actuates an event marker every time it fills with eluant and dumps its contents. The raw GPCdata thus consist of a trace of detector response proportional to the amount of polymer in solution and the corresponding elution volumes. A typical GPCrecord (gel permeation chromatogram) is shown in Fig. 4.25. A normal practice is to use a set of several columns, each packed with porous styragel of a different porosity depending on the range of molecular sizes to be analyzeod. For example, a series of columns that contain 106, 105, 10a, and 500 A pore diameter particles should, in principle, be capable of fractionating a molecular weight distribution that encompassesthis entire range. Data Interpretation

and Calibration

A’GPCchromatogram, as shown in Fig. 4.25, can yield a plot of the molecular weight distribution, since the detector response on the ordinate can be transformed into a weight fraction of total polymer while suitable calibration permits the translation of the elution volumeaxis into a logarithmic molecular weight scale (explained below). A baseline is drawn through the recorder trace, and chromatogram heights are taken for equal small increments of elution volume. An ordinate corresponding to a particular elution volumeis then converted to a weight fraction by dividing by the sum of the heights of all the ordinates under the trace. This process normalizes the chromatogram. Wenow consider how the elution volume axis of a raw chromatogram, such as shown in Fig. 4.25, can be translated into a molecular weight scale. This necessitates a calibration of the particular GPCcolumn for the particular polymer-solvent system used. Such a calibration requires the establishment of a relationship between the volume of solution eluted (or, equivalently, the elution time for a given flow rate of solution) and molecular weight of monodisperse fractions of the same polymer. The main problem encountered in this task is that monodisperse or very narrow distribution samples of most polymers are not generally available. However, such samples are available for a few specific polymers. A notable example is polystyrene for which anionically polymerized samples of narrow mole-

PolymerMolecular Weight~

299

¯ ’----" E[utionvolume Figure 4.25 A typical gel permeation chromatogram. The lower trace with short vertical lines is the differential refractive index while the uppercurve is an absorptionplot at a fixed ultraviolet frequency.The short vertical lines are syphon dumpsnumberedconsecutively from the time of injection of the sample. The units of the ordinate dependon the detector, while those of the abscissa can be in terms of syphonvolumes(counts) or volumeof solvent.

cular weight distributions (MWD) with polydispersity index less than 1.15 are commercially available in a wide range of molecular weights (103 to 106). Using such narrow MWD samples, a polystyrene calibration of molecu01ar weight versus etution volume can be easily obtained for the given GPC column (or columns) and the given GPCsolvent. A problem that would then remain is to establish a relationship for the particular GPC column between the elution volume and molecular weight of some chemically different polymer. Wenowconsider these two steps in sequence. Whena narrow MWD polystyrene sample, as described above, is injected into the GPCcolumn, the resulting chromatogramis narrower than that of a whole polymer, but it is not a simple spike because of band-broadening effects and because the polymer itself is not truly monodisperse. However, the distribution being very narrow, no significant error is committed by assigning the elution volumecorresponding to the peak of the chromatogram to the molecular weight of the polystyrene sample. Thus, a series of narrow

Chapter4

300

MWD polystyrene samples used with the particular GPCcolumn and the GPCsolvent, yields a set of GPCchromatogramsas shown in Fig. 4.26: The peak elution volumes and the corresponding molecular weights provide a calibration curve (Fig. 4.27) for polystyrene in the particular GPCcolumn and solvent used. The next step is to translate this polystyrene calibration curve to one that will be effective for another given polymer in the same apparatus and solvent. This technique is called a universal calibration. To extend the calibration to other polymers, a calibration parameter that is independent of the chemical nature of the polymer, that is, a universal calibration parameter, is required . Such a parameter has been found [19] experimentally to be the product of the intrinsic viscosity and molecular weight (i.e., [~/]M). Thus, as shown in Fig. 4.28, with tetrahydrofuran used as the solvent, the logarithm of the product [r/]M plotted against elution volumeprovides a single curve for a wide variety of polymers. Such a single curve for different polymers is not obtained, however, on simply plotting log Magainst elution volume. The experimental finding that [r/]M is the same function of elution volume for many different polymers suggests that a universal calibration procedure maybe possible. A theoretical validity of this experimental observation is obtained from a consideration of the hydrodynamicvolume of the polymer. This may be shown as follows: Consider the flow of a fluid through a tube. If the fluid wets the tube wall, its velocity will be zero at the wall and greatest at the center of the tube as shownin Fig. 4.29(a). Now consider a particle, of diameter much smaller than the dimensions of the tube, suspended in such a flowing fluid, as in Fig. 4.29(b). Impingement on the particle of fluid flowing at different rates causes the suspended body not only to move down the tube but also to rotate as shown. Since the particle surface is wetted by the liquid, its rotation brings the adhering fluid from a region with one velocity into a volume element that is flowing with a different speed. The consequent readjustments of momenta cause an expenditure of energy that is greater than what would be required to keep the same volume of fluid movingwith the particular velocity gradient, and this results in an increase in viscosity of the suspension relative to that of the suspending medium. As long ago as 1906, it was shown by Einstein [20] that the -viscosity increase is given by the expression r/r

= (r//r/o)

= 1 + co qb

(4.135)

where r/r is the relative viscosity, r/ the viscosity of the solution, r/o the viscosity of the pure solvent, ~b the volume fraction of the suspended material, and co is a factor that depends on the general shape of the suspended particle. According to Eq. (4.135), as the overall size or volume of the suspended particle increases (due to imbibition of solvent), so the volumefraction ~b and the relative viscosity. The effects of a dissolved polymer are similar in some respects to those of the suspended particle described above. It maybe noted, however, that the volume and shape of the real polymer coil in the suspension is not

Polymer Molecular Weights

301

Increasing molecular weight

~ Elution

volume

Figure 4.26 Gel permeation chromatography elution curves for polymer standards having very narrow molecular weight distribution.

10o ~o

E 1.0--

0.1 100

I I I 120 I~0 160 3) Peakelution volume,Ve(cm

180

Figure 4.27 A typical polystyrene standard calibration curve (M, vs. V~) for GPC.

302

Chapter 4

109

108

~ 7 10

~0 Polystyrene( [in ~- ¯ Polystyrene (’comb*) -- A Polystyrene(’star’) ~ X Poly(methy[ methacrytate (linear) - I-I Poly(vinyl chloride)

6 10

5 10

"~.

¯ Poty (butadiene) ~" ¯ Poly (phenyl siloxane) ¯ Poly (styrene-co-methyl." methacrylate ~

~:[ \

Q Poly (styrene-g-methylmethacrytate) -+ Poty(styrene-~.-methy[methacrytate) (comb) t

18

t

f

20

22

I

2Z,

I

26

I

28

3O

ELution volume (5mt counts,THFsolvent)

Figure4.28 A universal calibration curve for several polymers in tetrahydrofuran. (Drawn with data from Ref. 19.)

constant but changes continuously due to rotations about single bonds and motions of chain segments, and because some of the solvent inside the coil can drain through the macromolecule. Nevertheless, the time-averaged effects of the real, solvent-swollen polymer can be taken to be equal to that of equivalent, impenetrable spherical particles. For spherical particles, the shape factor w in Eq. (4.135) is 2.5 and this equation then becomes

r//r/o-

1 = 2.5¢

If all polymer molecules exist in solution as discrete entities,

(4.136) without overlap,

PolymerMolecular Weights

303

Radial position,r

Tube wall

\

,/ (a)

(b)

Figure 4.29 (a) Variation of the velocity of laminar flow with respect to the distance r from the center of a tube. (b) Rotation of a sphere suspended in flowingfluid.

and each solvated molecule of a monodisperse polymer has an equivalent volume (or hydrodynamic volume) V and molecular weight M, then the volume fraction ¢ of solvent-swollen polymer coils at a concentration c (mass/volume) ¢ = cVNAv / M (4.137) where NAyis Avogadro’s number. Combination of Eqs. (4.135) and (4.137) yields 2.5 V NAy

Implicit in this equation is the assumption that the contributions of the individual macromolecules to the viscosi~ increase are independent and additive, which is true when the polymer molecules are infinitely far from each other. But this condition will not be satisfied at ~ite concentrations where the disturbance of flow caused by one suspended macromolecule can interact with that ~omanother solute molecule. ~en effects of polymer concentration ~n only be eliminated experimentally when the solution is ve~ dilute. Of ~urse, for a ve~ dilute solution, ~ -- ~o will be nearly zero. ~erefore, ~scosities are measured at not too low concentrations and these data are used to extrapolate the left-hand side of Eq. (4.138) to zero concentration conditions. Then, [~]~lim1(~~o~° )~-0-c

limV-=(2"~v)

~0(4.1a9)

304

Chapter4

Multiplying both sides by

Mand taking logarithm gives

log ([~7] M) log (2.5 Nay) q- log (!i~rr~ V) (4.140) The product [r/]M is thus seen to be a direct function of the hydrodynamic volume of the solute at infinite dilution. Since studies of GPCseparations haveshown that polymers appear in the eluate in inverse order of their hydrodynamicvolumes in the particular solvent, it maythus be stated that two different polymersthat appearat the sameelution volumein a given solvent and particular GPCcolumn at a given temperature have the same hydrodynamic volumes and hence the same [r/]M characteristics; that is, log ([r/]~Mz) = log ([r]]~Ms) (4.141) where the subscripts :c and s indicate the unknownpolymer (i.e., polymer with unknownmolecular weight) and the standard polymer, respectively. If each intrinsic viscosity term in Eq. (4.141),is replaced by its MHSexpression [Eq. (4.120)], one obtains for the two polymers at equal elution volumes: a,+l as+l

log (Kzt; ) = log (KsM;

Solving for

/4.142)

log M~gives

1+as logMs (4.143) + ~ I + az This equation describes the elution volume calibration curve for ~[z. The elution volume (V~) that corresponds to a GPC peak in the unknown polymer is used to obtain a value of log Ms from the polystyrene standard curve. (Fig. 4.27) that has been obtained in the same column and solvent, and Mz is then calculated from Eq. (4.149). An alternative procedure simply to choose a number of,values of Ve and construct a new calibration curve for the polymer under study from the standard curve such as Fig. 4.27 and Eq. (4.143). The above procedure presupposes that the MHSconstants, Ks, as, K~, and az are known. Values of K~ and a~ are available in the literature for a variety of solvents (Table 4.3), and so, in manycases are values of and az. However, if the desired MHSconstants are not available for the polymer under study in the GPCsolvent or for the standard in the solvent to be used, they can be easily determined by measurementof the intrinsic viscosity, as described 6arlier. logMz

= 1(1)logKs + az

Problem4.23 A series of narrow distribution polystyrene standards dissolved in chloroform were injected into a GPCcolumnat 35°C yielding a set of chromatograms. The following data of peak elution volumes and corresponding sample molecular weights were reported [21]: -~ ~s x I0 (g/mo 0 867 670 411 160 98.2 51 19.8 10.3 -3.7 V~ (cm 3) 122.7 126.0 129.0 136.5 141.0 147.0 156.5 162.5 170.0

305

Polymer Molecular Weights

Assuming that universal calibration is valid, construct a calibration curve for the molecular weight-elution volume of poly(dimethylsiloxane) in chloroform at 35°C. The MHSconstants in chloroform at 35°C may be taken as: for polystyrene K = 4.9x 10-a cma/g, a = 0.79, and for poly(dimethylsiloxane) = 5.4x10 -a cma/g, a = 0.77. Answer: A semilogarithmic plot of Mversus V~gives the polystyrene calibration curve (see Fig. 4.27) for the given GPCcolumn, solvent and temperature. The MHSconstants are Polystyrene : Ks = 4.9×10 -3 cm~/g, t~ = 0.79 Poly(dimethylsiloxane) : K= = 5.4x10 -a cma/g, a~ = 0.77 Substitution of these values in Eq. (4,143) gives the expression log M= = -0.0238 + 1.0113 log M~ To construct an elution calibration curve eM= vs. V~)for poly(dimethylsiloxane), various values of Ve are assumed and corresponding to each Ve value the corresponding M~is first obtained from the polystyrene calibration curve (Fig. 4.27) and then M=from the above expression. A semilog plot of M=versus V~ gives the required calibration curve (Fig. 4.30).

100 ~0

I0

0.1 100

I I I 120 1/~0 160 Peak etutionvoturne,Ve 3) (crn

Figure 4.30 Elution calibration ane). (Problem 4.23).

180

curve (M= vs.

V~) for poly(dimethyl silox-

306

Chapter 4

Problem4.24 A sample of poly(dimethylsiloxane)

in chloroform in injected into the GPC column set of Problem 4.23 and the refractive index difference (Ah) between the eluted solution and pure solvent was measured as a function of elution volume (V~), which yielded the following data: sAh × l0 0.6 3.4 12.4 15.0 9.9 3.0 0.4 V~ (4-mL

count)

40

39

38

37

36

35

Using the calibration curve obtained in Problem 4.23, calculate Mn, M~and the polydispersity index of the polymer. (Assume that Ah is proportional to concentration and the proportionality factor is independent of molecular weight.) Answer: The molecular weight poly(dimethylsiloxane) corresponding to each elution volume is determined from the elution calibration curve for this polymer in Fig. 4.30. The corresponding weight fraction wi is computed from the refractive index difference by the relation

The results are tabulated below. Elution volume (4-mL count) 34 35 36 37 38 39 40 ~ = ZwiMi -~n ----

-3 Mi × 10 (from Fig. 4.30) 182 117 76 50 33 21 14

wi -Ahi/~Afii 0.089 0.067 0.221 0.336 0.277 0.076 0.013 = 54×103

1 / ~ (wi/Mi)

Polydispersity

index

gmo1-1

wi Mi 1620 7851 16834 16780 9154 1598 187 ~ = 54 × 103

(wi/Mi) × 106 0.049 0.573 2.914 6.712 8.406 3.624 0.957 23.236

-= 43 × 103 g mo1-1 = Mw/ Ms = 1.25

Some disagreement exists about the type of molecular weight average that is given by the peak elution volume in GPC_~_chromatogram. Some investigators assume that th___e peak maxima represent Mnbut others consider that the method gives Mw. It has been proposed [22] that the GPC elution peak maxima correlate with the root-mean-square molecular weight, 1/2, ~rms = (Mn. Mw) better than with ~w- On the other hand, a calibration procedure. [23] that replaces Eq. (4.143) logM~

--

l+a~

log-K~.f(¢~)q-

+~

logM~

(4.144)

307

Polymer Molecular Weights where

~i

= (2ai-

1)/3

leads to a good correlation

1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23.

of

M~owith the peak maxima.

F. W. Billmeyer, Jr., Textbook of Polymer Science, Wiley, NewYork (1971). B. Kronberg and D. Patterson, Macromolecules, 12, 916 (1979). T. L. Hill, Statistical Mechanics, Chap. 6, McGraw-Hill, NewYork (1956). A. Einstein, Ann. Physik, 33, 1275 (1910). P. Debye, J. Appl. Phys., 15, 338 (1944). P. Debye, J. Phys. Colloid Chem., 51, 18 (1947). P. Doty and J. T. Edsall, Advances in Protein Chemistry, vol. VI, pp. 35-121, Academic Press, NewYork (1951). P. J. Flory, Principles of Polymer Chemistry, Cornell Univ. Press, NewYork (1953). B. H. Zimm, J. Chem. Phys., 16, 1099 (1948). D. Margerison and G. C. East, An Introduction to Polymer Chemistry, Pergamon Press, NewYork (1967). M. L. Higgins and J. J. Hermans, J. Polym. Sci., 8, 257 (1952). (a) M. Bandrup and E. Immergut (eds.), Polymer Handbook, Wiley, New York (1975); (b) G. Meyerhoff, Fortschr. Hochpolymer-Forsch., 3, 59 (1961). L. Ubbelohde, Ind. Eng. Chem., Anal Ed., 9, 85 (1937). W. J. Moore, Physical Chemistry, 4th ed., Prentice Hall, EnglewoodCliffs, N. J. (1972). M. L. Huggins, J. Am. Chem. Soc., 64, 2716 (1942). E. O. Kraemer, Ind. Eng. Chem., 30, 1200 (1938). G. V. Schult.z and E Blaschke, J. Prakt. Chem., 158, 130 (1941). J. C. Moore, J. Polym. Sci., Part A 2, 835 (1964). Z. Grubisic, P. Remppand H. Benoit, Polymer Lett., 5, 753 (1967). A. Einstein, Ann. Physik, 19, 289 (1906). J. V. Dawkins and M. Hemming, Makromol. Chem., 176, 1777 (1975). M. Kolinsky and J. Janca, J. Polym. Sci., Chem. Ed., 12, 1181 (1974). H. Coil and D. K. Gilding, J. Polym. Sci., A2, 8, 89 (1970).

EXERCISES 4.1. If a polymer sample contains an equal number of moles of species with degrees of polymerization x = 1, 2, 3, 4, 5, 6, 7, 8, 9, and 10, what are

308

Chapter 4 the number-average and weight-average degrees of polymerization ? [Ans. DP,~ = 5.5; DP,o = 7]

4.2. Three lots of polyethylene, A, B, and C are’to be blended to achieve a weight-average molecular weight of 210,000 and polydispersity index (PDI) of 3.5. Howmuch of each lot is needed to make up a shipment of 100,000 kg ? Lot A B C [Ans. 4.3.

Weight-average mol. wt. 500,000 250,000 120,000

PDI 2.50 2.00 2.50

A = 21750 kg, B = 5660 kg, C = 72590 kg; (Total 100,000 kg)]

One gram of polymer A (DP,~ = 1000, DP,, = 2000) is mixed with g Of polymer B (DP,, = 2000, DP~, = 5000). Calculate the degree of polymerization of the mixture that would be derived from osmotic pressure measurements at several concentrations. [Ans. DP,~ = 1500]

4.4. A sample of PVC compound which contains 2 wt% calcium stearate [Ca(OOC(CHz)16CH3)2] as a lubricant is found to have M,~ = 20,000. is M,~ of the balance of the PVCcompound ? [Ans. M,~ = 57,480] 4.5. A polymer with true molecular weight averages Mn = 430,000 and M~, = 1,000,000 is contaminated with 3%by weight of an impurity with molecular weight 30,000. What would be the average molecular weights determined by light scattering and membraneosmometry ? [Ans. LS: 9.7x105; MO: 3.0x105] 4.6. Both polymers A and B have the most probable distribution (polydispersity index = 2.0). Also DP,~ > 100 for each of them. The polydispersity index increases to 4.0 when 200 g of A is mixed with 100 g of B. What is the ratio of the weight-average degree of polymerization of A to B ? [Ans, 0.158 or 6.34] 4.7. By a fractional precipitation a polymer A with most probable distribution (I = 2.0) and an osmotic molecular weight of 150,000 is separated into two fractions B and C with molecular weights, respectively, 250,000 and 325,000, determined by light scattering. Calculate the weight of B that would be obtained from 200 g of initial polymer A. If both B and C have the same polydispersity, what is the value of the polydispersity index ? [Am. WB= 66.7 g; 3" = 1.97] 4.8. A polymer P having number-average and weight-average molecular weights of 120,000 and 300,000, respectively, was cut into fractions A and B by fractional precipitation. If A and B have number-average molecular weights of 90,000 and 130,000, respectively, what are the weight fractions of A and B obtained from the initial polymer P ? If A and B have the same polydispersity, what is the polydispersity index ? [Ans. WA= 0.19; WB= 0.81; 1" = 2.45]

Polymer Molecular Weigh~

309

4.9. By fractional precipitation, 200 g of polymer P with the most probable distribution and a number-average molecular weight M~= 150,000 is separated into two fractions, A and B, the former weighing 68.0 g. Light scattering of A gives a molecular weight of 250,000. If the polydispersity index of A is the same as that of B, what is the number-average molecular weight of B ? [Arts. (M~)B = 143,300] 4.10. In a measurement for end-group analysis, 0.8632 g of a CTPBsample dissolved in 1:3 mixture of ethanol and toluene consumed 5.2 ml of 0.1240 N alcoholic potassium hydroxide solution in titration using phenolphthalein as the indicator. Calculate the molar mass of the polymer. [Ans. -1] 2672 g tool 4.11. A linear polyester was synthesized from a mixture of diacid and glycol with carboxyl to hydroxyl ratio greater than unity. A sample of the polyester (3.5 g) requires titration with 23 mLof N/50 KOHto reach a phenoiphthalein end point. By vapor-pressure osmometry the molecular weight of the polymer was determined to be 12,000. Calculate the average carboxyl functionality of the polymer (that is, average number of carboxyl groups per polymer molecule). ~lns. 1.6] 4.12. To determine the average hydroxyl functionality of a sample of polyester polyol of M~ = 3,000 about 1 g of the dry sample was treated with bromoacetyl bromide (BrCH2COBr)to convert hydroxyl to bromoacetyl end groups. The treated polymer was found to contain 4.88% Br by elemental analysis. Estimate the average number of hydroxyl groups on each molecule of the polyol.

[Ans.2.01

4.13. A polyol for polyurethane foam was synthesized by polymerizing propylene oxide by using glycerol as the initiator. For analysis a sample of this polyol was reacted with phenyl isocyanate to convert the hydroxyl groups to urethanes which were then analyzed for nitrogen. The nitrogen content of the treated polymer was found to be 1.523%. Estimate the hydroxyl equivalent weight and molecular weight of the polyol. [Ans. OHequiv. = 800 g (tool OH)-1; ~n = 2,400.] 4.14. The molecular weight of a polymer determined by an osmotic pressure measurement in a theta solvent is 20,000. What osmotic pressure (atm) would be expected at a concentration of 1.20 g/dL and 30°C. Would there be a difference in molecular weight if the osmotic measurement were made in a good solvent ? [Ans. II = 0.015 atm; no difference.] 4.15. An osmotic pressure measurement of a 0.22 g/100 cmz solution of poly(vinyl chloride) in toluene at 25°C in the apparatus shown in Fig. 4.5 indicated difference of 7.1 mmin the heights of the solution and solvent levels. (a) Whatis the osmotic pressure (atm) of the solution ? (b) If the second virial coefficient for poly(vinyl chloride) in toluene is I~2 = 200 cruZ/g, calculate M,~ of the polymer. [Density of toluene at 25°C = 0.867 g/cmZC.] [Ans. (a) 5.96× -4 at m; (b) 1, 34,000]

310

Chapter 4

4.16. A polyisobutylene sample has M~= 400,000. The second virial coefficient of the polymer in ehlorobenzene solution at 25°C is I’2 = 94.5 cma/g. Calculate the osmotic pressure in g/era 2 of 0.30 g/alL solution of this polymer in chlorobenzene at 25°C. Comparethis with the value calculated for an ideal solution. [Chlorobenzene density at 25°C is 1.11 g/cruZ.] ~4ns. -2] I-[ = 0.246 g cm-2; I-[(ideal) = 0.190 gcm 4.17. A polystyrene sample was dissolved in benzene to obtain solutions of different known concentrations. Measurement of osmotic head for these solutions at 30°C with a static equilibrium osmometer produced the following data: -1) Concentration, e (g dL 0.260 0.540 0.755 0.920 1.025

Osmotic head, h (cm CtHt) 0.140 0.516 0.966 1.389 1.730

Determine for the polymer (a) M~, (b) the second virial coefficient, (c) the X value in benzene at 30°C. [Density of benzene at 30°C = 0.868 g cm-~.] ~4ns. (a) ~n = 1,972,000; (b) A2 = 5.0×10-4 tool cm3 g-2; (c) X = 0.45] 4.18. Calculate the fraction of light of ,k = 5460 _~ that would be scattered in a 1-cm path length through a 2.0 × 10-4 g/cm3 solution of a polymer of ~,o = 5.2× 104 in a solvent of refractive index 1.3688. The specific refractive index increment of the polymer solution is 0.120 cm3/g. ~4ns. L/~o -5 = ]1.74×10 4.19. For a polymer (0.14 g) dissolved in dimethylsulfoxide (100 mL) the ratio the intensity of scattering at 90° to that at 0°, i.e., (~90/~0) is measuredto 0.01 with incident light of wavelength 546 nm. The solvent has a measured refractive index (h) of 1.475 at 21°C and dh/dc of 1.0. The turb idity ~can be obtained from the relation r = kfi2

(igo/io)

using the equipment constant value k = 0.100 cm-1 . supplied with the light-scattering photometer. Determine the apparent molecular weight for the polymer sample. [Note: The molecular weight is called "apparent" since it is for a single point and not extrapolated to zero c.] [n,~. 1.15 × 104] 4.20. Light scattering results for a polystyrene sample dissolved shown below : -a c, g dm 1.760 3.708 6.244 7.736 12.43 Rg~ × 10~, -~ cm 5.31 8.43 11.24 -7 cm2 mol g-~. Determine the Given: K = 2589×10 molar mass of the polymer. [Ans. -t] 1,48,000 g tool

in benzene are 10.230 13.80 weight-average

311

Polymer Molecular Weights

4.21. By measurement of light scattering (A = 436 nm) from toluene solutions of a polystyrene sample [D. Rahlwes and R. G. Kirste, Makromol. Chem., 178, 1793 (1977)], the following results were obtained for the Rayleigh ratio, /~(0), at various concentrations and scattering angles c × 103

3)(g/era --~-~--! 0.20 0.40 0.60 0.80 1.00

45°

2.47 ,I 1.53 4.40 2.84 5.91 3.96 7.07 4.91 7.98 5.71

R(O) × 10~ -~) (cm 75° 105° 0.86 1.65 2.37 3.02 3.61

0.56 1.09 1.59 2.05 .2.49

135° 0.428 0.839 1.23 1.60 1.96

°150 ,0.395 0.775 1.14 1.49 1.82

Construct a Zimmplot using the data and determine the weight-average molecular weight of the sample. Determine also the second virial coefficient F2. (The refractive index of toluene is 1.4976 and the specific refractive index increment of polystyrene-toluene solutions is 0.1121 cm~-/g.) [Ans. ~o = 6.1×106; I~2 = 373 cm3 g-l] 4.22. The results of light-scattering studies [D. Rahlwesand R. G. Kirste, Makrotool. Chem., 178, 1793 (1977)] for A = 4360 ~ on a styrene/a-methyl styrene block copolymer in toluene solution were as follows: c × 103 (g/cm3) °i !15 0.20 11.91 0.40 i 3.55 0.60 !4.95 0.80 16.16 1.00 17.19

° 45 1.47 2.78 3.94 4.96 5.87

R(O) 10 4 (c -I) ° °105 75 1.01 1.95 2.80 3.59 4.31

0.725 1.41 2.05 2.66 3.22

°135 0.577 1.13 1.65 2.15 ~.62

°150 0.537 1.05 1.54 2.01 2.46

The refractive index of toluene is 1.4976 and the specific refractive index increment of the polymer in toluene is 0.1263 cm3/g. (a) Construct a Zimmplot of the data and evaluate the weight-average molecular weight of the polymer. (b) Evaluate the second virial coefficient A2. (c) Evaluate the dissymmetry coefficient z4s and the root-mean-square endto-end distance of the polymer in toluene solution. -3 g/ cm (d) Calculate the turbidity of the solution of c = 0.2× 3. -4 3 [Ans. (a) ~ = 3.5×106; (b) A2 = 1.1×10 cm tool g-2; (c) z~5 ----- 2.63; = 150 nm; (d) ~- = 3.3×10-3 -1] cm (~"~)a/2 4.23. From the light scattering data of a polystyrene sample given in Problem 4.i7, determine the radius of gyration of the polymer in solution.

~,~. 480_~] 4.24. The relative flow times (t/to) chloroform are given below:

of a poly(methyl methacrylate) polymer

Chapter 4

312 Concentration (g/dL) 0.20 0.40 0.60

t/to 1.290 1.630 2.026

(a) Determine[r/] by plotting r/,p/c and r/inh against c. ---080 Co) Find M. for this polymer, given that [r/] = 3.4 x 10-8 M~" (dlJg) [Arts. (a) [~7] = 1.32 dL/g; (b) M. = 545,000] Showthat in the limit of infinite dilution, the reduced viscosity (r/w/c) and inherent viscosity (r/i,~h) are equal. The following data on intrinsic viscosities and GPCpeak elution volumes at 25°C for standard polystyrene samples dissolved in tetrahydrofuran have been reported [M. Kolinsky and J. Janca, J. Polym. Sci., Chem. Ed., 12, 1181 (1974)]: -3 51 19.85 10.3i 5.0 867 411 173 98.2 ~,o × 10 27.6 14.0 8.8 i 5.2 [r/] (cm3/g) 206.7 125.0 67.0 43.6 3) 219 234 I 253.5 149 157 177 186.5 199.5 V~ (cm Construct an elution volume calibration curve for poly(vinyl bromide) tetrahydrofuran at 25°C given that the MHSconstants for this system are K = 1.59 x 10 -2 cm~/g and a = 0.64 [A. Ciferri, M. Kryezewski, and G. Weil, J. Polyra. Sci., 27, 167 (1958)]. 4.27. A sample of poly(vinyi bromide) in tetrahydrofuran was injected into the same GPCcolumn used in Exercise 4.26. The refractive index difference between the eluted solution and pure solvent was measured and plotted against the elution volume. The result was a broad peak the maximumof which occurred at an elution volume of 180 cm3. Calculate the average molecular weight of the polymer both according to (a) correlation__ derived in the preceding exercise and (b) Eq. (4.144) that correlates M~with peak maxima. [Ans. ~] (a) 2.20x10~; (b) 2.13×10

Chapter

5

Condensation Polymerization

(Step-Growth)

INTRODUCTION As already discussed in Chapter 1, condensation polymerizations, also known as step-growth or simply step polymerizations, are merely classical organic reactions that are used to produce linear macromolecules starting from bifunctional monomers, or to produce polymer networks from mixtures of bifunctional and multifunctional monomers.Step polymerizations, as explained previously in Chapter 1, occur by consecutive reactions in which the degree of polymerization and average molecular weight of the product increase as the reaction proceeds. Usually, although not always, the reactions involve the elimination of a small molecule such as water, and hence the name condensation polymerization is used as a general term. Most polymers with heteroatoms (O,N,S) in the chain are produced step polymerization reactions. A numberof different chemical reactions may be used to synthesize such macromolecules by step polymerization. These include esterification, amidation, urethane formation, aromatic substitution, and others. Polymerization usually proceeds by the reactions between two different functional groups, for example, hydroxyl and carboxyl groups, or isocyanate and hydroxyl groups (see Table 1.2). Step-growth polymerization reactions are of great economic importance. While phenol-formaldehyde resins (Bakelites) made by condensation reactions were amongthe very first synthetic polymersto be used as thermosets, i.e., cross-linked materials, the major industrial expansion came with the development of polyamides and polyesters, both as textile fibers, and as moldable materials of good mechanical strength. More recently, high-performance polymers made by polycondensation, such as polycarbonates, polyethersulfones, polyetheretherketone, and polyphenylene sulfide (see Table 1.3) have found a rapidly expanding 313

314

Chapter 5

market. Step-growth polymerizations may be described as a stepwise or progressive conversion of a monomerwith two reactive end groups to higher molecular-weight homologues, which themselves retain two reactive end groups. It may take place either by a polycondensation reaction, whereby a low-molecular-weight by-product is formed along with the polymer, as is exemplified by polyesterification : nHOOC (CH~)~COOH + nHO(CH~)~OH HO-[-OC(CH~)~COO(CH~)uO-]~-,~H + (2n- 1)H.~O or by a polyaddition reaction in which the total reactants are incorporated in the polymer chain (and no by-products are formed), as is typified polyurethane .formation : (n + 1)OCN (CH.~)~NCO + n HO(CH~)uOH OCN-[--(CH~_)~NHqO(CH~)~O-,C, NH-]-~-~(CH~)~NCO

o

The kinetics of polycondensation and polyaddition reactions follow the same general scheme, but both differ sharply from the kinetics of addition or chain polymerization. All step polymerizations fall into two groups depending on the type of monomers employed. These are represented in a general manner by the equations : nA-B n A-A

---~ + n

B-B

+A-B--)-~-~ --~ -(-A-AB-B-)-~-~

(5.3)

(

)5.4

In subsequent treatment, a condensation polymer formed from a monomer of the type A-B will be referred to as type Ipolymer and the polymerization will be called type I condensation, while a polymer formed from a mixture of A-A and B-B type monomers will be termed a type IIpolymer and the process referred to as type H condensation. The synthesis of polyamides illustrates both groups of polymerization reactions : n H~N-R-COOH -~ H+NH-K-C0-)w-,~OH

+ (n-l)H~O

(5.5)

nH~N-R-NH2 + nHOOC-R’-COOH H-(-NH-R-NHCO-R’-CO-)v=-~. OH + (2n - 1) H20 (5.6) The characteristics of these two polymerization reactions are very similar. Step polymerization proceeds by a slow increase in the molecular weight of the polymer. Consider the synthesis of a polyester from a diol and a diacid: HO-R-OH + HO~C-R’-CO~H ~ HO-R-OOC-R’-COzH (Monomer + Monomer ~ Dimer + By-product)

+ -(5.7)

315

Condensation(S~ep-Grow~h) Polymerization

2HO-R-OOC-R’-CO2H -~ HO-R-OOC-R’-COO-R-OOC-R’-CO~H + H=O (Dimer + Dimer --~ Tetramer + By-product) (5.8) HO-R-OOC-R’-CO2H + HO-R-OH --~ HO-R-OOC-R’-COO-R-OH + H~O (Dimer + Monomer ~ Trimer + By-product) (5.9) HO~_C-R’-CO2H

+ HO-R-OOC-R’-COOH ~ HO2C-R’-COO-R-OOC-R’-CO2H (Monomer + Dimer ---+ Trimer + By-product)

+ HzO (5.10)

and so on. Anytwo species in the reaction mixture can react with each other. Step polymerization can therefore be expressed by the general reaction: n-mer

+ m-mer

~ (n

+ m)-mer

(5.11)

Thus, the polyesterification reaction mixture at any instance consists of various-sized diol, diacid, and hydroxyacid molecules. Any OH-containing molecule can react with any COOH-containingmolecule. This is a general characteristic of step polymerization. A comparative account of the differences between step polymerization or condensation polymerization on the one hand and chain polymerization or addition polymerization (see Chapter 6) on the other hand is given in Table 5.1. REACTIVITY

OF FUNCTIONAL

GROUPS

Generally condensation reactions, such as (5.7)-(5.10), are reversible, that the eliminated water must be removed if a high polymeric product is to be formed. The rate of a step polymerization is the sum of the rates of reaction between molecules of various sizes, that is, the sum of the rates for reactions such as (5.7)-(5.10). To describe the course of these reactions in terms of reaction kinetics would seem at first sight to be a very complicated task. However, fortunately it is possible to introduce simplifying approximations that make the kinetic problem tractable. Consider, for example, a polyesterification in which a dibasic acid condenses with a glycol, as shown by reactions (5.7)-(5.10). No matter what the reactants happen to be (which may be dimers, trimers, tetramers, pentamers .... high polymers), the chemical reaction in each step is the same and may be written as vww-COOH

+ HO-wwv --~ ww-C-O-~vvv

+ H,O

(5.12)

Thekinetic analysis of polycondensation withinnumerable separate reactionsthusbecomes greatly simplified if oneassumes that(1)thereactivities

316

Chapter 5

Table 5.1 Comparison of Step-Growth and Chain-Growth Polymerizations Step-growth/condensation

polymerization

Chain-growth/addition

polymerization

1. Monomersbearing functional groups such as -OH, -COOH, -NH2, -NCO, etc., undergo step polymerization.

Monomers with carbon-carbon unsaturation undergo polymerization.

2. The growth of polymer molecules proceeds by a stepwise intermolecular reaction (at a relatively slow rate), normally with the elimination of small molecules as by-products of condensation, such as H20, HCI, NH3, etc., in each step. The molecule never stops growing during the . course of the polymerization.

Each polymer molecule increases in size at a rapid rate once its growth has been started. When the macromolecule stops growing (due to termination reaction) it can generally not react with more monomers(barfing side reactions).

3. Monomerunits can react with each other or with polymers of any size. Growth occurs in a series of fits and starts as the reactive species of a monomeror polymer encounters other species with which it can form a link.

Growth of a polymer molecule is caused by a kinetic chain of reactions involving rapid addition of monomer to an active center that maybe a free radical, ion or polymercatalyst bond. The active center is produced by some external source (energy, highly reactive compound, or catalyst).

4. At any moment during the course of polymerization, the reaction mixture consists of molecules of all sizes ranging from monomer, dimer, trimer, etc., to large polymer depending on the extent of conversion.

At any moment the reaction mixture essentially consists of full-grown, large polymer molecules, unreacted monomer molecules, and a very low concentration (10-s - 10-~ -~) mol L of growing chains (i.e., possessing an active center) of intermediate sizes.

5. Molecular species in the intermediate stages of growth can be readily isolated. Molecular weight slowly increases throughout the process and it is only at a very high range of conversion that polymer molecules of very high molecular weight are obtained.

Molecular species in the intermediate stages of growth cannot be isolated. Conversion of monomerto polymer increases with time but the molecular weight of the polymer remains more or less unchanged with the progress of reaction.

Condensation(S~ep-Crow~h)Polymerization

317

Table 5.1 (contd.) Step-growth/condensationpolymerization Chain-growth/addition polymerization 6. Control of molecularweight is achieved by using a stoichiometric imbalance of the reactive functional group or a calculated amountof an appropriate monofunctional monomer.

Control of molecularweight is achieved by employingappropriate concentrations of initiator and monomer and temperature of polymerization, and for a given set of conditions by using calculated amountsof a chain transfer agent.

7. Backboneof polymerchains contains heteroatomssuch as N, O, S, etc., at regular intervals due to condensed interunit links.

Usually the backbone of polymer chains consists of-C-C-linkages and other kinds of atomssuch as O, N, S, etc., mayappear in the side groups.

of both functional groups of a bifunctional monomer(e.g., both hydroxyls of a diol) are the same; (2) the reactivity of one functional group of bifunctional reactant is the same irrespective of whether or not the other functional group has reacted; and (3) the reactivity of a functional group is independent of the size of the molecule to which it is attached. These simplifying assumptions, often referred to as the concept of equal reactivity of functional groups, make the kinetics of step polymerization identical to those for the analogous reaction of small molecules. Thoughthe assumption of equal reactivity is, at first sight, startling, there appear to be two major reasons for the validity of this assumption. 1. Contrary to the fairly commonmisconception that lower diffusion rates of larger molecules reduce the reactivity of the attached groups, the observed reactivity of a functional group is, however, dependent on the collision frequency of the group (not on the diffusion rate of the whole molecule), since in the final analysis, it is the encounter of functional groups that results in chemical reactions. The collision frequency is the number of collisions one functional group makes with other functional groups per unit time. Thoughlarger (and heavier) polymer molecules will diffuse more slowly through the solution, the functional groups attached to the chain ends may encounter each other at much higher rates (than suggested by the masses of the molecules) due to the high frequency of changes in polymer chain conformations. The collision rate and hence reactivity of a chain-end functional group would therefore be expected to be approximately independent of chain size. 2. The reduced mobility of the polymer molecule due to increase in size,

318

Chapter 5

however, alters the time distribution of collisions of the functional groups attached to chain ends. A lower diffusion rate means that any two functional groups will undergo more total collisions before diffusing apart. In other words, two potential reactants (functional groups) will not come together rapidly as diffusion becomesslower, but once brought together will not separate rapidly either, thereby increasing the number of collisions per encounter. Thus not only does the time between encounter increases as the molecular size increases (i.e., as the diffusion rate decreases), the average number of collisions per encounter also increases. A consequence of these two effects is that the observed rate constant would be independent of molecular size if the chemical reaction is very slow compared to diffusion. Onthe other hand, if the chemical reaction is very fast relative to diffusion, the observed rate constant would depend on molecular size. The former situation is usually obtained in step polymerizations, since reaction between two functional groups in a step polymerization occurs only about once in every 1013collisions [1]. It is therefore a valid approximation to look upon polyesterification, polyamidation, or polyurethane-forming reactions, and so on, as general reactions of the forms shownby Eqs. (5.13) to (5.15), respectively, in which the respective reaction rate constant does not depend on the sizes of the molecules to which the functional groups are attached: vww-COOH + HO--vwvv w~-COOH

RATES

O ~ vww-C-NH-v~v

+ H~N-wwv

vww-NCO + HO-wvvv

O ~ wvw-C-O-vww

--~

+ H~O + H~O

O vww-NH-C-O-vww

OF POLYCONDENSATION

(5.13) (5.14) (5.15)

REACTIONS

Since the reactivity of two functional groups (or rate constant) is independent of the size of the molecule to which they are attached, it is possible to measure the polycondensation reaction rate simply by determining the concentration of functional groups as a function of time. For example, in a polyesterification reaction, the general reaction at any time t is as shownby Eq. (5.13) and, as the reaction proceeds, the functional groups -COOH and -OH disappear at the same rate. The rate of the polyesterification reaction can then be defined simply as reaction rate --

- d[cooH] dt

(5.16)

Condensation(Step-Growth)Polymerization

319

and the reaction rate can be measured easily by titration of the unreacted carboxyl acid groups during polyesterification. Let us consider the polyesterification of a diacid and a diol to illustrate the general form of kinetics of a typical step polymerization. Like simple esterification, polyesterification is an acid-catalyzed reaction [2] O vww-C-OH

OH ÷ + H ~ wwv--C-OH

(5.17)

(I) followed by reaction of the protonated species (I)-with the alcohol to yield the ester (III): OH I vww-C-OH +

OH

~.-OH ~- ~-C-OH +1 ww-OH (II)

(~)

(5.18)

OH

0 k~ + wwv-C-OH ~ wwv-C-O-wwv +l ~6 wwv-OH (II) (III)

+ H~O + H

(5.19)

In the above equations, the wavylines (vvvvv) are used to indicate that these equations apply to all acid or alcohol species in the reaction mixture, i.e., monomer,dimer, trimer, r~-mer. The rate of a step polymerization is usually most conveniently expressed in terms of the rate of disappearance of the reacting functional groups. Thus, the rate of polyesterification, P~, can be expressed as the rate of disappearance of carboxyl groups :

- [COOH] R~-

dt

where [COOH]represents the concentration of unreacted carboxyl groups. The polyestefification can thus be followed experimentally by titrating for the unreacted carboxyl groups with a base during the course of the reaction. Polyesterification reactions, like manyother step polymerizations, are equilibrium reactions. However,for obtaining high yields of high-molecularweight product such polymerizations are usually run in a manner so as to continuously shift the equilibrium in the direction of the polymer. In the case of a polyesterification this is usually accomplished by removal of the water, which is a product of the reaction of species II (Eq. 5.19). Underthe

Chapter5

320

conditions the kinetics of the polymerizationcan be treated by considering the reactions in Eqs. (5.18) and(5.19) to be irreversible, that is, k4 and are nonexistent. (A brief consideration of the kinetics of a reversible step polymerizationis given on p. 336.) For the usual polyesterification, kl, k2, and k5 are large compared to k3. So whenthe reaction is run under nonequilibrium conditions by continuous removalof the by-product water, the polymerization rate can be considered to be synonymous with the rate of formation of species II: + -- d[COOH] dt ]- ]g3[C(OH)2][OH

Rp -

(5.21)

4-

where[OH]and [C(OH)2]represent the concentrations of hydroxyland protonated carboxyl groups, respectively. Theconcentrationterms are in units of molesof the particular functional group per liter of solution. Equation (5.21) is inconvenient in that the concentration of protonated carboxyl groups is not easily determinedexperimentally. A moreconvenient expression for the polymerizationrate i~ obtained by substituting for [C(OH)2] from the equilibrium expression for the protonation reaction (Eq. 5.17): + K-

]gl

__

[C(OH)2]

(5.22)

+1 ]~2 [COOH][H Combinationof Eq. (5.21) with Eq. (5.22) yields + --d[COOH] ] kl/¢3 [COOH] [OH] [H _ (5.23) dt k2 Twodistinct kinetic situations arise from Eq. (5.23) dependingon the sourceof H+, that is, on whetheror not a strong acid such as sulfuric acid or p-toluene sulfonic acid is externally addedas a catalyst. Uncatalyzed (Self-Catalyzed)

Polyesterification

The term uncatalyzed is to be understood here to meanthat the reactions are carried out in the absenceof externally addedstrong acid catalyst. In the absence of an added strong acid catalyst, the diacid monomeracts as its owncatalyst for the esterification reaction. Assumingthat [H+] is proportional to [COOH],Eq. (5.23) can be written d[COOH] (5.24) - k[cooml2 [OHI dt wherek is the overall rate constant. For equimolarinitial concentrations of the diacid and diol, [COOH] = [OH] = C and Eq. (5.24) may then be simplified

321

Condensafion (S~ep-Grow~h) Polyrneriza~ion

dC dt or

3-- kC

dC

aC

- kdt

1

1

(5.25) (5.26)

Integration of Eq. (5.26) yields 2kt _

C2 Co

(5.27)

whereCois the initial concentrationof functional groups,that is, C ---= Co att = 0. It is convenienthere and for manyother purposesas well to introduce a parametercalled the extent of reaction anddesignatedby p, whichrepresents the fraction of functional groups initially present that have undergone reaction at time t. Thus, Co - C (5.2s) ivCo TheconcentrationC at time t of either hydroxylor carboxylgroupsis then given by C = Co- Cop = Co(1-p) (5.29) Combinationof Eqs. (5.27) and (5.29) yields 1 -- 2C:~t + I (5.30) (1 - ~)2 A plot of 1/(1 -- p)2 vs. t should thus be linear. However,whenplots are madeof experimental values of 1/(1 -- p)2 versus time, it is found that Eq. (5.30) is not obeyedfrom p = 0 up to p = 0.80 (i.e., for first 80%of the esterification of -COOH and -OHgroups) [3]. Above80% conversionthe integrated rate expression(5.30) is obeyedvery well. It may be noted that the deviations below80%conversion are also observed for simple esterifications such as whenthe dicarboxylic acid is replaced by a monocarboxylicacid. Apparently, the major reason for deviations from Eq. (5.30) in the initial region below80%conversion is that there occur large changes in the polarity of the reaction mixture as the polar carboxylic acid groups are converted to the less polar ester linkages. At high conversions the ester product is the solvent and the reaction mediumbecomesessentially invariant. The prevalent view [4] (although not universal) amongpolymer chemists is that the kinetics of condensationpolymerizationhave meaning only for the last 20%of the reaction whenthe reaction mediumhas become essentially invariant and the true reaction rate constantsare to be obtained from the linear plots of 1/(1 -- /))2 versus time in this high conversion region.

322

Chapter 5

Problem5.1 -Equimolar mixture of 1,10-decanediol

and adipic acid was p01y merized under mild conditions to 82% conversion of the original carboxyl groups and the resulting product was further polymerized at a higher reaction temperature without any externally added catalyst, yielding the following data [5] for condensation of the residual carboxyl and hydroxyl end groups. The reverse reaction of hydrolysis was avoided by removing the water of condensation by passing a stream of dry nitrogen through the mixture. Temperature 190°C Time (min) % Reaction 0 0 30 20.6 60 39.0 90 50.2 150 61.2 225 66.8 300 71.5 370 74.4 465 77.2 510 78.2 550 78.8 600 79.6 660 80.6 730 81.7 800 82.5 Data from Ref. 5.

Temperature 161°C Time (min) % Reaction 0 0 20 9.1 40 16.0 100 31.6 150 41.1 210 47.9 270 52.5 330 57.0 390 60.0 45O 62.6 510 64.6 550 65.5 700 69.2 840 71.9 880 72.4

Determine the rate constants and the activation energy for the uncatalyzed reaction.

Answer: Temperature 190°C At t = 0, conversion (p) of the original COOHgroups = 0.82 1 1 -- 5 -- 0.3086 x 102. (1 - p)~ (1 - 0.82) At t = 30 min, conversion (5o) of the original COOH groups = 0.82 + (1 - 0.82) ×0.206 = 0.857 and 1 1 --- 0.49 x 102 ~ (1 - p)2 (1 - 0.857) Values of 1/(1 - p)2 are similarly calculated for other t values and plotted against t in Fig. 5.1 which shows that Eq. (5.30) is obeyed very well in the later stages of conversion. Thus, 2C~ok "1 = slope = 1.23 min

Condensation(Step-Growth) Polymerization

323

Molar mass of adipic acid (C6H1004) 146 g mo1-1 -~ Molar mass of 1,10-decanediol (C10H2202)= 174 g mol Considera mixture of 1 moI adipic acid and I tool 1,10-decanedioI. Total mass = 0.32 kg (2 molescarboxyl or hydroxylgroup) Co = [COOH]o =[OH]o = (0.32 -1 = 6.25 tool kg -~) 2Therefore, (1.23mol rain k~9°°c = 2 (6.25 kg-1) = 1.57×10_2 kg2 tool_ 2 min_ ~ Temperature 161°C The kinetic data for 161°Care similarly plotted in Fig. 5.1, from which slope = 0.428 rain 2[COOH]2ok = -1 -~) (0.428 min k~s~0c = 2 (6.25 mol 2kg-1) = 5.48 × 10-3 kg2 tool -2 -~ min From the Arrhenius expression ln [klgooc/k161oc] E =

k = Aexp(-E/RT),

- -(~3~°K46~K) (1.987 cal mol-1-~-)E In (k~ooc / k161oc)(1.987 cal mo1-1°K-l)

(4341_OK 46~°K) = 14.5 × 103 cal mo1-1

The rate constants obtained by the above method for uncatalyzed polyesterification of adipic acid with different glycols are Shownin Table 5.2. The Arrhenius parameters A and E of the equation k A exp(-E/RT) are also presented in Table 5.2 for those reactions that have been studied kinetically at more than one temperature. Note that the concentration units of the rate constants are in terms of "moles per kilogram," which is a more convenient measure of concentration than the usual "moles per liter" because the volumeof the system decreases significantly due to reaction. Catalyzed Polyesterification If a strong acid, such as sulfuric acid or p-toluene sulfonic acid, is added to a polyesterification system, [H+] in Eq. (5.23) then represents the

324

Chapter 5

10

96.8

31

96.5

95.9 161~C 95.0

92.8 0

I

0

I

~00

I

i

800

i

= 1200

82.0

Time (min)

Figure5.1 Plot of Eq. (5.30) for the later stages of uncatalyzed polyesterification of adipic acid and 1,10-decamethylene glycol (Problem 5.1). (Note that the reaction time of zero corresponds to 82%conversion of the original COOH groups present.)

concentrationof this addedcatalyst. Since the catalyst concentrationremains constant duringthe courseof the polymerization[see Eqs. (5.17) and (5.19)], Eq. (5.23) can be written d[COOH] _ k’[COOHI[OH] (5.31) dt where the three rate constants kl, k2, k3, and the catalyst concentration have been collected into a single rate constant k’. Polyesterification in the presence of externally added strong acid catalyst is usually referred to as catalyzed polyesterification and that in the absence of strong acid catalyst as self-catalyzed or uncatalyzed polyesterification. For a typical polyesterification [5], k’ of the catalyzedreaction [Eq. (5.31)] is close to 2 orders of magnitude larger than k of uncatalyzed reaction [Eq. (5.24)]. Assuming k’ >> k, the

325

Condensation (Step-Growth) Polymerization

Table5.2 Kinetic Parameters fo~ Uncatalyzed Polyesterification

of Adipic Acid

awith Several Glycols Glycol HO-(CH2)~-OH HO-(CH2)10-OH HO-(CH2)12-OH HO-(CH~)2-O-(CH2)2-OH Data from different sources.

A (kg 2 mo1-2 min-1) 4.8 × 104 4.7 × 102

E (kcal mo1-1) 14 11

k at 202°C (kg:~ -1) mo1-2rain -,~ 0.005 0.0175 0.0157 0.0041

rate of polymerization in the presence of strong acid catalyst represented by Eq. (5.31). For equimolar initial concentrations of diacid and diol

can thus be

[COOH] = [OH]= C and Eq. (5.31) is simplified

dC - 2k’ C dt Integration t = t yields

of Eq. (5.32)

between

k’t Combination of Eq. (5.33)

C = Co at/;

1 C

1 Co

with Eq. (5.28)

- p) --

(5.32) = 0 and C = C

(5.33)

leads

Cok’t + 1

(5.34)

which is a description of the dependence of the conversion on reaction for catalyzed polyesterification.

time

Problem5.2 Howwould you modify Eq. (5.33) for a case of polyesterification where k is not negligible relative to Answer: For the atypical situation where k is not negligible relative to k’, the kinetic expression for conversion as a function of time must be derived starting with a statement of the polymerization rate as the sum of the rates of the catalyzed and uncatalyzed polymerizations : 2 k’[COOH] - d[COOH]/dr = k[COOH] 3 + (P5.2.1)

326

Chapter 5

For equimolar initial

concentrations of diacid and diol,

[COOH] = [OH] = C and Eq. (Pill) is simplified -dC/dt = kC 3 2+ k’C

(P5.2.2)

Integration of Eq. (P5.2.2) yields [5] ~ [C(~Co + ~1 + ~ ~ k’t = k~ In[co(kC + k,)] ~- ~oo

(Ps.za)

The natural log term on the right side of Eq. (P5.2.3) is the contribution of the uncatalyzed reaction. Its relative importance increases as k/k’ increases. When k/k’ is very small, Eq. (P5.2.3) converts to Eq. (5.33).

Problem5.3 Show that for the external acid-catalyzed

polymerization of an equimolar mixture of a diacid and diol-the time required to go from 98% to 99% conversion is nearly same as the time to reach 98%conversion from start of polymerization. Answer: From Eq. (5.34), For p = 0.98, Therefore,

t

t0.gs

P k’Co(1 - p) 49 -- k’Co and for p = 0.99,

t0.99 - t0.gs =

50

99 "~0.99

~ --

~-- t0.98

Problem5.4 In another series of experiments similar to those in Problem 5.1 the low-molecular weight polyester corresponding to 82% conversion of the original COOHgroups were further polymerized at 161°C in the presence of p-toluene sulfonic acid (0.004 mol per mol of polymer) yielding the following conversion data [5] of the end hydroxyl and carboxyl acid groups: Time

(min) % Reaction 0 0 5 34.6 10 54.7 15 65.5 20 70.8 30 77.9 Data from Ref. 5.

Time

(min) 40 50 60 75 90 105

% Reaction 82.9 85.7 87.9 90.1 91.5 92.6

Determine the rate constant for the catalyzed polyesterification.

327

Condensation (S~ep-Grow~h) Polymerization Answer: Let p = conversion of the original COOH groups. At t = 0, p = 0.82, 1/(1 -- p) = 1/(1 - 0.82) = 5.550. At t = 5 rain,

p=0.82 + (1 -- 0.82)x0.346

=0.8823, 1/(l--p)

=8.49.

Values of 1/(1 -p) are similarly calculated for other t values and plotted against t in Fig. 5.2 which shows that Eq. (5.34) is obeyed very well in the later stages of conversion. Thus, Co k’ -1 = slope = 0.67 min Co = [COOH]o = 6.25 mol kg -1 (see Problem 5.1) k’ = (0.67 min-1)/(6.25 molkg -~) -~ = 0.107 kgmo1-1rain

Comparison of Fig. 5.2 with Fig. -5.1 shows that the catalyzed polyesterification is significantly faster than the uncatalyzed reaction. Similarly, there is much greater rate of increase of the degree of polymerization Xn (see p. 330) with reaction time in the catalyzed polyesterification (Fig. 5.2) relative to the uncatalyzed reaction (Fig. 5.1). Catalyzed polyesterification is therefore economically more feasible.

100

8O

..t’ ~o

20

0

!

I

/,0

I

I

80 Time(rain)

I

I

120

Figure 5.2 Later stages of catalyzed polyesterification

of adipic acid and 1,10decamethylene glycol (Problem 5.2). (Note that the reaction time of zero corresponds to 82% conversion of the original COOHgroups present.)

328

Chapter 5

Second-order rate constants for some acid-catalyzed polyesterifications and polyamidations obtained by the above method are shown in Table 5.3. The Arrhenius parameters A and /3 of the equation k~ = A exp(-E/.RT) are also tabulated for those reactions that have been studied kinetically at more than one temperature. Most step polymerizations are carried out with close to stoichiometric amountsof the two reacting functional groups and Eqs. (5.30) and (5.34) then applicable. The kinetics of polymerizations involving nonstoichiometric amounts of A and B functional groups can also be handled [6,7] in a straightforward manner (see Problem 5.5). It becomes necessary to adopt this method only when the mole ratio of A and B functional groups is considerably different from unity.

Problem5.5 A nonstoichiometric mixture of a dicarboxylic acid and a glycol having 20 mol % excess of the latter and a carboxyl group concentration of 5.64 mol/kg, was polymerizedto about 80%esterification of the carboxylic acid groups. The mixturewas then further polymerizedin the presence of a strong acid catalyst whichyielded the followingdata of the concentration of carboxylic acid groupsin the mixture versus reaction time. Time (min) 0 5 10 15 20 25

[COOH](mol/kg) 1.24 0.78 0.52 0.38 0.27 0.20

Determine the reaction rate constant for the catalyst polyesterification. Answer: Stoichiometric

[COOH]o 1 imbalance, r -- [OH]o -- 1.2 -- 0.833

(P5.5.1)

where [COOH]oand [OH]o are the concentration of the original COOH and OH groups: [COOH]o= 5.64 mol kg-1, -1. [OH]o = 6.77 mol kg The polymerization rate, defined as the rate of disappearance of the functional groups present in deficient amount,is given by (P5.5.2) - d[COOHl/dt = k’[COOH][OH] wherek’ is the rate constant of the catalyzed reaction. The following stoichiometry holds: (P5.5.3) [COOH]o - [COOH]= [OHIo -- [OH]

329

Condensation (Step-Growth) Polymerizafion Combinationof Eqs. (P5.5.2) and (P5.5.3) followed by integration

[6] yields

I

([COOH]o [OH] ~ [OHio-- [COOH]oIn \[-~-~o[-~-0-~--~)

(P~.8.4)

= k’t

which is combined with Eq. (P5.5,1) to give In [COOH] -

-lnr

+

[COOH]ok’t

(P5.5.5)

To test Eq. (P5.5.5), [OH] is calculated from Eq. (P5.5.3) and In ([OH] / [COOH]) is plotted against t (Fig. 5.3). The plot is linear with a positive slope of 0.049 rain -~. Therefore, -1 0.049 min k’

= (0.049

rain -~)

-~ 1 2 ~g ~.64

tool

kg

= 0.04~ kg tool -~ -~ rain

Table5.3 Kinetic Parameters for Some Acid Catalyzed Polyesterifications

and

Polamidations Reaction system

Catalyst

T k’ (°C) (kg mo1-1 -1) min

A E (kg mo1-1 (kcal min-1) -1) tool

HOOC-(CH2)4-COOH + HO-(CH2)2-OH

p-Toluene sulfonic acid (0.4%)

109

0.013

HOOC-(CH2)4-COOH + HO-(CH2)~0-OH

p-Toluene sulfonic acid (0.4%)

161

0.097

H2N-(CH~)~0-COOH

m-Cresol (solvent)

176

0.011

1.4 x 1013

31

H~N-(CH~)I~-COOH

m-Cresol (solvent)

175

0.012

1.7 x 101~

29

Data from different sources.

330

Chapter 5

2.0

,~, ,.~ 1.5 o,.9_,0 I.% 1.0

0.5 l

0

5

l

I

10 15 20 Time(rain)

I

I

25

Figure 5.3 Test of Eq. (P5.5.5) for nonstoichiometricpolyesterification (Problem

Number-Average

Degree

of Polymerization

The number-average degree of polymerization of the reaction mixture, Xn, is defined as the total number(ATo)of monomermolecules initially present, divided by the total number (AT) of molecules present at time

_ AT0 N

(5.35)

For hydroxyacids (which possess a hydroxyl and a carboxyl group on the same molecule) or stoichiometric mixtures of diol and diacid, the numberof unreacted carboxyl groups is equal to the total numberof molecules present in the system. This is obviously true for hydroxyacids. For stoichiometric mixtures this is true because each molecule larger than a monomerwill on the average have a hydroxyl at one end and a carboxyl at the other end, while each diacid monomermolecule contains two carboxyls and each diol monomercontains no carboxyls, which makes an average of one carboxyl per molecule. Thus Eq. (5.35) can be written -~,,

= Cole

(5.36)

where Co is the initial (at t = 0) concentration of hydroxyl or carboxyl groups and C is the concentration at some time t. CombiningEqs. (5.28)

33I

Condensation (Step-Growth) Polymerization and (5.36) one obtains ~n --

1 1-p

(5.37)

This equation is sometimesreferred to as the Carothers equation and is applicable to all step polymerizationsrepresentedby Eqs. (5.3) and (5.4) systemscontaining stoichiometric amountsof A and B groups. Note that X,~ given by Eq. (5.36) or (5.37) is the number-average degreeof polymerization of the reaction mixtureandnot just of the polymerthat has beenformed. The number-averagedegree of polymerization Xn of a polymer is defined as the average numberof structural units per polymer chain. The residue from each monomer in the polymerchain is termed a structural unit (or a monomer unit). Thus, for polyester synthesized from a hydroxyacid monomer,the repeating unit of the chain consists of one structural unit, whichis the residue of the monomer. For the polyester of a diol anddiacid, on the other hand, the repeating unit of the chain consists of two structural units, one each of the diol and diacid. Recall that the average degree of polymerization DPnis d_~efined as the average numberof repeating units per polymerchain; thus, Xn is not necessarily equal to DPn.For example, for an average polyester chain represented by H-(-O-R--CO-)loo-OH

(iv)

both Xn and DAPnare equal to 100, while for an average chain represented by H-{-O-R-O0C-R’-CO-)loo-OH

(v) Xn = 200 and DPn = 100. The number average molecular weight, Mn, is related to Xn by M, = X,.Mo

+Meg -

(1

-p)

+Meg

(5.38)

whereMois the molecularweight of the monomer residue in structure (IV) and the meanmolecular weight of the two monomerresidues in structure (V). Megis the molecular weightof the.end groups of the polymer.Taking the example of polyesterification of adipic acid, HO2C(CH2)4CO2H, and ethylene glycol, HOCH2CH2OH, the repeating unit is

332

Ghapter5

and one-half of its weight or 86 is Mo. The end groups are H- and -OH, and Meg is thus 18. For even a modest molecular weight polymer the contribution of Me9to Mnis negligibly small, and Eq. (5.38) becomes ~,,

= ~,~.Mo

-

Mo (5.39) l-p By combining Eq. (5.38) with Eqs. (5.30) and (5.34), the dependence of the molecular weight on reaction time for catalyzed and uncatalyzed polyesterifications, respectively, is obtained as shownby Mn = Mo (1 + 2Co2kt) 1/2 --]- Meg (5.40) ~,~ = Mo (1 + Cok’t) + Meg (5.41) The kinetic expressions (5.30) and (5.34) are obeyed for esterifications above ,~ 80%. For conversions of this magnitude the values of t are usually sufficiently large that unity in the parentheses 0f Eqs. (5.40) and (5.41) may be neglected and so also Meg, which has a value of 18. The approximate equations ~n (uncatalyzed) "~ Mo[COOH]o1/2 (2k)l/2t (5.42) (5.43) Mn (catalyzed) ’~ Uo [COOH]ok’t where [COOH]ois the concentration of the carboxyl group at t = 0, are then obtained. These equations may be used with the rate constant data of Tables 5.2 and 5.3 to construct curve_~_s of number-average molecular weights as a function of time. Evidently, Mnincreases muchfaster with t in catalyzed polyesterification than in uncatalyzed polyesterification. While most of the kinetic relationships derived in the preceding sections have referred to polyesterification reactions between a dicarboxylic acid and a glycol, extension to other step-growth polymerizations of bifunctional monomers can be done in a straightforward manner. EQUILIBRIUM Closed

CONSIDERATIONS

System

Since most step polymerizations involve equilibrium reactions, it is important to analyze how the equilibrium affects the extent of conversion and molecular weight. An important consideration is whether or not an equilibrium polymerization will yield a high-molecular-weight polymer if carried out in a system where none of the products of the forward reaction are removed. Such a system is referred to as a closed system. In this system, the concentration of products (polymer and usually a small molecule such as water) build up until the rate of the reverse reaction becomesequal to the forward-reaction (polymerization) rate. The reverse reaction is usually

333

Condensation(S~ep-Growth)Polymerizafion

referred to as a depotymerization reaction (other terms such as hydrolys/s or g~ycolysis may be used in specific systems). The polymer molecular weight is determined by the extent of forward reaction when equilibrium is established. Consider the polyesterification of an equimolar mixture of diacid and diol, represented by the equilibrium reaction K vww COOH + wvw OH ~ vwv COO vwvv + H20 (5.44) in whichtheinitial hydroxyl groupandcarboxyl groupconcentrations are both Co. The concentration of ester groups [COO]and [H20] at equilibrium are both peCo where Pe represents the extent of reaction at equilibrium. So the.concentrations of hydroxyl [OH] and carboxyl [COOH]groups at equilibrium are each Co(1 - Pe). The equilibrium constant for reaction (5.44) is given by 2 K = [cool [H20] = (peCo) (5.45) 2 [COOH] [OH] [Co (1 -- Ve)] which simplifies to K

--

(1

P~

(5.46)

Solving for Pe by the general solution for a quadratic equation yields

K+x/ P~

-

K-

The root Pe = (K -1- V~) / (K values greater than one. The useful root is

(5.47)

1

has no meaning as it yields p

1

+

~

(5.48) Equation (5.48) yields the extent of conversion at equilibrium as a function of the equilibrium constant. Substituting for p from Eq. (5.48) in Eq. (5.37) yields an expression for the degree of polymerization as a function of K: ~,~ = ~+ 1 (5.49) Equation (5.49) indicates the limitation imposed by equilibrium on the synthesis of a high-molecular-weight polymer. Thus, according to this equation even for a high equilibrium constant of 104, a degree of polymerization of only about 100 can be obtained in a closed system. A consideration of the equilibrium constants for various step polymerizations [4, 8-12] readily shows that polymerizations to obtain high-molecular-weight polymer cannot be carried out as closed systems. For example, K for a polyesterification

334

Chapter5

is typically no larger than 1-10, K for a transesterification is in the range 0.1-1, and K for polyamidation is in the range 102-103. Even for K = 103, ~,~ is only about 32. Open Driven

System

The inescapable conclusion that follows from the discussion in the preceding section is that to synthesize a high-molecular-weight polymer a step polymerization must be carried out as an open driven system, that is, at least one of the products of the forward (polymerization) reaction [Eq. (5.19)] be removedfrom the reaction mixture so as to drive the reaction to higher conversions required for a higher molecular weight [cf. Eq. (5.37)]. It usually more convenient to remove the small molecule by-product (such as H20, HC1, CHaOH)rather than the polymer from the reaction system and this can be done by a combination of higher temperature, reduced pressure, and purging with inert gas. (By-product HC1can also be removed by having a base present in the reaction system to neutralize the hydrogen chloride.) The extent to which the concentration of the small molecule by-product must be reduced in order to achieve a desired molecular weight can be calculated from the equilibrium constant and the equation relating the degree of polymerization to molecular weight. Thus, Eq. (5.45) can written as

p [H20](5.50) [COO] [H20]= [Co(1 pCo[H20] K = [COOH][OH] -p)]2 = o(1 and combined with Eq. (5.37) to yield [HzO] --

KCo KCo X,~ (X~ - 1) - X2,~

for large

-X~

(5.51)

where Co = [COOH]o= [OH]°. Equation (5.51) which applies equally to A-B and stoichiometric A-Aplus B-B polymerization, indicates that the concentration of H20 in the reactor system must be greatly reduced to obtain high Xn values. It is also seen that the lowering of [H20] necessary to achieve a particular Xn is less, the more favorable the equilibrium (that is, the larger the K value). Thus the synthesis of polyainides (with typical K values > 102) is clearly easier from the equilibrium viewpoint than polyester synthesis (K ,’-, 0.1-1). It should be understood that the desired approach with an open driven system is to control the [H20] so as to obtain the required degree of polymerization and not simply to lower [H20] as muchas possible.

Condensation(Step-Growth)Polymerization

335

[-COO-][H20] Problem 5.6 The equilibrium constant K = [-COOH][-OH]for the esterification reaction of decamethyleneglycol and adipic acid is of the order of unity at 110°C. In the polymerization in equimolar proportions at 110°C, what weight ratio of dissolved water of condensationto polymerwouldlead to an equilibrium value of X. = 50 at 110°C ?

Let y = [-COOH] yv[COOn]o [-COO-] p [COOH]o Then, K=YT.OH]__. =U [COOH]= [COOH]o (1 We setX~

= 1/(1-p)

=

P)

(P5.6.1)

or p = 0.98

FromEq. (P5.6.1): 50py = K = 1 y = 1/[(50)(0.98)] = 0.0204 mol/mol Repeating unit of polyester: -[-O-(CH2)10-OOC-(-CH2)4-CO-]-1 Molar mass of repeating unit = 284 g tool Average molar mass of two monomer residues in repeating unit, Mo= 284 g mo1-1 = 142 g mo1-1 2 For ~, = 50, average molar mass, ~ = 50x142 or 7100 g mo1-1 Due to stoichiometric balance between COOH and OHgroups there is an average of one COOH group per molecule. Thus the water content of the reaction mixture = (0.0204 tool H20/mol COOH)(18g/mol H~O) = 5.2×10_~ g/g (7100 g/tool COOH) (It is hard to achieve such lowwater contents in practice.)

Conveniently one often carries out step polymerizations at temperatures near or above the boiling point of water for purposes of obtaining desired reaction rates with the added advantage of facilitating water removal. Driving an equilibrium in the forward direction, however, requires considerable effort, since the by-product (water or hydrogen chloride or another small molecule) must diffuse through and out of the reaction mixture. Diffusion is difficult since typical step polymerization systems are fairly viscous at high conversions. Under such conditions, the polymerization can become diffusion-controlled with the rate of reaction being controlled by the rate of

336

Chapter5

diffusion of the small molecule by-product [13], and consequently, neither Eq. (5.31) for the rate of conversion nor Eq. (5.43) for the molecular weight will then be appropriate expressions. Reversible

Polymerization

Kinetics

Although reversible or equilibrium polymerizations would almost always be carried out in an irreversible manner as an open driven system, it is interesting to consider the kinetics of polymerization for the case in which the reaction is allowed to proceed in a reversible manner. Consider reversible polyesterification with externally added strong acid catalyst represented by the equilibrium reaction k wvw COOH + vww OH = vvwv COO vww + H20 (5.52) k’ in which the initial hydroxyl group and carboxyl group concentrations are both Co. Since the concentration of the acid catalyst remains constant throughout the course of polymerization we have the rate equation

a[coo]--

a[COOH] --

k[COOH][OH] -- k’[COO][H20] (5.5a) dt dt The concentrations of ester groups [COO]and [H20] are both pCo, where p is the fractional conversion of carboxyl and hydroxyl groups. The concentrations of these two groups are each (1 -- p)Co. The rate equation then obtained as

1 ap

-- k(1 - p)2 _ k’p: Co dt This can be easily integrated to obtain [14]

(5.54)

where Pe is the equilibrium conversion given by [cf. Eq. (5.48)1

pe = ~/(,/-~ + 1)

(5.56)

and K is the equilibrium constant given by

K = ~/k’

(5.57)

CombiningEqs. (5.72) and (5.73), p can be obtained P = 1 + ~where 7 =

kCot.

v~[1 - ~xp(-2~/v~)] (V~-

1)exp(--2"r/v/-~)

(5.58)

337

Condensation(S~ep-Grow~h) Polymerization

Equation (5.58) enables the extent of conversion in a closed system below the upper limit of equilibrium conversion to be calculated as a function of time, when the rate constant and the equilibrium constant are known. It should be noted that Eq. (5.55) is applicable to reversible polymerization where the reacting monomershave equimolar initial concentrations and none of the reaction products is initially present in the reaction mixture.

Problem 5.7 Compute the conversion that would be obtained in 1 h for reversible polyesterification in an external acid catalyzed equimolarsystemwhere Cok = 4 x 10 -4 s -1 and K = 1. Comparewith the conversion that would be obtainedin irreversible polyesterification. Answer: T = Cokt = (4× 10-4 s-1)(3600

s) = 1.44

FromEq. (5.58): 1 - exp(-2 x 1.44) p = = 0.47 1+1-0 For irreversible polymerization,Eq. (5.34) gives 1/(1 - p) = 1.44 + 1 = 2.44 p = 0.59

MOLECULAR

WEIGHT

OF

STEP-GROWTH

POLYMER

The molecular weight of a polymer is of prime concern from the practical point of view, for unless a polymeris of sufficiently high m, olecular weight (> 10,000) it will not have the desirable strength characteristics. It is important to consider the main factors that affect the polymer molecular weight. The dependence of Xn, and hence molecular weight, on the extent of reaction in condensation polymerization is shown by Eq. (5.37). From the data presented in Table 5.4 it is evident that to produce a high-molecularweight polymer (Xn > 100) it is necessary to allow the reaction to proceed to a very high degree of conversion (p > 0.98) or, in other words, to product that contains a very small numberof chain ends. In general, this wili be possible only when equal concentrations of the two reactive functional groups are maintained throughout the course of the reaction. This means that not only must the reaction be started with stoichiometric reactant ratios, but the system must be free of impurities that contain the same functional groups.

338

Chapter 5

Table 5.4 Effect of Extent of Reaction on NumberAverageDegree of Polymerization % Reaction (px 100) 50 75 90 95 98 99 99.9 99.99

Numberaverage degree of polymerization (X~) from Eq. (5.37) 2 4 10 20 50 100 1,000 10,000

The data in Table 5.4 reveal that Xn, and hence molecular weight, increases very rapidly with conversion in the high conversion range (p 0.99). An increase in conversion__ from 0.990 to 0.999, for example, leads to a 10-fold increase in Xn; thus, it is not practical to try to control polymer molecular weight by adjusting the level of conversion in factory scale step-growth polymerization. An alternative procedure would be to provide, deliberately, an imbalance in the ratio of the two types of functional groups in the feed. Control

of Molecular

Weight

In the synthesis of polymers, one is usually interested in obtaining polymers of specific molecular weight for optimum properties. Molecular weights higher or lower than the desired molecular weight are equally undesirable. Even if the desired molecular weight is obtained by quenching (e.g., by cooling) at the appropriate conversion (Table 5.4), polymers obtained this manner from stoichiometric amounts of bifunctional monomers(e.g., diol and diacid) are unstable in that subsequent heating during processing leads to changes in molecular weight because the end groups (e.g., -OH and -COOH)can react further with each other leading to chain extension. This situation can be avoided and molecular weight can be controlled by adjusting the concentrations of bifunctional monomersso that they are slightly nonstoichiometric, tha~ is, one of the reactants is present in slight excess. The polymerization then proceeds to a point at which the deficient reactant is completely used up and all chain ends possess the same functional group--that is, the group that is in excess. Further polymerization is then not possible, and the polymer is stable to subsequent molecular weight changes.

339

Condensation(Step-Growth)Polymerization

For example, the use of excess diol in the polymerization of a diol with a diacid [Eq. (5.1)] yields a polyester (VI) with hydroxyl end groups which are incapable of further reaction, since the diacid has been completely reacted : HOOCRCOOH+ HO R’OH(excess) H-(-OR’O OCRCO-)n-ORtOH (5.59) (VI) The use of excess diacid accomplishes the same result; the polyester (VII) this case has only carboxyl groups at both ends whenthe diol is completely reacted : HOR~OH ÷ HO OCRCOOH(excess) HO-(-OCRC OOR’O-)n-OCRCOOH (5.60)

(wi) Another methodof achieving the desired molecular weight is by addition of a small amount of a monofunctional monomer. The monofunctional monomer limits the polymerization of bifunctional monomersbecause its reaction with a growing polymer chain produces chain end devoid of a functional group and hence incapable of further reaction. For example, the use of benzoic acid in reaction (5.5) yields a polyamide (VIII) with nonreactive phenyl groups thereby limiting the polymerization: H2NRNH2

+ H O2CR~CO2H + C6HsCO~H C6H~CO-(-NHR NHCOR’CO-).-OH (5.61) .(VIii) In practice, monocarboxylic acids, such as acetic acid or lauric acid, are often used to achieve molecular weight stabilization of polyamides. Quantitative

Effect

of Stoichlometric

Imbalance

The different reactant systems employed in step polymerizations can be classified into Type 1: Stoichiometric imbalance in A and B functional groups in the polymerization of bifunctional monomers A-A and B-B (e.g., diol and diacid or diamine and diacid) where B-B is present in excess. Let ]VAAo= number (tool) of A--A molecules initially

present.

NBBo= number (mol) of B-B molecules initially

present.

NAo= number (moO of A functional groups initially

present.

NBo= number (tool) of B functional groups initially Therefore, NAo = 2NAAoand NBo = 2-NBBo.

present.

Stoichiometric

imbalance ratio,

r =

NAo/NBo = NAAo/NBBo.

340

Ghapter ,5

(The ratio r is defined so as to have a value equal to or less than unity, but never greater than unity, i.e., B groupsare those in excess.) Extent of reaction, p = fraction of A groupswhichhave reacted. Therefore, the fraction of B groups whichhave reacted = rp. Fraction of unreacted A groups = (1 -- p). Fraction of unreacted B groups =(1- rp). Numberof unreacted A groups = NAo(1 - p). Numberof unreacted B groups = NBo(1 - rp). Total number of polymer chain ends = NAo(1-- p) h- NBo(1rp) Total number of polymer molecules = [NAo(1 -- p) + NBo(1rp) ]/2 Total numberof monomermolecules initially = NAAo + NBBo = (NAo + NBo) /2 Therefore,

NAo(1 + l/r) 2

(5.62) Whenpolymerization is 100%complete, i.e., p = 1.00, Eq. (5.62) becomes ~n--

1 + r

(5.63)

whichgives the limiting value of Xn and hence molecular weight. Whenthe two bifunctional monomersare present in stoichiometric amounts,i.e., r = 1.00, Eq. (5.62) reduces 1 Sn --

- p)

whichis the sameas Eq. (5.37).

Problem5.8 Calculate the feed ratio of hexamethylene diamineandadipic acid that shouldbe employed to obtain a polyamideof approximately 10,000molecular weightat 99%conversion.Whatis the identity of the endgroupsof this product? Answer:. Formulaweight of repeating unit -[-HN(CH2)6NHCO(CH2)4CO-]= Mo = ½ ×226 = 113 X~ = M~/Mo = 10,000/113

= 88.5

341

Condensation (S~ep-Crowfh) Polymerization From Eq. (5.62),

with p = 0.99 l+r

= 88.5 1 + r - 2r(0.99) r = 0.9974 The polymerization is carried out with either COOH/NH2or NI-I2/COOH = 0.9974. For COOH/NH2= 0.9974, all end groups will be NHz; for NH2/COOH = 0.9974, all end groups will be COOH. Xn

=

1000 500

200

p=0.995

100

20 1 2 3 ~ Mote percent excess B--B over A-A I I I ! I 1.000 0.9615 0.9901 0.980h 0.9709 Stoichiometrie ratio,r 0

I

0

I

I

I

I

Mote percent excess B with NBB 0.5 1 1.5NAAo= 2 °

Figure 5.4 Dependence of the number-average degree of polymerization ~,, on the stoichiometric imbalancefor different extents of reaction p in the polymerization of A- A with B - B.

342

Chapter 5

Figure 5.4 shows plots of Xn versus the stoichiometric imbalance, for several values of p in accordance with Eq. (5,62). The stoichiometric imbalance in this figure has been expressed both as the ratio r and the mole percent excess of the B -B reactant over the A-Areactant. Note, for example, that 1 mole percent excess of B-B over A-Acorresponds to r = (2x 100)/(2× 101) = 0.9901 and, similarly, 0.1 mole percent excess B-B A-Acorresponds to r = (2×100)/(2×100.1) = 0.999. The plots in Figure 5.4 showhow r and p must be controlled so as to obtain a particular degree of polymerization. In practice, however, there is no complete freedom of choice of r and p values in a polymerization; thus, complete control of r is not always possible for reasons of economy and difficulties in the purification of reactants. Manypolymerizations are, moreover, carried out to less than 100 percent completion (i.e., p < 1.00) for reasons of time and economy. Type 2 Addition of small amounts of a monofunctional reactant, for example, B, to an equimolar mixture of A-A and B-B. Two approaches may be used. Approach I uses a stoichiometric imbalance r, which is now redefined as

NAo

r = VBo+2N where NI~° is the number of monofunctional Bo molecules initially

(5.64) present

and other 2NBBo. [Note terms that are the ascoefficient defined above; 2 in front thus, ofNAo NI~= 2Nnno Eq. and (5.64) NBo = ° in required since one B molecule has the same quantitative effect as one excess B-B molecule on limiting the growth of a polymer chain.] Thus, for 1 mole percent excess B and NAAo= NBBo, r = 100/(100 + 2xl) ---- 0.9804. With r defined as by Eq. (5.64)~ Eq. (5.62) is applicable calculation of X,~ versus p (see Fig. 5.4). Problem 5.9 A heat-resistant aromatic polyamide of M,~ = 24,100 yielded the following composition, on hydrolysis: 39.3%by weight m-aminoaniline, 59.8%terephthalic acid, and 0.9%benzoic acid. (a) Write the formula for this polymer. (b) Calculate the degree of polymerization and extent of reaction polymerization. (C) Calculate the effect on the degree of polymerization if the polymerization had been carried out with twice the amountof benzoic acid. Answer:

(a) Consider a 100 g sample of polymer and convert wt.%composition to mole composition.

343

Condensation (Step-Growth) Polymerization ra-aminoaniline (C~HsN2) = (39.3 g)/(108 g -1) = 0. 3639 tool Terephthalic acid (CsI-l~O4) = (59.8 g)/(166 g -1) = 0. 3602 tool Benzoic acid (CzH602) = (0.9 g)/(122 g -~) = 0. 00 74 mol

Since the mole ratio of diamine and diacid is nearly 1:1, the repeating unit is

0 Formula weight (C14HloN~O~) = 238

~)

Mo = ½ (Formula weight of repeating X, = 24,

100/119

unit)

= } × 238 = 119

= 202.5

From Eq. (5.64): 2 × 0.3639 2 × 0.3602 + 2 × 0.0074

= 0.9899

From Eq. (5.62) Xn

1 + 0.9899 1 + 0.9899 - 2(0.9899)p p = 1.0001 ~--

= 202.5

Formula of the polymer: The degree of polymerization X~ = 202.5 means an average of 101 repeating units with OHas one end group and C6H5CO-as the other :

(c) With 2x0.0074 = 0.0148 tool C6HsCOOH, r

=

X,~ (at

2 × 0.3639 2 × 0.3602 + 2 × 0.0148 V = 1.000)

= 0.9704

1 + 0.9704 = 1 + 0.9704 - 2(0.9704)(1)

344

Chapter 5

Approach H uses average functionality

fay defined by

fay -- ENi fi

(5.65)

where Ni is the number of moles (either per unit volume or in the whole reaction vessel) of species i. Note that fay represents the average number of functional groups per molecule in the reaction mixture. Equation (5.65) holds strictly whenfunctional groups of opposite kinds are present equal concentrations, i.e., for stoichiometric mixtures. In nonstoichiometric mixtures, the excess reactant does not enter the polymerization in the absence of side reactions and should not be counted in calculating fay. Let us consider a polymerization system in which NAo < NBo, where NAo and NBoare number of equivalents of functional groups of types A and B, respectively, present initially. In this case, the numberof B equivalents that can react cannot exceed NAo,and therefore, fav-

2NAo (5.66) EN~o (Thus, fay represents the average numberof useful equivalents of functional groups of all kinds per molecule present initially in the reaction mixture.) Let No

= total number (mol) of monomers(of all types) present tially. N = total number (mol) of molecules (monomers plus polymers of all sizes) whenthe reaction has proceeded to an extent p. _No - ~r = number of linkages formed at the extent of reaction p. (This follows from the fact that every time a new linkage is formed the reaction mixture will contain one less molecule.) Since it takes two functional groups to form a linkage, moles of functional groups lost in forming (No - _N) moles of linkage = 2(No - _N) Extent of reaction,

p =

No. of functional groups used No.of functional groupspresent initially

_ 9,(No- N)

(5.67)

whence _N = ½(2No- _Nopfav)Therefore, from Eq. (5.35):

Xn=--_N° = _N (2_No

_No --

_Nopfav)/2

_

2 --

2

Pfav

(5.6s)

Condensation (Step-Growth) Polymerization

345

Problem 5.10 For some applications nylon-6,6 produced by polycondensation of equimolar mixture of hexamethylene diamine and adipic acid is difficult to process if its Mnexceeds 20,000. Howmuchof acetic acid per mol of adipic acid should be added so that Mn does not exceed this value? Use both the approaches I and II for the calculation. Answer: Formula weight of repeating unit Mo = 226/2

= 113

~ _ 20,000 113 Approach I

_ 177

-[-HN(CH2)6NHCO(CH2)4CO-]-

=

From Eq. (5.62): limX, -- 177 r-~l 1 + r - 2r whence r = 0.9888 Let mol (5.64): of acetic acid be added for every mol of adipic acid. From 2 x Eq. r -Therefore,

2+2xx x = 0.0113

-

1 l+x

- 0.9888

So for every mol of adipic acid 0.0113 mol of acetic acid is to be added. Approach II From Eq. (5.68), 2 lim~,~ = p-*l 2- 1 x_f~v Hence, fay = 1.9887

-- 177

Let x tool of acetic acid be added for every tool of monomer. Component H2N(CH2)6NH2 HOOC(CH2)4COOH CH3COOH Tothl amine equivalents

mol 1 1 x Total 2+x

_f 2 2 1

Equivalents 2 2 x

= 2

Total acid equivalents = 2+x fay 2 x equivalents of limiting group total moles whence x = 0.0114, in agreement with result

2x2 = 1.9887 2 -t- x obtained by Approach I.

346

Chapter 5

’I~]pe 3 Addition of small amounts of a monofunctional reactant, for example, B, to A-B type monomers, such as hydroxyacid or aminoacid. Approach I uses Eq. (5.62) with r redefined r = NAo where 2_N’I~ ° has the same meaning N,o q- 2N~ as in Eq. (5.64) o

(5.69) and NAo

NBo = Number of groups of one type (A or B) = number of A-B molecules. (Bifunctional A-A or B-B monomers can also be employed to control molecular weight in this polymerization.) The plots in Fig. 5.4 are applicable. Approach H uses the average functionality as described above under Type 2.

Problem 5.11 Poly(ethylenoxy benzoate), a fiber forming polymer produced the interchange reaction nHOCH2CH20-~--~o-OCH3 ~ --(--OCH2CH20-~)--~O--)w-.

+ nCH3OH

is too viscous to spin conveniently when its M, exceeds 50,000. Howwouldyou adjust the polymerization recipe so that Mndoes not exceed this value ? Answer: Formula weight of -/cOCH~CH20-(~CO-)- = 164 ~. _

50,000 -- 305 164 To ensure that X. does not exceed this value, a monofunctionalimpurity like (~C-OCH3

or

(~x/--C-OH

may be added.

FromEq. (5.62) l+r lim X, = r~ 1 +r - 2r Let x tool (5.69), r =

-- 305,

whence

r = 0.9935

of ~}-COOCH3 be added for every mol of monomer.From Eq. 1 1+2x

= 0.9935,

whence x = 0.0033 mol

347

Condensafion(Step-Growth)Polymerization

Alternatively, from Eq. (5.68)i 2 whence fay = 1.9934 = limX~ p-~l 2 -- fay- = 305, Let x mol of monofunctionalimpurity be needed for every tool of monomer.Then, :omponent

mol

Equivalents

f

OH(l), CHACO0(1) OH = 1, CHACO0= 1

;HaOOC~CH~CH~OH

CH3COO(1)

;H3OOC-~

CHaCOO= x OH=1, CHaCOO = 1 +x

l+x useful equivalents 2 ×1 = 1.9934, -~ moles 1 -t- x

MOLECULAR IN LINEAR

WEIGHT DISTRIBUTION POLYMERIZATION

whence x = 0.0033

(MWD)

The molecular weight distribution (MWD)has been derived by Flory a statistical approach based on the concept of functional group reactivity independent of molecular size. The derivation given below applies equally to A-B (Type I) and stoichiometric A-A plus B-B (Type II) polymerizations. Consider, as an example, an A-B type polymerization x HO-R-C-OH --~ H-(--O-R-(~-)-x-OH O 6 The polymer chain is

+ (x-l)H20

O O O O II II li II HO-R-C-O-R-C-O-R-C-O-R-C-O ........ 1 2 3 4 (IX)

O R-~-O-H x

Probability that a given group has reacted = fractional extent of reaction (p). Probability that a given group has not reacted = 1 - p. In a polymer consisting of x monomerresidues, the number of ester linkages = (x - 1). Probability that the molecule contains (x - 1) ester groups pz-1. Probability that the xth carboxyl group is unreacted = (1 -- p). Therefore, the probability that the molecule in question is composed of exactly x units = pX-l(1 -- p) = nx (5.70)

348

Chapter

where nz is the mole or number fraction of molecules in the polymer mixture which are x-mers, and is also given by

nx = N /N. Here Nz is the number of molecules which are x-mers and N is the total number of molecules at the extent of reaction p. However, N = _No(1 -p), which is the same as Eq. (5.68) with fay Equation (5.70) therefore becomes N~ = No-(1 - p)2p~-X (5.71) Let Mo= formula weight of a monomerthat has reacted at both ends, i.e., -R-COO- (the end groups OH and H having reacted). Therefore, for an x-mer (structure IX) the formula weight = xMo (neglecting unreacted ends OHand H). Total weight of all molecules = NoMo(neglecting unreacted ends OH and H). Hence the weight fraction Wx of x-mers is ~U3x --

N~xMo

--

N~x

(5.72)

Combining Eq. (5.72) with Eq. (5.71) one obtains

= x(1 - p)2p -I

(5.73)

Equations (5.70) and (5.73) give the differential number and weight distribution functions, respectively, for linear step polymerizations at the extent of reaction p. These distributions are usually referred to as the most probable or Flory distributions. Combined or cumulative weight fraction Iz of all polymer molecules having degrees of polymerization up to and including x is obtained by integrating Eq. (5.73); this yields I~ = 1 - [1 + (1 - p)x]p ~ (5.74) Equations (5.70), (5.73), and (5.74) are plotted in Figs. 5.5, 5.6, 5.7, respectively. It is seen from Fig. 5.5 that nx decreases steadily as x increases for all values of p. This means there are more monomer molecules (x = 1) present than there are molecules of any other species (x > 1). This situation, however, refers to number of molecules; on a weight basis, these monomermolecules are of negligible importance as can be seen from weight fraction distribution plots (Fig. 5.6). These plots also show that at any given extent of reaction a certain x-mer has the largest weight fraction. For example, at 95%reaction (/9 = 0.95), the polymer consisting of 39 units (x = 39) occurs in the largest amount by weight. can calculate from Eq. (5.73) by elementary calculus the extent of reaction necessary to give the maximum yield by weight of a particular x-mer, i.e., w~(max), and also the value of x at which wz reaches maximum.

349

Condensation (Step-Growth) Polymerization

P=O’95’~n=20

3.2 2.4 1.6 -

0

L~p=0.98,

50

~n=50

100

150

200

250

X

Figure5.5 Mole fraction or number fraction distribution of reaction mixture in linear step-growth polymerization for conversions of 95%, 98%, and 99%[Eq. (5.70)]. (From Ref. 15 with the permission of the American Chemical Society, Washington, D.C.)

3.2

~p=0.90,

~w=19

~o 2.4 x

1.6~p=0.95, [ \\ p.8 0

~w=39

r//~-~~.~_p 50

=0.99, 100

150

~w=199 200

250

Figure 5.6 Weightfraction distn’bution of reaction mixture in lineal step~rowth polymerization for conversions of 90%, 95%, 98%, and 99%[F~q. (5.73)]. (From ReL 15 with the peEnission of the AmericanChemicalSociety, Washington,D.C.)

Chapter 5

350

0"6 t~’~//~’~

P = O" 98

IX

0

200

400

600

800

1000

Figure5.7 Integral weight fraction distribution of reaction mixture in linear stepgrowth polymerization for conversions of 95%, 98%, and 99%[Eq. (5.74)]. Ref. 16.)

(After

It is seen from Fig. 5.7 the integral curves are progressively displaced toward greater chain length with greater extent of reaction (/9) and this shift becomes more marked as p moves closer to unity (i.e., 100%reaction).

Problem5.12 Calculate the extent of reaction necessary to obtain the maximumyield by weight of species consisting of 100 monomerunits in step-growth polymerization of a stoichiometric mixture. Derive first a general expression to obtain this. Answer:

To determine the value of p at which w= reaches a maximum, one has to find the value of p at which the derivation of w~ with respect to p is zero. From Eq. (5.73): dw-~5= = x(x - 1)(1

dp

- p)2p=-2 _ 2z(1 p) p=-I =

Thus the value of p at which w= is maximumis x-1 x+l 100 - 1 .... 0.98 100 + 1

p =

(P5.12.1)

351

Condensation(Step-Growth)Polymerization

Problem5.13 Showmathematically that the maximum of the weight distribution function at high conversionsin a mostprobabledistribution is located at the number average degree of polymerization X,. Answer: To determine the value of x at which w~ reaches maximum, one has to find the value of z at whichthe derivative of w~with respect to ~c is zero. ThusfromEq. (5.73): 0 (P5.13.1) At high conversions with p close to unity one can set q = 1 - p and lnp ---- ln(1 - q) ---- -q + ...... . For maximumw~ at a givenp, one thus derives from Eq. (P5.13.1), 1 = X~ [cf. Eq. (5.77) derived below] whichsignifies that, at high eonversions, maximum yield by weight is obtained for species containing the same numberof monomerunits as the average degree of polymerizationof the reaction mixture. Note: An alternative way to derive Eq. (P5.13.2) is to calculate dlnw~/dz and set it equal to zero.

Breadth

of

MWD

The number-average degree of polymerization is given by -~,~

_ ExN~ _ ExN~ _ ~_,xn~ (5.75) EN= N where the summations are over all values of x. Equation (5.75) may compared with Eq. (4.7a) for DP~.Note that n~ in Eq. (4.7a) represents the average number of repeating units in polymer chains, while nz in Eq. (5.75) represents the average number of monomer residues in polymer chains. Combination of Eqs. (5,70) and (5.75) then gives -~,~ = ~_, xp z-~ (1 - p) (5.76) Evaluation of the series summation in Eq. (5.76) at fixed p yields (see Appendix5.1)

X. = - p) x=l

- (1-

--

p p)2

1 --

(5.77)

which is the same result obtained earlier [see Eq. (5.37)]. The weight-average degree of polymerization, Xn, is given by [cf. Eq. (4.12a)]

Xw =

(5.78)

352

Chapter 5

Combination of Eqs. (5.73) and (5.78) gives -~w ~--- E 332px-1 (1 - p)2

(5.79)

Evaluation of the series summation in Eq. (5.79) at fixed p yields (see Appendix5.1) ~-I=(1-p)~(I+p)-(I+p) -~o=(1-p)2~..x2p

(i - p)3

The breadth of the molecular weight distribution

X~, Xn (I -I-

(I - p)

(5.80)

curve is then given by

(5.81)

p)

The ratio Xw/Xn is synonymous with the ratio M~o/M,,discussed Chapter 4. The z-average degree of polymerization, Xz, is given by

Xz - E z~ w~

in

(5.82)

An expression of Xz in terms of p is then obtained by combining Eqs. (5.73) and (5.82) and performing series summation (see Problem 5.15): Xz = (I + 4p + p2)

(i-

(5.83)

Problem 5.14 In a polymerization of H2N(CH2)10COOH to form nylon-ll, 95%of the functional groups is knownto have reacted. Calculate (a) howmuch monomer (in terms of weight fraction) is left in the reaction mixture, (b) weight fraction of the reaction mixture having a number-averagedegree of polymerization equal to 100, and (c) the extent of reaction at whichthe 100-merhas the maximum yield by weight. Answer: (a) FromEq. (5.73), for x = 1 and p = 0.95: W1 = ° (1)(1 - 0.95)2(0.95) = 2.5 X 10-3 (i.e., 0.25%by weight) (b) From Eq. (5.73), for x = 100, p = 0.95: wl00= (100)(1 "" - 0.95)2(0.95) = 1.56 × 10-3 (i.e., 0.156%by weight) From Eq. (P5.12.1), for x = 100: Pi0o = (100 - 1)/(i00

i) = 0 .9 8

Problem 5.15 In the polymerization of equimolar mixture of H2N(CH2)rNH2 and HO2C(CH2)4CO2H to form nylon-6,6, 99%of the carboxylic acid groups are

353

Condensation (Step-Growth) Polymerization known to have reacted. this stage.

Calculate

M~, M,,, and M~of the reaction

Mo = ½× Formula weight of repeating = 113

unit

mixture at

-[-HN(CH2)6NHCO(CH2)4CO-]-

From Eq. (5.77) X~

=

1 - 100, 1 - 0.99

~

=

X~Mo

=

11,300

From Eq. (5.80) X~

= (1 + 0.99)/(1-

0.99)

M~

= XwMo = 22,487

=

From Eq. (5.73) ~X2LOz ~---

~’~X3(1 -- p)2pz-X (i - p)2 ~-~x3pZ-1

Using the appropriate series 2l+4p+p

~

X2~

summation from Append~ 5.1,

-

(P5.15.1)

(l+v)/(1-p)

(P5.15.2)

~

From Eq. (5.80): Exw~

=X~=

Substituting Eqs. (P5.15.1) and (P5.15.2) in Eq. (5.82), Xz

(1+p)(l-p) 2 __1 +4(0.99) +(0.99) (1 + 0.99)(1 - 0.99)

Mz = X..Mo

A--A

plus

Stolchiometrlc

-- -- 298.50

= 33,730

B--B

Polymerization

Mixture

of A--A

and

The derivations given in the previous Section for A-B type polycondensation (type I) hold also for A-A plus B-B types of polycondensation (type II) reactions when there are precisely equivalent proportions of A-A and B- B reactants and the principle of equal reactivity of all functional groups of the same chemical type is applicable. Now, however, x represents the combined number of both types of units in the polymer chain. Therefore, in place of Mo in Eq. (5.38) one has to use the average weight of units A

Ghapter 5

354

and MBor one-half of the repeat unit in the polymer chain. If, moreover, there are more than 2 monomertypes in the reaction mixture, a weighted average of all units is to be used.

Problem5.16 In a synthesis of polyester from 2 moles of terephthalic acid, 1 mol of ethylene glycol and 1 mol of butylene gl__ycol, the__reaction was stoppedat 99.5%conversion of the acid. Determine (a) M~and Mwof the polyester; and (b) molefraction and weight fraction of species containing 20 monomer units. Answer: -1 MA= -OCC6H4CO-----

132 g tool MB~= -OCH2CH20- = 60 g mol-1 -~’B~ = "O(CH2)40-= 88 g mo1-1 Mo = (2MA + MB, + MB~)/4 = 103 gmo1-1 (a) FromEq. (5.77), M,

= X~Mo

-

103 g mo1-1 Mo -1 - p (1 - 0.995)

-~ -- 20, 600 g tool

FromEq. (5.80), Mo(1 + p) (1 - p) (103 g mol-~)(X + 0.995) (1 - 0.995)

M~ = X~Mo

-

41,100 g mo1-1

(b) FromEq. (5.70), noo = (1 - 0.995) (0.995) (°°-1) = 0.0045 FromEq. (5.73), woo = (20)(1 - 0.995)2(0.995) (2°-1) = 0.0005

In type II condensations three types of molecular species are present: A-AB -- B-wwvvww-A-AB -- B Type IIAB and

A-AB - B-vwwwwv-A-- AB - BA-A Type IIAA

B - BA- A-vwwwvwB - BA- AB -- B Type IIBB

Condensation(Step-Growth)Polymerization

355

All molecules containing an even number of units will necessarily be of type IIAB, while those containing an odd number of units will be of type IIAA or IIBB. In a stoichiometric mixture in which the numbers of A--A and B-B units are exactly equal, there will be as many even-z molecules (i.e., molecules with even number of units) as odd-z molecules (i.e., molecules with odd numberof units), and the latter will be equally divided between types IIAA and IIBB. The molecular weight distribution given in the previous section for type I polymers (made from A--B monomers)applies here also; only the alternation of end groups as described above will occur between successive even and odd values of z. Nonstolc2u’ometric

Mixture

of A--A and B--B

If an excess of either A-Aor B -B is employedin the reaction, the relative numbers of the three types of species described above will be altered. In particular, there will be more odd-z molecules (including both odd subtypes together) than even-z molecules. This is readily evident from the fact that there are more B than A end groups when the B- B unit is the one present in excess. At completion of the reaction, i.e., PA= 1 in this case, only type IIBB molecules will remain in the mixture. Case

I:

r # 1,

p = 1

Consider reaction of A-A with B-B wherein an AB bond is formed by reaction of an A group with a B group. Assumeno other reaction takes place and all groups A and B are equally reactive irrespective of the size of the molecules to which they are attached as end groups. Let the initial number (tool) of A groups be NAo and that of groups be NBo; at a later time there are NA and NB groups. Let NAo < NBo. Therefore, A-A is the limiting reactant which can be completely reacted. Define the stoichiometric ratio, r = NAo/NBo Define the extent of reaction, PA ~ (NAo -- NA) /NAo For the given case, r is fixed < 1 and all the A groups have reacted (i.e., PA = 1). Chain molecules are formed, of which one kind - the pentamer - is: Probability B--BA-~AB--BA--AB--B of reaction rxl lxl rxl lxl (1 - r) The probability that a B group selected at random has reacted is PB = pANAo/ NBo= r, if all A and B groups are equally reactive and if there are no other possible reactions. The probability that a B group has not reacted = 1 -- r. The probability of formation of the chain illustrated above, the pentamer,

356

Gbapter ,5

can be deduced as follows: Starting from the left end group B (the probability of the end group being B is 1 since all Agroups have reacted) and movingtoward the right the probability that the next B group has reacted is r (as explained above) and that it has reacted with Ais r X since B can react only with A. The reacted A group is followed by another A group whichtherefore meansthat the next group is a reacted B and the probability is 1 x 1. Thenext groupto the right is a B and the probability that it has reacted with an A group is r x 1. Yet another A group will therefore follow to be connectedto a reacted B with probability 1 × 1. The probability of finding the next B group unreacted is (1 -- r). Therefore the probability of selecting a pentameris P5 = (rxl)(lxl)(r× 1)(111)(1 - r) -and the correspondingprobabilities for monomer and trimer are

P1 = (1 - r) P3 = r(1 - ~) In general, Pz = v(x- 1)/2 (1 - v) (x is odd as no even numberspecies can exist.) Since the probability of picking an i-met at randomis equal to the numberor mole fraction of them, the probabilities P can be equated to molefractions (numberfractions) n, i.e.,

,~x___,.(~-1~/2. (1 - ~-)(x is

(5.S4)

The number-averagedegree of polymerization is defined as

x~ - ~=~xn~_ E x~(~-~/~.(1 - ~’) x=l

(5.s5)

Define ~=2z - 1, so that if ¯ = 1~3~5r.., ~ = 1~2~3~.... Then fromEq. (5.85),

= (~ - ") (1 -,)2

(see Appendix5.1)

l+r

(5.86)

Alternatively, this expressioncan also be derived from

(5.87)

357

Condensation(Step-Growth)Polymerization where No --

number (mol)

of mers = NAo/2 + NBo/2 = NAo(1

11)12 and

N= number (mol) of species on completion of reaction (PA = = number (moO of pairs of ends

= (NBo Therefore,

--

NAo)/2

= NAo(~--

X,~- No

1)/2 (5.88)

_ .l+r 1-r

N

which is identical with Eq. (5.86). We now turn to weight fractions of .various species x = 1, x = 3, x = 5, ¯ ..... . The weight fraction wz of x-mers is N~ x Mo n~ N x

_

_

where N~ = number (moO of species which are z-reefs = mole fraction of x-mers (nx) x total moles of species after reaction (N) and Mo= average formula weight of reefs. Substituting for N/No from Eq. (5.88) into Eq. (5.89), w~ = x.r (~-~)/~. Define x ~ 2z-1~ 2~ 3~ --.. Thus,

1,

so that

~-~XWx = ~(2Z-

~ (1 - r)

(5.90)

+

if

x = 1,3,

1)2r (z-l)

5,...,

then

z = (5.91)

(~--

which yields, using appropriate equations from Appendix5.1, o~ Exwx

[4(1 + r) = I-Gr-~ 4 1-r "~

Since

~’x~l

(1 r) 2 + 1----~ 4 1-r + l+r l+r 4

1-r

X,~

(1Z/-

r) (5.92)

X Wx,

4

and

1 ] (1 - 2r)

4

1

+r

1-r

l+r

(1

+r)

(5.93)

l+r

2

+

~

(5.94)

Chapter 5

358

For r = 1, this

reduces

to X~o/Xn = 2 [cf.

Eq. (5.81)

for

p = 1].

Problem5.17 A synthesis of poly(ethylene terephthalate)

is made by reacting one mole of dimethyl terephthalate and two moles of ethylene glycol. Calculate (a) number average degree of polymerization, (b) weight-average degree polymerization and (c) polydispersity index of the product, if the reaction carried to 100% exchange of the methyl group forming the ester link between glycol and terephthalate. (d) What are the mole fractions and weight fractions monomer,trimer, and pentamer in the reaction product ? Answer: Consider A-----COOCHa, B--OH For the given composition, r = ½ (a)~rom Eq. (5.8S),

X. = (1+ ½)/(1

(b) From Eq. (5.93),

X~o = 1

---- 3

4

4 l+l

1

= 17/3 (c)

= X~,/X,~

+ ~

1 1 --~ 1 + ½

= 17 /9

(d) From Eq. (5.S4): From Eq. (5.90):

nl = 1/2, wl 1/6,

n3 = 1/4, n5 = w3 1/4, w5 = 5/24.

Note: These values would not be correct, however, unless special precautions were taken, because in the ester interchange reaction by which methanol is eliminated, some ethylene glycol would be distilled over with the methanol due to its volatility, whereas the trimer and all higher molecular weight species are effectively nonvolatile.

Problem5.18 Assuming that from the polymerization

system of Problem 5.17 all excess ethylene glycol is distilled out along with the by-product methanol, what would be (a) X,, and (b) ~o of the reaction product Answer

(a) Zn

Ezc’~I

z~n°~=3 X12 ~ nz ~n-~3

xn z --

1(nl)

1 - nl (I + r)/(l - r) - (l--r) 1 - (I-r) = 5 (since

(b)

¢x~

W

1 - wl

r = 1/2).

E~=I zw= - l(Wl) 1 - wl

359

Condensation (S~ep-Growth) Polymerization From Eq. (5.90),

wl = 1×(½)°× (3/2) From Eq. (5.92), 4 1-r Therefore,

X~ --

4 l+r

1 - r l+r

(17/3) - (1/6) = .--33 1 - (1/6)

17 3 = 6.6

Problem5.19 In Problem 5.17, estimate the degree of polymerization of species which has the maximumyield by weight at the completion of the condensation. Answer: Substituting

x = 2z - 1 in Eq. (5.90),

To determine the value of z at which the weight fraction w~ reaches a maximum, one has to find the value of z at which the derivation of w~with respect to z is zero : dw x

__

dz

(1

-

,/.)2

(z-l)

[ 2~ -

2zlnr - ln r]

(1 + r)

z

=

= 0

-

Therefore, x ---- 2z - 1 = -2/lnr For r = 0.99, x = 199

Case

2:

r ~ 1,

(P5.19.1)

p ~ 1

In the more general case of incomplete reaction (p ¢ 1) and nonequivalence of reactants (r ~ 1), all three of the previously mentioned species, i.e., type IIAB, type IIAA, and type IIBB are present. The number-average degree of polymerization is given by Eq. (5.62). Expressions for number distribution, weight distribution, and average molecular weights are derived below. For nonstoichiometric mixtures, extents of reaction must be considered separately for the A functional groups, PA, and for the B functional groups, PB- However, these parameters are algebraically related, provided the only possible reaction is between A and B groups to yield AB links: 2NAAo

PA = 2NBBo

PB or

PA/PB

= NBBo/NAAo

where NAAoand NBBoare number (mol) of molecules

initially

(5.95) present.

360

Cl~apter 5

If x is the total number of A-Aand B-B type units and y is the numberof A-Atype units in a molecule,the three type II species identified abovecan be described by the followingalgebraic relations: Type IIAB :

x = 2y

Type IIAA :

x = 2y -- 1

Type IIBB :

x = 2y + 1

(5.96)

Besides these three types of species present in a reaction mixture, account has to be taken of the unreacted B-Bmoleculessince they are not included in any of these categories. Considering a chain molecule, for example, a pentamer in the type IIAA category A--AB--BA--AB--BA--A 1 2 3 4 5

6

For this molecule, x = 5 and y = 3. Probability of 1 being an unreacted m

No. of unreacted Agroups No. of unreacted Aand B groups

(1 (1 - pA)+ VBBo (1 -- pB)] (1 - PA) (1 -- p.~,) q- (NBI3o/NAAo)(1PB) (1 -- PA (1 - PA) (p A/PB)(1 -pn + p~ -- 2php~

(5.97)

The probability of 2 being reacted is PAand of being reacted with B is PAX1, since A reacts only with B. Similarly, the probabilities of 3 being reacted with A, 4 with B, and 5 with A are PB X 1, PAX 1, and PB × 1, respectively. Theprobability of finding the next Agroup unreacted is (1 -- PA)- Therefore, the probability that a type IIAA molecule composedof y A-A units is

(1

Pu = PA + PB- 2pAPB

(5.98)

361

Condensation (Step-Growth) Polymerization

Similarly for a IIBB type molecule containing y A-Aunits, (1 -- pB)2FYA+lp~ 3

(5.99)

PY = PA + PB -- 2pAPB and for a type IIAB molecule containing y A-Aunits,

py = 2(1-pA)(1- PAPB

(5.1oo)

PA + PB -- 2pAp~ The probabiliWfor unreacted B-Bmolecules is similarly

~ = P* (~ - P~)~ (~.~0~) PA~ P~ -- 2pap~ The mole or numberfraction of molecules containing y A-Aunits can be expressed as (1 pn)2p~-lp~ (1 pB) PA + PB -- 2pnp~ +

PA + P~ -- 2p~p~

2(1 - pA)(1 p~)pUAp~ PA + P~ -- 2pAPB

(5.102)

and the mole (number) fraction of unreacted B-Bmolecules (5.103) nBB = pA(1 -- p~)2/(pn + pB -- 2pAP~) Equations (5.102) and (5.103) give the differential numberdistribution functions for polymerization of nonstoichiometric mixtures of A--Aand B-B. In Eq. (5.102) the first term represents T/,IIAA," the molefraction of type IIAAspecies; the second and third terms represent, respectively, nlIBB and nIIABTocalculate the weightdistribution function, it is necessaryto introduce the formula weights of the base units, MAand MB(i.e., monomeric reactants that have reacted at both ends), which are not identical with the molecular weights of the monomers.For instance, the base units in poly(ethylene terephthalate) are respectively O-CH2-CH2-O and OC-C6H4CO, as comparedwith the monomers,ethylene glycol and terephthalic acid. In calculations, MAand MB will be used for both reacted and unreacted monomer units. The error introduced by this will, however, be significant only at very low molecularweights. Anexpression for weightdistribution function can be derived fromthat of numberdistribution function Eq. (5.102). Let N be the total moles all species at PA, PBextents of reaction, Theweightfraction of type IIAA species will thus be given by W,~.A = (n,U~A)N[yMA

+ (y-

1)Ms]/(NA~oMA

+ NBBoM~) (5.104)

362

Chapter 5

wherenllAA, is the mole fraction of type IIAAspecies in the reaction mixture. It is easy to showthat N JVAAo+ -N’BBoMB

NAAo(1-- PA) + NBBo(1-- PB) NAAoMA+ NBBoMB (1 - PA) + (NBBo/NAAo)(1 -MA + (NBBo/NAAo)MB PA + PB -- 2pApn (5.105) pBMA + pAM~

Substituting Eq. (5.105) and the first term of Eq. (5.102) for nIIAA Eq. (5.104), weget 1)MB WIL~A= (1 - pA)2 p~-lp~[ [yMA ~-~]~A+ +(yp-~---~

(5.106)

Theweightfractions of the remainingtwo types of polycondensatemolecules, included in Eq. (5.102) can be similarly expressed as: WIIBB

=

[yMA + (y ++ pAMB 1)MB" (1 - pB)2 p~A+Ip~ [ -~BI~

= -

a)(1

+

(5.107) (5.108)

The weight fraction of unreacted B-Bmolecules is also similarly derived from Eqs. (5.103) and (5.105): WBB= 2MB pA(1 -- pn) pBMA + pAMB

(5.109)

It maybe noted that Eq. (5.106) refers to molecules containing y A--A units and (y -- 1) B-Bunits, Eq. (5.107) to molecules with y A-Aunits and (y -t- 1) B-Bunits, and Eq. (5.108) to species having y of each of units. Equation(5.109) accounts for B-Bunits left over. Thesumof weight fractions of all species containing y A-Aunits, wy, is thus given by wy = WIIAA + W~IBB+ WIIAB (5.110) wherethe right hand terms are given by Eqs. (5.106)-(5.108). The number-averagemolecular weight, Mn, can be obtained directly from -Total weight Mn= Total moles at a given conversion

363

Condensation(Step-Growth)Polymerizafion NAAoMA

+ NBSoMB

NAAo(1 -- PA) + NBBo(1 -= pBMA + pAMB PA + PS -- 2pAps [usingEq. (5.95)]

(5.111)

The weight-average molecular weight, -~fw, can be calculated (5.106)-(5.109)

from Eqs.

-~w (psM~ pAM~ ¯ -- 1 1+PAPS -- PAPS \pBMA ++~AMB]

(1

4pAPB-hIA MB - PAPB)(PB-~/’A q- PAMB)

For stoichiometric mixture of A-A and B-B, PA = PB = P above formulas reduce to the Schultz-Flory distribution: M,~ ~o

=

1 Mo-1-p

= Mo 1 +p 1-p

and the

(5.113) (5.114)

wherein Mo is the average formula weight of the units, (MA + MB)/2 (or is the weighted average of all units if there are more than 2). The weight distribution function of A-Aunits (i.e., wv vs. y) in the reaction mixture can be obtained from Eq. (5.110). Note that it impossible to represent the weight distribution wz as a function of x (total number of units) by a single curve since the value of the weight fraction alternates between odd (i.e., IIAA+IIBB types) and even (i.e., IIAB type) members, the former exceeding the latter if r ~ 1. T he a verage curve, however, corresponds very nearly to that given by Eq. (5.73) with p replaced by pr1/2. This generalization includes Eq. (5.70) for number fraction and Eqs. (5.113) and (5.114) for number- and weight-average mofecular weights, respectively.

Problem 5.20 Consider synthesis of nylon-6,10 from hexamethylene diamine, H2N(CH2)6NH2,and sebacic acid, HOOC(CH2)sCOOH, where the mole ratio (r) of the diacid to the diamine is 0.99. At 98%conversion of carboxylic acid groups (.PA = 0.98) determine (a) M~and M,o of the product, and (b) sum of the mole fractions and of the weight fractions of species of degrees of polymerization 1 through 6 inclusive. Comparewith the corresponding values calculated as approximationsfrom equations for the stoichiometric case with PA replaced by rI/2. pA

364

Chapter 5

Answer: Let A-A represent

the diacid

MA =-HN(CH2)fNH- ----/~-/B

and B-B the diamine.

114 g mo1-1

= -OC(CH2)sCO- ~ 168 g mo1-1

PA = 0.98, r = 0.99 PB = rpA = 0.9702 (a) Substituting the values in Eq. (5.111), ~,~ = 5,660 g mo1-1 -~ From Eq. (5.112), ~,~ = 11,180 g mol (b) Sum of mole fractions through 6 is given by

F n, =

+

+

y=l

z=l

of molecules exhibiting

degrees of polymerization

+ -BB

The first term on the right hand side is calculated from Eq. (5.102), the second term from Eq. (5.98) and the third term from Eq. (5.100); nBBis calculated from Eq. (5.103). This gives 6

~’~.n~

= 0.0942

+ 0.0072

+ 0.0211

+ 0.0179

= 0.1404 Sumof weight fractions of molecules exhibiting degrees of polymerization 1 through 6 is given by 6

2

= E ZOY q- (WlIAA)y=3 nt- (WlIAB)y =3 + ~BB E z=l y=l The first term on the right hand side is calculated from Eq. (5.110), the second term from Eq. (5.106), and the third term from Eq. (5.108); wBBis calculated from Eq. (5.109). This gives 5

~w~ = 52.17×10

-a

+ 8.64×10

-~

+ 31.49×10

-t

-4 + 5.31x10

= 0.0098 Approximate values can be calculated from equations for stoichiometric with PA replaced by pArU~: p = par’~ ~ = (0.9S)(O.99)V ~ = 0.9751

mixtures

-~ /V/o = (141 + 168)/2 -~ From Eq. (5.113): From Eq. (5.114):

or 141 g mol -141. g -~ tool M~ = 1 - 0.9751

~0 = (141 g mol_~) 1 + 0.9751 -~ = 11,184 g mol 1 - 0.9751

From Eq. (5.70) ~-~ ~-~n::

= 5663 g mol

= (1-0.9751)~-~(0.9751) = 0.1404

365

Condensation (S~ep-Growth) Polymerizaffon From Eq. (5.73) ~ w:~ = (1 - 0.9751) ~ ~-~ ~ x (0.9751) z=l

= 0.01~

Problem5.21 For the above problem, calculate the mole fraction of species which have (a) both -COOHend groups, (b) both -NH2 end groups, and -COOHand -OH end groups. What would be the corresponding values if the reaction mixture were a stoichiometric one ? Allswet: (a) PA = 0.98, r = 0.99, PB = rPA = 0.9702 FromEq.(5.102), takingthefirstterm, (1 - pn)2pB

~-, V=I

nlIAA = (PA "~ ~;’~--

2-~--APB)~/=1Z(PAPB)~-I

(1 - pA)2pB (PA + PB -- 2pmvB)(1 -- P~B) = 0.1622 Mole fraction

of species

(b) From Eq. (5.102),

with -COOHgroups at both ends = 0.1622

taking the se~nd term, (1 -- pB)~p~pB

~~nlIBB

= = 0.3460

From Eq. (5.103), PA (1 - 2pB)

= 0.0179 (vh + vB- 2pApB)

r~BB = Mole fraction = 0.3639

of species with -NH2 groups at both ends

= 0.3460 + 0.0179

(c) From Eq. (5.102), taking the third term, ~-~ r~IIAB V=I

2(1 -- PA) (1 PB)PAPB

(PA -[- PB -- 2PAPB)(1 -- PAPB) = 0.4739 Mole fractionof specieswith -COOHand -NH= end groups = 0.4739 Check:0.1622+ 0.3639+ 0.4739= 1.000 Stoichiometric case : Mole fraction of species with -COOHboth end groups = mole fraction of species with -NH2 both end groups

366

Chapter 5

= 0.25 Mole fraction

of species

with both -COOHand -NH2 end groups = 0.50

MULTICHAIN OR NONLINEAR STEP POLYMERIZATION Branching Discussions so far have been concernedwith the polymerization of bifunctional monomers,whichobviously form only linear polymers. If there are also one or more monomerswith more than two functional groups per molecule in the system, the polymer formed will be branchedinstead of linear. (Withcertain monomers, describedlater, cross-linking will also take place to form network structure in which a branch or branches from one polymer becomeattached to other molecules.) Consider polymerization of an A- B type monomer in the presence of a small amountof a monomerAI containing f functional groups. The value of f represents the functionality of the monomer; thus, if f -- 3, AI represents a trifunctional monomer A---~--Ahaving three A groups. If reaction A occurs only between A and B groups and neither A groups nor B groups are capable of reacting with each other, the product of polymerization of A-Bin the presence of trifunctional AI will lead to a branchedpolymerof structure (X). A careful consideration showsthat only one AI moleculecan .~--A- BA-BA-BA- BA- BA--~AB-AB-AB-AB-AB-AB-AB-AB-A ~

~x- Branch point A A B I A B I A I A B I A

(x) be incorporated into each polymermolecule. Further, cross-linked species cannot form, since branch chains from one molecule possessing functional

367

Condensation(S~ep-Crow~h)Polymerizafion

groups A at their ends cannot react with those from another. For example, e-aminocaproic acid may be condensed with a small proportion of tetrabasic acid R(COOH)4. If the condensation is carried close to completion, linear species HO-[-CO(CH2)rNH-]z-H virtually disappear in favor multichain polymer molecules, R[-CO(NH(CH2)rCO)~OH]4,the number of these being equal to the number of tetrabasic acid molecules in the system. (Molecules possessing cyclic structures are disregarded here.) Since each chain of the multichain polymer has carboxylic end group, linking of the chains leading to a cross-linked structure does not take place.

Molecular

Weight

Distribution

Molecular weight distribution in the above type of multichain step polymerization would be expected to be muchnarrower than for a linear polymerization. Consider, for example, polymerization of bifunctional monomer A--B with a small proportion of an f-functional substance A.f leading to polymer molecules of the type o[-A(B-A)r/]f, where it is understood that ~1, the numberof units in a chain, maydiffer for each of the f chains. Each chain is subject to the same statistical opportunities for growth as a linear molecule in ordinary bifunctional condensation. The difference, however, lies in the fact that the total number:c of units in a multichain molecule depends on the sum of the y values for each of its chains. For such a multichain molecule, a size :c muchlarger, or muchsmaller, than the average is less likely than in linear (monochain)polymerization, as this would require the f branches makingup a multichain molecule to be all very long or all very short, and such cooperation amongstatistically uncoordinated components will be comparatively rare. Molecular weight distributions for multichain polymers will thus be narrower than for ordinary monochain polymers. The distribution functions for multichain polymerization have been derived statistically [17] based on the principle of equal reactivity. According to this principle, at every stage of the polymerization process an equal opportunity for reaction is available to each functional group of a given chemical type, irrespective of the size of the molecule to which it is attached. The probability that a given functional group has reacted is then equal to the fraction (extent of reaction) p of all functional groups the same type which have reacted. Denoting the fractional conversion of A groups by p, in the aforesaid polymerization of A-B with a small proportion of Ay, the following expressions are derived [16] for distribution functions and average degrees of polymerization. The mole fraction of x-mers amongthe multichain molecules is given by (x+

f-

2)!

] p~-l(1

-p)Y

(5.115)

368

Chapter 5

The weight fraction distribution is given by

~x,s= (] - 1)! (x - 1)!/ For values of p near unity, Eq. (5.175) is approximated satisfactorily the equation

wx,s ~- (x f / f!)(-lnp)f+Ip~

(5.117)

which was derived earlier by Schulz [18]. The number-average degree of polymerization, Xn, and the weight average degree of polymerization, Xw, derived from the mole fraction size distribution nz,f are

(f.p + 1 - p) / (1 (f- 1)2p2 + (3f- 2)p+

(S.p+ 1 - p)(1-

(5.ns) (5.119)

The ratio of weight: to number-average degrees of polymerization is Xw__

x,,

_ (I-

1)2p 2 -t-

(3/-

2)p q- 1

(I.p + 1 - p)2

(5.19,0)

For linear polymerization, f = 1. Equations (5.118), (5.119), and (5.120) then reduce to Eqs. (5.77), (5.80), and (5.81), respectively.

Problem 5.22 Explain on the basis of Eq. (5.120) that in the polymerization of A-Bwith a small proportion of Af, the distribution becomessharper as f increases. Answer: Let Q represent the numberof equivalents of the multifunctional reactant AI in the reaction mixture per mole of monomerA-B. If the reaction were carried to completion, i.e., no unreacted B groups remaining, there wouldthen be obtained Q/f moles of polymer molecules, each molecule having f chains, and Q moles of A end groups will remain unreacted per mole of A-Bunits. The fraction, p, of A groups reacted at completionof the condensationis therefore given by (P5.22.1) p = 1/(1 + Q) Substituting p from Eq. (P5.22.1) in Eq. (5.120) yields X~

x.

1 -F

f (1 + Q)

(f + Q)2

For small values of Q, Eq. (P5.22.2) reduces approximately 1

xw/x. -- 1+?

(P5.22.2)

(P5.22.3)

Condensation(Step-Growth) Polymerizafion

369

Equation (P5.22.3) indicates that the distr~ution becomessharper as f increases. The case of f = 1 corresponds to "l~pe 3 stoichiometric imbalance in linear polymerization (see page 346). For f = 1, Eq. (P5.22.3) gives Xw/X, = 2, as wouldbe expected for most probable distribution in linear polymerization. For f = 2, which represents the linear polymer prepared by condensing to completion an A-B monomerwith a small amount of A-A(e.g., a dibasic acid), X~,/X,~ = 1.5. Thus, linear polymerization of A-Bwith small amounts of A-A leads to significantly narrower distribution than A-Balone or A-Aplus B--B.

Cross-linking Consider polymerization of a system consisting of A-B, e.g., HO-R-COOH AI (with f > 2), e.g., HO--R’--OH and B-B, e.g., HOOC-R"-COOH Because of the presence of B - B, branches from one polymer will be capable of reacting with those from another polymer molecule. This will lead to cross-linked polymer structure. Cross-linking can be pictured as leading to the structure (XI) in which two polymer chains have been joined together (cross-linked) by a branch. The branch joining the two polymer chains referred to as a cross-link. A cross-link can be formed wheneverthere are two branches (e.g., those indicated by arrows in XI) that have different functional groups at their ends (e.g., A and B groups in XI). Cross-linking will also occur in other systems involving reactants with functionalities greater than two. A few such systems are (A--A + By), (A-A + + By), (A.f + B.f), where, f > The cross-linking reaction is extremely important from the commercial standpoint. Because of network structure, cross-linked polymers cannot be dissolved in solvents and cannot be softened by heat; strong heating only causes decomposition. At high degrees of cross-linking, polymers have rigidity and are dimensionally stable under a wide variety of conditions. Polymers or resins that are transformed into a cross-linked product, and thus take on a ’set’ on heating, are said to be of thermosetting type. Since a cross-linked product cannot be melted or dissolved, these materials are prepared, by intent, in only partially polymerized state (prepolymers), so that they may be deformed or shaped in a mold while hardening by curing (cross-linking). A few commonexamples are illustrated in Figs. 1.41.10 in Chapter 1. The eros-links are usually formed between polymer or prepolymer molecules by foreign atoms or molecules, e.g., sulfur atoms in vulcanized rubber (Fig. 1.4) and styrene molecule in polyesters (Fig. 1.8), or by small chain segments as in polyurethanes (Fig. 1.9) and glyptal resins (Fig. 1.10).

370

Ghapte~5

.~,--A-BA-BA-T-AB-AB-AB-AB-AB-AB-BA~BA-~-A,,~ A A B B

Cross4ink

A B I A B I B,

A B-AB-BA~--AB-A B-AB- BA- BA-BA--LA A B ~ I B A A B ~ I B A A B

Onset of CIelatlon Cross-linking is marked by the occurrence of gelation at some point in the polymerization reaction. Phenomenologically,the gelpoint is defined as that point in the reaction wherethe systemloses fluidity and the viscosity becomes so large that an air bubble cannot rise through it. A gel is an insoluble polymer fraction in which polymer molecules have been cross-linked to each other to form a macroscopic molecule. The gel is in fact considered as one molecule. However,at the gel point not all the material is insoluble. Only the gelled portion (and it represents only a small percentage of the total reaction mixture) is insoluble while the other portion is soluble. The portion that is soluble is referred to as the sol, whereasthe part that is insoluble is referred to as the gel. In a cross-linking step reaction system, there occurs, at first, a general rise in viscosity and then a sudden enormous rise as the gel point is

Condensation(Step-Crow~h)Polymerization

371

approached. Also the reaction, as measured by the conversion of the reactive group, proceeds very slowly as the gel point is approached. These two factors makeit difficult to predict the gel point well in advance from the experimental conversion or viscosity data. Since gelation is irreversible and results in a loss of fluidity of the system, it is obviously important to be able to predict the onset of gelation. It is therefore important to understand the relationship between gelation and the extent of reaction. Average ~-bnctionallty

Approach

Carothers derived a relationship between the extent of reaction at the gel point and the average functionality of the reaction. As we derived earlier [see Eq. (5.68)], the number-average degree of polymerization Xn of the reaction mixture of monomersinvolving A and B groups is given by 2 -(.5.121) 2 - P.fav which can be rearranged to 2 2 P - fay X..fav (5.122) Equation (5.121), which reduces to Eq. (5.37) for fay = 2 and is referred to as the Carothers equation, relates the extent of reaction and the degree of polymerization to the average functionality of the system. At the gel point, where the number-average degree of polymerization becomesinfinite (Xn --~ cx3), the critical extent of reaction Pc is given by 2 p~ -(5.123) Equation (5.123) can be used to calculate the extent of reaction required to reach the onset of gelation in a mixture of reacting monomersfrom its average functionality. Thus, for a glycerol-phthalic acid (2:3 molar ratio) stoichiometric system, 2x3+3x2 fay = = 2.4 2+3 and Pc = 2/2.4 = 0.833 For applying Eq. (5.123) to nonstoichiometric mixtures, however, fay must be calculated as the average useful functionality of the reaction mixture. Problem5.23 Can the following alkyd recipe be reacted to complete conversion of the limiting reactant without gelation ? Pentaerythritol C(CH2OH)41.2 mol Phthalic anhydride C6H~(CO)200.5 mol Tricarballylic acid CH2(COOH)CH(COOH)CH2(COOH) 0.5

372

Chapter 5

A.svcer : Component

Moles

Pentaerythritol Phthalic anhydride Tricarballylic acid

Functionality Equivalents

1.2 0.5 0.5

Total

4 2 3

4.8 1.0 1.5

2.2

7.3

Total OHequivalents ---- 4.8. Total acid equivalents = (1.0 + 1.5) = 2.5. Eq. (5.66): Since the acid equivalents are in deficient

supply,

fay -- -2(2.5)-- 2.2727 2.2 (Notethatthisrepresents theaverage useful functionality of thereaction mixture, whiletheaverage functionality is 7.3/2.2 = 3.32.) 2 2 = 0.88 Pc = -= fay 2.2727 Since the acids are the limiting reactant, the conversion of the acids at the gel point (Carothers) = 0.88. So complete conversion of the acids is not possible without ’gelation.’

Consider again a nonstoichiometric

reaction

mixture consisting

NAA moles of A--A NBB moles of B-B NA~ moles of f-functional

Af with f > 2

Let B groups be in excess.

Then from Eq. (5.66),

fav=

The ratio

2(2NAA + f.Nn~) NAA + NB + NA I of A groups to B groups is given by r

The fraction given by

= 2NAA

+ f.NA, 2NBB of all A groups which belong to the reactant f.NA~

P = 2NAA + f.NA~ Combinationof Eqs. (5.124), (5.125), and (5.126) 4rf fay = f + 2rp + rf(l _ p)

of

(5.124)

(5.125) with f (> 2) (5.126)

(5.127)

Condensation(Step-Growth)Polymerization which can be substituted into Eq. (5.123) to yield p (1 -- P) + 1 +-

373

(5.128)

The extent of reaction Pc at the gel point refers to the extent of reaction of the A functional groups. The extent of reaction of the B groups at the gel point is rpc. [Note that Eq. (5.128) is applicable only to A-A + B+Af(f > 2) or B-B+A$(f > 2) systems with B groups being in excess over A groups, while Eq. (5.123) is applicable to all systems with fay > 2. Both Eqs. (5.123) and (5.128) are based on Carothers theory.] Statistical I.

Approach

Systems with one type of branch unit

Flory [1,15] and also Stockmayer[19] used a statistical approach to derive an expression for the prediction of the extent of reaction at the gel point. ~’he statistical approach assumes: (1) the reactivity of all functional groups the same type is the same and independent of molecular size; (2) there are no intermediate reactions between functional groups of the same molecule. Consider polymerization of bifunctional molecules A-A and B-B and trifunctional molecule A--g--A in a mixture, not necessarily in equimolar quantities. This will lead to~rifunctionally branched network polymer(XII) A-AB-BA-ABBA--~~X~ N. A-AB- BA-AB-BAIrA _A Chain section or

segment

B m ~ A B I B A

Nonbranchunit Branch point (or branch unit)

374

Chapter 5

Figure 5.8 shows schematically a portion of the trifunctionally branched network polymer (XII), Assumethat a chain section has been selected randomfrom the gelled polymeric structure and that it lies within the first envelope. In Fig. 5.8, this chain section in the first envelope happens to give rise to branch points, one at each end. The resulting four new chains happen to lead to three new branch points (on envelope 2) and one terminal nonbranch unit. The resulting six new chain sections happen to lead to two new branch points and four terminal nonchain units on envelope 3, and so on. The fact that the chain sections are shownas being equal in length is of no consequence and does not affect our analysis. Weare interested in the probability of such chain sections irrespective of their lengths. Let us define ee as the branchingprobability, that is, the probability that any functional group from a branch point leads via bifunctional units to another branch point rather than to a bifunctional non-branch unit. Then (l -- o~) is the probability that the functional group from the branch point leads to a nonbranch unit. Considering an ith envelope from the randomly selected chain section in envelope 1 in Fig. 5.8, suppose there are ~ branch points on the ith envelope, If all chain sections arising from these branch points ended in branch points on the (i + 1)th envelope, then there would be 2Y/branch points on the (i -t- l)th envelope. However, because of the branching probability o~, defined above, the expected number(1/-/+1) of branch points on the (i "-I- 1)th envelope wilt be 2Y/c~. The criterion for gelation continuous expansion of the network is that the number of chain sections emanating from the (i + 1)th envelope, i.e., 2Y/+1, be greater than the number of chain sections emanating from the ith envelope, i.e., 2Y/. In other words, the criterion is

2Y~+1> 216/ 1 or 2(2~ a) > 2I~/, that is, o~ > Whenoz < 5’ 1an infinite network will not be generated. Obviously, the critical value of a, namely, ac, when the branching unit is trifunctional (y = 3), 1 (5.129) OZc = i or generally, O~cwhere f is the functionality of the branch unit, i.e., the number of chain sections meeting at the branch point, which, in turn, is the functionality of the monomerwith functionality greater than 2. Otc is termed the critical branching coefficient for gel formation. If more than one monomerwith f > 2 is present, an average value of f of all the monomerswit5 f > 2 is used in Eq. (5.129). It should be noted that at o~ = c~c, only finite species are present, and only when c~ exceeds o~c by a finite amount does the theory permit the existence of infinite networks.

Condensation(Step-Crowfh) Pol~merizafion

5 3

¯

3~5

6

4

Branchpoint (or branch unit)

X Terminal non branch unit ~ Chain section

or chain segment

Figure 5.8 Schematicrepresentation of portion of trifunctionally branched network polymer. (From Ref. 15 with the permission of the American Chemical Society, Washington,D.C.) It now remains to compute the actual value of oe for a given polyfunctional system. Let the fraction of A groups on multifunctional (f > 9,) units be denoted by p [cf. Eq. (5.126)], i.e., p

=

Number of all Agroups(reacted and unreacted)on trifunctional units

Total numberof all groups(reacted and unreacted)in mixture (5.130) Nowconsider the following section (X_III) of the gel network shown (XII)

~ A~-- I,--A~B- BA--A B -BA--AB- BA--I,--A’~v~,’, Trifunctional ~ unit ~

~’ Branch point .~ (or branchunit)

(xm)

376

Chapter 5

Fraction of A groups reacted = PA. Fraction of B groups reacted = PB. Fraction of A groups on trifunctional units = /9. Fraction of A groups on bifunctional units --- 1 -- p. The probability that A at position 1 has reacted = iOA. Since A can react only with B, the probability that A at position 1 has reacted with bifunctional B = PA × I = PA. The probability that B at position 2 has reacted = PB. The probability that B at position 2 has reacted with a bifunctional unit and not a trifunctional unit = (1 -- p). Therefore, the probability that B at position 2 has reacted with A on a bifunctional unit = pB(1 -- p). The appropriate probabilities for positions 3, 4, and 5 are similarly obtained. The probability that B at position 6 has reacted with A on a trifunctional unit = PBP. Therefore, the probability o~ that the chain section shownin (XIII) has formed = pA[PB(1 -- p)pA]2pBP In general, these sections can be represented as A--~--A-(--B-BA--A-)-e-z B-BA--~--A A A (XIV)

(For the chainsection shownin (XIII), :e = 2)

Therefore, the probability o~(x) of obtaining the section shown in (XIV), containing x number of -(-B-BA--A-)units is given a(x) = pn[pB(1 ~pB". P)p A] Weare interested in a total probability, c~, that is independent of the length of the chain section, as reflected in the value of x. Accordingly, ol3

X----0

=

PAPBP

(5.131)

This is the probability that a functional group on a branch unit leads (via bifunctional units) to another branch unit. Either PA or PB can be eliminated by using the ratio r of A groups to B groups. Thus, All A groups NAo (5.132) All B groups NBo PA =

Reacted A groups NAo

ReactedB groups ,

PB =

NBo

(5.1,33)

377

Condensation(Step-Crowth) Polymerization

Since one A group can react only with one B group, and hence pANAo-pBNBo,it follows from eqs. (5.132) and (5.133)

r = PB/PA

(5.134)

Substitution for PAor PB from Eq. (5.134) into Eq. (5.131) yields (5.135) Let PA: Pc at the gel point. So from Eq. (5.135), at the gel point, ,~

=

-

~P~

-

(~.~6)

Combination of Eq. (5.135) with .Eq. (5.129) yields an useful expression for the extent of reaction (of the A functional groups) at the g~l point:

=

+

1

_

Equation (5.137) can be used for calculating the conversion at the gel point for a given feed composition, or for the rewrse situation, i.e., calculating feed stoichiometu for a desired gel point ~c).

Problem 5.24 Simplify Eqs. (5.135) and (5.137) for the following special cases of reaction mixtures: (a) two functional groups A and B are present stoichiometric quantities; (b) feed consists of only i (f > 2)andB-B with groups in excess over A groups; (c) feed consists of only l ( f >2)andB-B with A and B groups present in stoichiometric quantities; and (d) feed consists only of monomerswith f > 2. Answer: (a) Here NAo = NBo; r = 1; PA =PB = P. Equations (5.135) and (5.137) then become a =

p2p 1 - p2(1- O)

(P5.24.1)

and 1 Pc = [1 + p(f -- 2)]a/2 (b) to and

Here,

p = 1; r(= NAo/NBo) < 1;

(P5.24.2) Eqs. (5.135) and (5.137) reduce

~ = rv~ = p~lr

(P~.~4.z)

1 Pc = [r + r(y - 2)1V2

(P5.24.4)

Chapter

378 (c) Conditions of both case (a) and case (b) are present, Eqs. (5.135) and (5.137) become

r = p = 1. So

~ = p2

(P5.24.5)

and Pc = [1 + (/

i - 2)]1/2

(e5.24.6)

(d) Here the probability that a functional group on a branch unit leads to another such unit is simply the probability that it has reacted, i.e., a = p. Therefore, from Eq. (5.129), 1 f-1 Use an average f for all monomers with f > 2. Pc --

.(P5.24.7)

Problem5.25 A cross-linked polyurethane is to be made from a diisocyanate, 1,4trans-cyclohexane diisocyanate (CHDI), and a polymeric tetrol, R(OH)4, without addition of any water. Let the stoichiometric ratio of initial concentrations of NCOto OHgroups be denoted by r. (a) Calculate the critical extent of conversion of NCOgroups above which gelation would occur for r = 4. (b) What is the minimumvalue of r below which no gel can ever be obtained Answer: Let A represent the alcohol functions and B the isocyanate functions. All A functions belong to tetrafunctional (f = 4) base molecules; therefore p = r ----

[NCO]o/[OH]o

pANAo = pBNBo,

= NBo/NAo.

PA = pB(NBo/NAo)

= rPB"

Equation (5.131) then reduces to a = rp~ 3. The critical value at which gelation occurs is given by 1 f-1 To get gelation,

1 3 a > ac

Denoting the conversion of B functional group (PB) at the gel point 1

pc

-

1

and

r =

(a) For r = 4, Pc = 0.29. NCO. (b) For Pc ----less than 0.33.

1

Hence gelation

occurs above 29% conversion

1, r = 0.33. So gelation can never be obtained for any r value

Problem5.26 For a system composed of diethylene propanetricarboxylic

glycol (2.0 mol), 1,2,3acid (0.6 mol), and adipic acid (1.0 mol), determine

379

Condensation (Step-Growth) Polymerization number average degree of polymerization before gelation. expression to obtain this.

Derive first

a general

Answer: Consider a condensation polymerization system consisting A,~ (f > 2). Let NAo = total number of A groups present initially. NBo

of A-A, B-B and

= total number of B groups present initially.

p =

number fraction

of all A groups that belong to the reactant with f > 2.

Total number of monomer molecules present initially

No - NA°(1 --

,

~.

p) + NA°P NB°

-?-- ÷ -~-

(P5.26.1)

Since each new linkage formed involves one A group, number of linkages formed at the extent of reaction, PA, of A groups = pA]VAo. Since with each linkage formed there will be one molecule less, molecules reacted -- pANAo.

number of

Numberof molecules (N) present at the extent of reaction PA is related N = -No -- pANAo Number-average degree Substituting yields ~,~

of polymerization,

for No from Eq. (P5.26.1)

~ _

No No - pANAo

(P5.26.2)

and using r NAo/NBo, Eq. (P 5.26.2)

= f(1 - p + l/r) + 2p f(1 - p- 2pA + 1/r) + 2p

(P5.26.3)

This equation is not applicable above the gel point. For the given system,

f=3 P = 3x0.6/(2xl.0

+3×0.6)

= 0.4737

r = (2 × 1.0+ a × 0.6)/ (2 ×2.0)=0.950 From Eq. (5.137), Pc

=

1 [0.950 + 0.950 × 0.4737(3 - 2)]1/2

From Eq. (P5.26.3),

= 0.845

taking p = 0.845,

3(1 - 0.4737 + 1/0.95) + 2 × 0.4737 3(1 - 0.4737 - 2 x 0.845 + 1/0.95) ÷ 2 × 0.4737

= 9.25

Comment:The average degree of polymerization of the reaction mixture is thus not large at the gel point. It also does not increase rapidly at the gel point. This

Chapter 5

380

merelymeansthat at the gel point manysmall moleculesare still present; it does not preclude the form~itionof a fractional amountof indefinitely large structures at or beyondthe gel point (see Modelfor Gelation Process).

2.

Systems with different types of branchunits

Let us consider a more general ease where both A and B types of branch units occur and where there are also monofunctional and bifunctional A and B reactants, for example, polymerization of A + A-A + Ay + B + B-B + By. To derive a general expression for the conversion of functional groups at the gel point, let symbolsf, 0, and p denote, respectively, functionality (>2), fraction of functional groups belonging to bifunctional reactant, and fraction of functional groups belonging to multifunctional (f > 2) reactant, with subscripts A and B representing the type of functional group. Let r represent the stoichiometric ratio of functional groups such that r _< 1. Thus, fa = functionality of Ay. 0A = A~s belonging to A-A/total A’s. PA = A~s belonging to Ay/total A’s. fB = functionality of By. 0B = B’s belonging to B-B/total B’s. PB = B’s belonging to By/total B’s. Let NA and NB be the total number of A groups and B groups, and NA! and NB[ be the total number of Af and Bf molecules, respectively. The probabihty t~ that any group A belonging to a branch unit and selected at random is connected via a chain to another branch unit is obtained as follows. This state of affairs may come about in 4 possible ways shown in structures (a)-(d) below (assuming f

(a) A--~--A(B -- BA-A)nB -- BA ~-~---AA A~A(B A

-- BA-

B ~--B(AB

A)nB

-~B B

AB - B)nA-~I--A A

B ---~--B(A- AB - B)nA-AB B B where n is any integer from 0 to cxz.

(b)

(c)

(d)

Condensation(Step-Growth) Polymerization

381

The probability o~ is equal to the sum of the 4 individual probabilities of each of these 4 cases occurring in any selection taken at random, each case being equally valid for the purpose of evaluating ~. If the extents of reaction, PAand PB, are the fractions of A and B groups, respectively, that have reacted, then the 4 probabilities are given by equations written below : o~a = ~pA[pANA/(pANA 0

+ pBNB)][OAOBPAPB] n OBPBPA(5.138)

ab = ~,pA[pANA/(pANA + pBNB)][OAOBpAPB]nPB(5.139) o

n

ac = ~’:~pB[pBNB/(pANA 0

q- PBNB)][OAOBPAPB] n PA (5.140)

~ OAPAPB ad = ~],pB[p~NB/(pANA+ pBNB)][OAOnpApB] (5.141)

o [Note: The development of these expressions is similar to that of Eq. (5.131) except for the inclusion of the terms pANA/(pANA q- pBNB) and pBNB/ (pANA q- pBNB), which represent the probability of the multifunctional (f > 2) unit being of A and B type, respectively.] The sum of the above four probabilities gives the total probability o~: NAp2AOBPBPAPB + NAPAPBPA + NBpApBPB + NBOAp~3pApB (NAPA q- NBPB)(1 -- OAOBPAPB) [Note: ~o~n=oy,~ = 1/(1 -- y), (Appendix 5.1).] At the gel point [cf. (5.129)[,

y being constant:

(5.142) 0
(5.143) ~c = 1/(f -- 1) where f is the functionality of the branch unit. If there is more than one branch unit, f in Eq. (5.143) must be replaced by the appropriate average. For the present system, the average value of f is (NA~,fA -b NBIfB) / (NAI -1- NB~) so that instead of Eq. (5.143), we may write

NA, + NB,

~ = NA,(f~- 1) + ~V~,(y~-

(5.144)

Thus the final equation applicable at the gel point is

(NA~ + N~)(NApa + Nnp~)(1 -

= [N~,(y~- ~) + N~,(y~- ~)l ~p~(N~O~ N~O~)+ ;~;~(N~p~+ N~p~)] (~.~4~)

382

G1aapter5

where PAand/9B now represent conversions at the gel point. If B functional groups are in excess, r = NA/NBand PB = rpA (since NapA = -NBPB). It is convenient to express Eq. (5.145) in terms of conversion of the limiting functional group (A in this case). Thus, substituting rpn for PB in Eq. (5.145), and replacing pA by Pc to denote the gel point conversion of the limiting functional group, we have after rearrangement, (XW -t-

OAOBUV)rp2c

-t-

WYpc - UV =

where

U = NAf + NBf V = NAPA + W = NA,(fA -X = NAOBP2A Y = 2pApBNA if or Y "- 2pAp~NB if If

(XW q-

OAOBUV )

NBPB 1) + NB,(fn -+ N~OAp~ A is the limiting B is the limiting :

1) group. group.

O,

(5.147)

pc = UV / WY When (XW -IPc

= -WY

OAOBUV)~£ O, solving

Eq. (5.163)

we have

+ ~/W2Y ~ + 4UVr(XW + OAOBUV) 2r (XW + 0AOBUV )

(5.148)

Note that Eq. (5.148) gives the gel-point conversion Pc of the functional group which is the limiting reactant. The conversion of the other functional group is rpc, where r is the mole ratio of the two functional groups, such that r < 1.

Problem 5.27 Predict the conversions of both A and B groups at the point of gelation in the polymerization systems: (a) Aa (1 mol) + .44 (1 tool) B-B (4 moles); (b) A (2 moles) A-A (3moles) + A 3 (3 moles) + B (2 mo + B-B (1 tool) + B4 (3 moles); (c) A (4 moles) + A--A (30 moles) moles) + A5 (1 mol) + B (5 moles) + B-B (25 moles) + B4 (5 Answer: (a) A = 0, PA = 1, 0B = 1, PB= 0 1×3+1×4 fA -1+1 --3.5 NA~ = 1 + 1 = 2, NB~ = 0, NA = 7, NB = 8, r = 7/8 = 0.8750 U = 2+0 = 2 V = 7x1+8×0 = 7 W = 2(3.5 - 1) +0 =

383

Condensafion (Step-Growfh ) Polymerization X = 7xlxl2+8x0x0 Y = 2xlx0x7 = 0 From Eq. (5.148), -SxO+ Pc

So PA ----

= 7

~(5x0)

2 +4x2x7x0.875(Tx5

~

+Oxlx2xT)

2 x 0.875(7 x 5 + O)

= 0.6761 0.5751, PB = rPh ----

0.5916

0.3529 /13 x e2////e2x 1 + X 11 x+3X2 =~/~.~ 2 + 3+3X x 4)3) = 6/17 0.1250 2/16 PA ---- (3 x 3)/17 = 0.5294 PB(3 x 4)/16 0.7500 U=3+3=6 V = 17 x 0.5294 + 16 x 0.750 = 20.9998 W = 3(3- 1) +3(41) 3.7717 X = 17 x 0.125(0.5294) 2 ~ + 16 x 0.3529(0.75) r = NB/NA --- 16/17 = 0.9412 Y --- 2 x 0.5294 x 0.750 x 16 = 12.7056 Substituting the above values in Eq. (5.148) yields Pc = 0.5638. 0.5538 and PA = rPB = 0.5305.

0 A ---C°) ~Bt*

So PB =

(c) fA = (2 X 3 + 1 X 5)/(2 + 1) = fS=4 NAy = 2 + 1 = 3, NBz ---- 5 ~A ---- (30 x 2)/(4 X 1 + 30 x 2 + 2 x 3 + 1 x 5) = 60/75 = 0.800 0n(25X2)/(5X1+25x2+ 5x4) = 50/75 = 0.6667 r=l PA = (3 X 3.6667)/75= 0.1467 PB(5 x 4)/75 = 0.2667 U = 3 + 5 = 8, V = 75(0.1467 + 0.2667) = 31.005 W = 3(3.6667 - 1) + 5(4 - 1) = 23.00 X = 75 x 0.6667(0.1467) 2 + 75 x 0.80(0.2667) 2 = 5.3438 Y ---- 2 x 0.1467 x 0.2667 x 75 -- 5.8687 Substituting these values in Eq. (5.148) yields Pc = 0.7562. So PA = PB ---0.7562.

Model

for

Gelation

Process

Invariably in all experimental studies where the loss of fluidity has been taken as marking the gel point, the conversion at the observed gel point has been found to be higher than that (calculated) at the theoretical gel point. This is explained by the model proposed by Bobalek et al [20] for the gelation process, as shown in Fig. 5.9. According to this model, at the theoretical gel point, a number of macroscopic three-dimensional networks (gel particles) form and undergo phase separation. The gel particles so formed remain suspended in the medium and increase in number as reaction continues. At the experimentally observed gel point, the concentration of gel particles

Ghapter5

Phase inversion, physical First formation of Infinite network getation theoreticalgel poi,~ Increasein number of Gelled gel. particles ~ ¯-~ ’X__~resin

~

particles__

"~

~,

~

~

Molecularweightincrease

~T~

~

~,

oo

,

0

=

Figure 5.9 A model for gelation process. (After Ref. 20.) reaches a critical in viscosity. Molecular

Size

value and causes phase inversion as well as a steep rise

Distribution

The molecular size distributions for multifunctional polymerization leading to three-dimensional polymers are derived in a manner analogous to those for linear polymers, but with much more difficulty. The derivations have been discussed elsewhere [1,15,19], and only the results will be considered here. The results given below are based on three simplifying assumptions of ideal network formation : (1) all functional groups of the same type are equally reactive; (2) all groups react independently of one another; and (3) no intramolecular reactions (cyclization) occur. The simplest possible type of three-dimensional polymer is that formed by stepwise homopolymerization of a multifunctional monomeror by stepwise copolymerization of two monomers having the same functionality f and all functional groups possessing the same probability of reaction or extent of reaction /9. [That is, for reactions between two monomersA t, and By having the same functionality (f > 2), ’the monomers are to present in equimolar amounts in order that PA shall equal PB.] An example of such stepwise homopolymerizationis the etherification of pentaerythritol. An example of stepwise copolymerization of the above type would be the condensation of a trihydric alcohol with an equimolar proportion of a

~ondensa~ion(S~ep-G~ow~h) Polymerization tribasic acid, all three functional groups on each monomerbeing equally reactive. Or, both reactants might be tetrafunctional. For such polymerizations, the number Nz, the number or mole fraction nz, and weight fraction wz of x-mer molecules in the polymerization system are given, respectively, by

N~= No~t (f;(:x : --~ x)!/ ¥ 2)~]ot

z-1

(1 o~) fx-2z+2

x)t fa/2) cex-l(1 nz = x!(fx- 2x(f~ q- -2)!(1]

(5.149)

(5.150)

(Yx-~)!/ ~o~= (~_ 1)!(/~2~ + ] ~-1 ~1- ~)~-~+2(5.1Sl) where No is the initial number (moles) of monomer(s) and c~ is branching coefficient or branching probability, defined earlier as the probability that one arm of a branching unit (i.e. unit having a functionality greater than two) leads to another branching unit. An x-mer molecule with x > 3 can be constructed in various ways from x f-functional units (f > 2). Equations (5.149)-(5.151) combine all of these (geometric mers) into a single function (Nz., nz, wz) regardless of the particular way in which x units are connected. The number- and weight-average degrees of polymerization in polymer systems of the above type are given by

X,, =

No No -

_

Nofp/2

1

1 - o~f

/2

(5.152)

-x~ = E~w~ = (1 +~) 1-- (y -- 1)c~

(5.~53)

x~ (1 + ~)(1 - ~f/2)

(5.1~4)

~,~

= 1

--

(f-

1)a

Note that Eqs. (5.149), (5.150), and (5.151) are similar in form to (5.71), (5.70), and (5.73), respectively, derived for linear systems and reduce to the latter on substituting f = 2 and o~ = p. Similarly, Eqs. (5.152), (5.153), and (5.154) reduce to Eqs. (5.77), (5.80), and respectively, for liner systems on substituting f = 2 and ~ = p. For a system containing only monomers with f > 2 the branching coefficient o~ is simply equal to p, the extent of reaction [see Problem 5,24(d)], or the probability of reaction for any given functional group; all o~’s in Eqs. (5.149)-(5.151) can thus be replaced by p. These equations

386

Chapter 5

afford calculation of the molecular size distribution weights at various stages of condensation.

and average molecular

Problem5,28 The reaction of phenol with formaldehydeto phenolic resins is ve~ complicated. Assumingfor simplicity that in phenol-formaldehydecondensations with excess (3 : 1) formaldehyde,the phenolis a trireactive structural unit and the formaldehydemerelysupplies the interunit linkage, calculate the weight-average molecular weight of the phenolic resin at 30%conversion of the functional groups. Whatis the weightfraction of species containing 5 phenol residues at this conversion? 2~[LS wet:

Consider the monomeras OH HOH2C

_~_

CH20H

CH20H with Mo(CgH1204) 184 g mo1-1 an d f = 3. FromEq. (5.153), for c~ = p = 0.30, 1+0.3 = = 3.25 1- (3- 10) x0.3 (3.25) (184 g mo1-1) - 3.25 × 3 × 0.3 (18 g mol-’)/2 = 563 g One water molecule is eliminated by condensation of two methylol

(Note groups.) FromEq. (5.151),

= (5 [

(3x5-

5)’3

+2)!]

(3~(5-2x5+2) (0.3)(s-~) (1 - 0.3)

= 0.06

The weight distribution described by Eq. (5.151) is plotted in Fig. 5.10 showing variation of the weight fraction wz with numberof units or degrees of polymerization for different f values at a fixed a = p. It clearly shows that for the stepwise polymerizations, the weight fraction distribution broadens out progressively with increasing functionality at comparable extents of reaction. Figure 5.11 shows molecular size distribution as a function of the extent of reaction in a simple trifunctional condensation (f -- 3). In contrast to the weight-fractlon distribution for linear condensation polymers

387

Condensation(Step-Crowth) Polymerization

0.5 ~ 0.4 ~: 0.3

f=2

~¢- 0.2 ._~ ~ 0.1 0

[

I

2

3

I

4

~ s

I

.

5 6 7 8 9 10 11 12 Numberof units, x

Figure 5.10 Molecular size distribution curves, w~ vs. z, according to Eq. (5.151) for step-growth polymerizationof monomers with different values of f a fixed value of t~ = p = 0.3. (After Ref. 21.) (see Fig. 5.6), the curve shows a monotonic decrease with increase size z. Thus, monomersalways are present in greater amount, even on a weight basis, than species of any other size; in general, z-mers are more abundant than (z q- 1)-mers. Moreover, as the reaction progresses, i.e., ~ increases, the distribution curve falls less steeply (becomingincreasingly broad) with increasing x. The progressive change in polymer composition as the reaction progresses can be shownin another way by plotting weight fractions of various polymer sizes against the extent of reaction p = c~. Plots of this type are shown in Fig. 5.12, calculated again from Eq. (5A51). As would expected, the amount of monomerdecreases continuously as the reaction progresses. Dimer begins to form at the start of the reaction, reaches a maximumamount near p = ot = 0.2, and then decreases as the reaction continues. Trimer formation does not set in at once, owing to the fact that it is a secondary product formed from dimei’. The w3 curve thus reaches a maximumsomewhat later (near p = ot = 0.3) and then decreases. The w4 curve for tetramers is similar; the maximumcomes later. In no case, however, does the maximumoccur beyond the gel point ot = o~c = 0.5 [see Problem5.24(d)]. The average degrees of polymerization calculated from Eqs. (5.152) and (5.153) are plotted in Fig. 5.13 for f = 3. As the reaction progresses, the number average increases, but very slowly as compared to the weight average. Thus, at the gel point the number average molecular weight has increased only four-fold; i.e., there are one-fourth as manymolecules as

388

Chapter 5

0.30 (:X =0.25

~ 0

4

.40

8 12 16 Numberof units,x

20

Figure 5.11 Molecularsize distribution, w=vs. ~, according to Eq. (5.151) for step-growthpolymerization of trifunctional (f = 3) monomers at various stages reaction, denoted by cz = p. (FromRef. 15 with the permission of the American ChemicalSociety, Washington,D.C.)

1.0 0.28

,w2

~ 0.20

0.6 0.4

~ 0.12

o.o 0

0.2

0.4

0.6

0.8

1.0

Figure5.12 Weight fractions of various firdte species (w=) and of gel (w~) as a function of a = p. w~ is calculated from Eq. (5.151) and w~ from Eq. (5.159). (FromRef. 15 with the permission of the AmericanChemicalSociety, Washington,D.C.)

389

Condensation (Step-Growth) Polyrneriza~ion

I

13

Ix

1

0

0.2

0.4 ~ (--p)

Figure 5.13 Number- and weight-average degrees of polymerization as a function of c~ (= /9) for a trifunctional polymerization. The portions of the curves after the gel point (a = p = 0.5) are for the sol fraction only. The species in the sol decrease in average size because the larger, branched species are preferentially tied into the gel. (From Ref. 15 with the permission of the American Chemical Society, Washington, D.C.)

32

0.8

c 24

0.6

0.2 ,’----~--~X 100

l

I 200

0

Time (minutes)

Figure5.14 The course of polyesterification

of a mixture containing tricarballylic acid, succinic acid, and diethylene glycol with r = [COOI-I]/[OI-I] = 1.002 and p = 0.404. The oberved gel point (ave) is 0.894 while Pc calculated from Eq. (5.137) is 0.843. (From Ref. 15 with the permission of the American Chemical Society, Washington, D.C.)

Chapter5

390

were present initially. The weight-average molecular weight, however, runs to infinity at the gel point. This is true, as shownby the progression of the melt viscosity to infinity as the gel point is approached (Fig. 5.14), though precisely at the gel point only finite species are present, according to the distribution equation. (The theory permits the existence of infinite networks only when o~ exceeds the critical value by a finite amount.) The breadth of the size distribution thus increases with increasing conversion up to the gel point. The broadening_of the distribution with increasing conversion can also be noted b_~v the Xw/Xn value. For trifunctional polycondensations, the va__lue of Xw/-~n at the gel point (ce -- p = 0.5) is enormous, since Xwis infinite, while Xn has a finite value of 4 (Fig. 5.13). Finally a = p = 1, only gel is present (Fig. 5.14), i.e., the whole system is converted into an insoluble, infusible mass of one giant molecule and Xw/Xn becomes equal to unity. The discussion above related to polymerization of monomerA_f with f :> 2 or a mixture of monomers of the same functionality f(> 2) present in equivalent amounts. For the condensation of f-functional units with bifunctional units, A-~I~---A A

+ B--B

where A and B are present in equivalent amounts, Stockmayer [19] derived the number distribution equation N,,,e = fNopn-~ (1 - p)* pn+t--1 (1 -- p)~"~--2’~+2

- n + e)! × n!CA(fn

(5.155)

- 2n + 2)!

where N,~,i is the number of polymer molecules containing n numbering ffunctional and g bifunctional units; Nois the numberof f-functional units in the entire polymeric mixture, p is the fraction of A groups on f-functional units, and p again is the extent of reaction. Stockmayer [18] similarly derived an expression for the distribution of "n~ ~ 8-mers" formed from three reactants, one of them f-functional and the other two bifunctional, e.g., A~A

+

A--A

÷

B-B

A A typical exampleis the polyesterification of a mixture of a tribasic acid, a dibasic acid, and a glycol. The equations derived for such systems are difficult to apply, inasmuchas they contain morethan one size variable, e.g., the molecular weight depends on all three parameters n~ £~ s, representing the numbers of one trifunctional and two bifunctional units. Whenthe proportion of branch units is small and, hence, the average number of AA and B-B units between branch units is large, the distribution equations

391

Condensation (S~ep-Crow~h) Polymerizaffon can be replaced

[19]:

approximately

by a more tractable

distribution

n + 1)!~ ~. = 2(1- ~)~n!(f,~ (f,~-- n2,~ +

function

(~.~6)

wherewnis the weightfraction of moleculescomposed of n branchunits, regardless of individual the branching probabili~

lengths of chains be~een branch units, and a is calculable from Eq. (5.135); ~ is defined

~ = ~ (~ - ~)~-~

(~.~7)

Note that Eq. (5.156) does not give the actual size distribution but rather "complexity" distribution, though the similari~ be~een this equation and the size distribution equation (5.151) for the simple f-functional case apparent. Furthermore, these expressions are somewhat inaccurate, owing to the disregard of intramolecular reactions. This approximation becomes more serious the larger the molecule; thus, as a increases toward the gel point and larger species are formed, the error due to this approximation becomes more important.

Problem5.29 A polyesterification

system consists of diethylene glycol (1 tool), tricarballylic acid (0.144 mol), and adipic acid (0.784 mol). What weight fraction of the polymeric mixture has (a) only linear chains, and (b) 5 or less chains molecule at 40%conversion ? Obtain an estimate of the corresponding values at the gel point, disregarding intramolecular reactions. Answer: Let

COOH ~ A, OH ~ B 0.144 x 3 ÷ 0.784 x 2 lx2 PA----- PB f = 3, 0 = (0.144×3)/(0.144×3 + 0.784×2) At 40% conversion (p =0.40), from Eq. (5.135): (0.40)2 (0.216) a = = 0.0395 1 - (0.40)2(1 - 0.216) From Eq. (5.157), fl = (0.0395)(1 - 0.0395) (3-2) = 0.0379 From Eq. (5.156), for f =

= 0.216

(2~

~. = 2(1 or.

w~ = 2(1

-

[(~- 1)/2]! [(z+3)/~]!

where z, the number of chains per molecule, is related (z is necessarily an odd integer).

(~5.29.1) to n by z = 2n + 1

392

Chapter5

For linear chains, z = 1. So from Eq. (P5.29.1), wl = 2(1 - 0.0395) 2 01!(0.0379) 0! 2! = 0.9226 Therefore, weight fraction of linear polymersin the polymericmixture = 0.9226. Similarly, w3 = 0.0699 and w5 = 0.0066. However, w2 = w4 = 0. Therefore, weight fraction of polymerspossessing 5 or less chains per molecule = 0.9226 + 0.0699 + 0.0066 = 0.9991. At gel point (3-2) = 0.25 a = ac = 1/(f-1) = 0.5, /~ = (0.5)(1-0.5) From Eq. (P5.29.1), wl = 0.25, w3 = 0.125, w5 = 0.078 Therefore, weight fraction of linear chains = 0.25 and weight fraction of polymers composedof 5 or less chains = 0.25 + 0.125 + 0.078 = 0.453.

Post-gel

Relations

Gelation in polymerization, which can be observed by a sudden loss in fluidity of the reaction mixture, takes place long before all the reactants are bound together forming one giant molecule. This is evident from the fact that if one were to attempt to dissolve in a suitable solvent the gelled polymermass just as it exists at its gel point, all but a small portion would dissolve. Only a few large molecules are required to induce gelation. The portion that is soluble is referred to as the sol, whereas the part that is insoluble is referred to as the gel. It may be noted that the curves in Fig. 5.12 representing the weight fractions of various species calculated from Eq. (5.151) continue through the gel point (o~ = 0.5) without discontinuity. This wouldbe expected as the derivations of the distribution equations (5.149)-(5.151) place no restriction on c~ or p except that it be in the physically real range from zero to unity. However, as stated earlier, the derivations of Eqs. (5.149)-(5.151) neglect the possibility of intramolecular reactions, which is probably not a serious consideration in the pregel period. But the infinite network will contain intramolecular linkages (which, in fact, are responsible for the many circuitous connections and, hence, network structure of the gel fraction), which will becomemore abundant in the gel fraction as o~ or p increases beyoadthe gel point. Equations (5.149)-(5.151) are thus valid only for finite species beyond the gel point. If Eq. (5.151) for wz is summedover all values of x, the weight fraction of sol (ws) would result. Whenthe units are trifunctional (f = 3), this yields (1 - (~)3 (5.158) W8 -~ E Wx -O~ 3 all finitex Since the sum of the weight fractions of gel wg and sol ws must be unity, we obtain an expression for the gel fraction as

Condensation(S~ep-Grow~h ) Polymerization

~g ~ a

393

(~ - ~)~

(~.~)

~3 and o~ ---- p. The weight percentage of gel plotted in Fig. 5.12 is calculated from this equation. Its formation commencesabruptly at the critical point (o~ = 0.5), rises rapidly, and proceeds to 100 per cent as a ---- p goes unity, i.e., as the reaction proceeds to completion. The size distribution of the sol fraction for ot > 0.5 is the same as prevailed for the entire mixture when p ---- 1 -- o~. In other words, the distribution functions for a > 0.5 are identical with those at the corresponding lower value 1 -- o~ except that they are reduced by a factor ws, the total weight fraction of sol present at that value of o~. The dispersity of the size distribution in the sol above the gel point thus becomes less until, at ce = 1, only monomeris left in a vanishingly small amount. The reduction of Xw/Xnshown for values of o~ increasing above 0.5 is another way of expressing the same phenomenon. Problem 5.30 A mixture of equivalent amountsof glycerol and tricarballylic acid is condensedto the extent of 65%conversion. Assumingequal reactivity of functional groups,calculate (a) weightfractions of sol and gel in the mixture, and (b) weight fractions of the monomer and the dimer in the sol fraction. Answer: Since both monomersare trifunctional (a) From Eq. (5.158), (1 - 0.65) -- 0.156,

(0.65)

(b) From Eq. (5.151),

(f = 3), a = p = 0.65.

Wg = 1 - 0.156 = 0.844

wl (in mixture)

[2!3](0.65)o(1- 0.65)3 = 0.0429

w2(in mixture)

= [4!3] (0.65)(1- 0.65)’= 0.0293 LI!4!J

Hence, wl (in sol) = 0.0429/0.156 = 0.275 wz (in sol) = 0.0293/0.156 = 0.188 Alternatively, take a = 1 - 0.65 = 0.35 and calculate from Eq. (5.151): Wl

=[2!3] (0.35)o(1_ 0.35)3=0.~.~5 Lo!3!J

W2

=

[4’3](0.35)(1--0.35)’ [1!4!1

= 0.187

wl and w2 directly

394

Chapter

Problem5.31 Stoichiometric polymerization of 1,3,5-benzenetriacetic acid with 1,10-decanediol was conductedbeyondthe gel point to measurethe massfraction of sol w~at different extents of reaction p, yielding the followingdata [22]: 0.726 0.809 0.877

0.752 0.130 0.036 Calculate the sol fraction by Flory’s equation to comparewith the experimental ms values. Answer: Let Az = 1,3,5-benzenetriacetic acid and B2 = 1,10-decanediol. p = 1, r = 1, PA = PB = P From Eq. (5.135), a = p2. For p = 0.726, a = (0.726) 2 = 0.527. 3(1 - 0.527) ws ---0.723 (cf. exptl. 0.752)

(0.527)

Similarly, for io = 0.809, ws = 0.147 (cf. exptl. 0.130) and for p = 0.877, ws = 0.027 (cf. exptl. 0.036)

RE, CURSIVE APPROACHFOR AVERAGE PROPERTIES Molecular weight distributions in step-growth polymerization described so far in this chapter are all based on the probability approach of Flory [1,15] and Stockmayer [19]. Starting with the assumptions of equal reactivity of functional groups and no intramolecular reactions, they used combinatorial arguments to derive expressions for the distribution of all Species as a function of the reaction extent and then used these distributions to calculate the average properties (Mn, Mw, and PDI). For cases of practical importance these distribution functions becomequite complex [19]. Derivations based on a variety of other approaches have also been given for the molecular weight distributions, in both linear and nonlinear step polymerizations [23-26]. However, a knowledge of the average properties of the distributions (Mn, Mw, and PDI) is often sufficient for many practical purposes. Macoskoand Miller [27,28] have developed a most useful approach for obtaining the average properties directly without resorting to the necessity of first calculating the molecular weight distribution. The approach utilizes the recursive nature of a step polymerization and assumes a Markov growth process in which the reaction with a polymer chain end depends only on the functional group at the end of that chain and not on the identity of any group further back from the chain end (this is the

395

Condensafion(Step-Growth) Polymerization

situation in almost all step polymerizations). The method is demonstrated below for a few ideal linear and nonlinear step-growth polymerizations. The method retains the three simplifying assumptions of Flory, namely (1) all functional groups of the same type are equally reactive; (2) all groups react independently of one another; and (3) no intramolecular reactions occur in finite species. A system satisfying the above assumptions is referred to as an "ideal" system. However, some departures from these assumptions, such as unequal reactivity and substitution effects and some aspects of intramolecular loops, can also be treated by the recursive method[29,30]. The recursive approach uses an elementary law of conditional expectation. Let A be an event and A its complement. Let Y be a random variable, E(Y) its expectation (or average value) and E(Y [ A) is conditional pectation, given that the event A has occurred. P(A) is the probability that event A occurs. Then the law of total probability for expectation is [311:

E(Y)

- E(YI

A) P(A)

q- E(Y I A) P(A)

(5.160)

This law is discussed in most introductory books on probability theory. Linear

Step-Growth

Polymerization

The simplest step-growth linear polymer is of the ABtype. Typical examples of AB polymerization are the step polymerization of HORCOOH and H2NRCOOH.For example, the polymer from HORCOOH is H-{--OR,C,--)-(--OR-q+ ......... -{--0 R-: ,~--)-OH O O 0 Symbolically, the polymer may be shown as In

-(--A~B

Out

Out

In

) ( A* --B-~---A

--B

--)--

Picking an A group at random, we define the ’in’ direction from the chosen A toward the B group of the same mer unit. ’Out’ is then the opposite direction from the chosen A toward the remainder of the chain on the ~k’ side of the met. Similar definitions apply to the ’in’ and ’out’ directions associated with B groups. We begin by asking for the expected weight of the polymer chain attached to a randomly chosen A group (indicated by *) looking ’out’, i.e., E (W,~Ut). Since the A group is chosen at random, ut is a r andom variable. W~ut equals 0 if A has not reacted. If A has reacted (with of the next mer unit) then W,~ut equals WI~n, the weight attached to B looking into B’s parent molecule: W~ut

= /

if A does not react 0 n Wd if A does react (with B)

(5.161)

396

Chapter~

ByEq. (5.160),

"t -- E (w~ut[ Adoes not react) P (A does not react) E (W,~

+ EI = 0(1-

p)

A reacts) P (A reacts) + E(Wgn)p

(5.162)

wherep is the extent of reaction of A groups, i.e., the fraction of all A groups that have reacted: p = (gAo -- NA) / gAo (5.163) Here NAorepresents the initial moles of A Wpegroups and NAequals the moles after somereaction time. Theexpectedweightattached to a B loo~ng’in’ is equal to the weightof an ~ mer plus the expected weight attached to an A loo~ng ’out’. Thus, E(W~ n) = M~ + E(W~ ~t) (5.164) and the repetitive nature of the simple polymermolecule leads us back to the starting situation. If we begin instead by as~ng for the expected weight attached to a B group loo~ngout, we arrive at a parallel set of 2 equations by the same arguments:

E(W = E(W2")v E (W~~) = MAn + E (W~~t)

(5.166)

Therecursive nature of these chains is thus clear. Choosinga starting point somewhere along the chain and movingalong the chain in somedirection will alwayseventually result in reachinganother position statisti~lly equivalent to the starting point. Solving Eqs. (5.163) to (5.166) gives ,~ (w~ut)

+ E (WI~ut) = 2MAB(1 P_~)

E (w~n) - E(VI/’I~n) : 2- /~AB (’ ~__~ p)

(5.167) (5.168)

Averagemolecularweightsare calculated by recognizingthat the weightaverage molecular weight will be given by the sumof the weight of an AB merunit plus the expectedweightsattached to each armlooking ’out.’ Thus, ~w = MAB + E(W2 ut) + E(WI~ ut) (5.169) Alternatively, Mwis obtained by picking an ABmer unit or an A group at randomand then finding the expected weight of the molecule of which it is a part; that is,

= E(Wi.) + E(Wut)

(5.170)

Condensation(S~ep-Grow~h)Polymerization

397

andsubstituting forS (WAn) fromEq.(5.166) leadsagainto Eq.(5.169). CombiningEq. (5.169) with Eq. (5.167) gives (1 + p’~

(5.171)

The weight-average degree of polymerization, X~o, is given by ~,0_ M,~

_ l+p (5.172) M~ 1 -p which gives the well-knownresult of Eq. (5.80) derived earlier through size distributions. In the above calculations, we have chosen molecules at randomby picking mer units at random. This is a weight-averaging process for molecules, since, the larger the molecules, the proportionately larger chance it has of being chosen. If instead we pick chain ends at randomto choose molecules, and ask for the expected weight attached to the end group looking ’in,’ we obtain a ’number-averaged’ quantity, since the smaller the molecule, the proportionately larger chance it has of being chosen. If we pick end groups, however, we must statistically weight the E (Win) by the mole fraction of each type of end group, that is, the mole fractions of unreacted A and B (denoted by ~zA and ~ZB); thus: -~, = ~,,,E(WI" (5.173) ) + ~,E(W/~") Since there are equal numbers of A and B ends in this simple case, #A = I~B = 1/2, and we get from Eq. (5.168):

and Xn = -~n/MAB = 1/(1--p) Thus, the well-knownresult of Eq. (5.77) is recovered. Problem 5.32 Use the recursive approach to derive expressions for ~, and M~for a polymerization system composed of equimolar amounts of A-A and B-B. To keep the systemas simple as possible take the molecular weights of the two structural units as equal (denoted by Mo). Answer: Letthesystem A-A+ B -B react untilsomefraction PAof the,42sandfraction PB. of the B’s have reacted. Since A-Aand B-B are in equimolar amounts and A + B is the only type of reaction, PA= PB = P" The polymer resulting from the A-A+ B-B polymerization is shownbelow: In Out Out In -(-A--A--)-(--B--B ) ( A--A--)-<--B--B--)-6-A--A)-

398

Chapter5

By application of Eq. (5.160), the following equations can be written for the system: n) Z (WA. °ut) = (1 -- PA) X 0 q- PA E (W]~ a) = pE (W~ (P5.32.1) n) ut) ut ) E(Wt~ = MBB+ E(WI~ = Mo + E(WI~ (P5.32.2) n) m) = pE(Win E(W~ut) = (1 - PB) × 0 pBE(Wf~ (P5.32.3) E (W~n) = MAA+ E (W2 ut) = ut) Mo + E (W,~ (P5.32.4) Let WAA be the total molecular weight of the molecule to which randomly chosen A-Abelongs. Let WBBbe similarly the weight for a randomly chosen B-B. Then, E (WAA) = E (W~n) + Z (W~ ut) = ut) Mo + 2E (W,~ (P5.32.5) E (WBB) = E (W~) + E ut) = Mo + 2 E (W~ut) (P5.32.6) Solving Eqs. (P5.32.1) through (P5.32.4) and substituting into Eqs. (P5.325) (P5.32.6) gives + p) E(WAA) = E(WBB) Moq- 1-2pMo(1 p~ (P5.32.7) To find the weight average molecular weight, we pick a unit of mass at random and computethe expected weight of the molecule of which it is a part [another application of Eq. (5.160)]. Thus, M~, = wAAE (WAA) + WBB(WBB) where WAA and WBBare the weight fractions of A--A and B-B units; WAA = WBB= 1/2. Therefore, ~0 = ½E (Wan) + ½E (WBB) (P5.32.8) whichon substitution from Eq. (P5.32.7) yields M~,

= Mo +

2pMo(1 + p) 1 - p2

= Mol +/9 1-p and X~ = (1 + p) / (1 The numberaverage molecular weight is given by

(P5.32.9) (P5.32.10)

(P5.32.11) n) n) Solving Eqs. (PS.32.1)-(P5.32.4) for E and E (W and substi tuting in Eq. (P5.32.11), --Mn = 7E1 (w~n) + ½E(W~n)

~,,

= Mo (~--~_1

and X’--’~

1 = ~ 1-p

p)

(P5.32.12) (P5.32.13)

399

Condensation (S~ep-Growth ) Polymerization

Note : The expressions for M~, X~,, M~, and X~ derived above for the simple case of equal molecular weights of structural units are identical with the corresponding expressions for the A-B system. Expressions, though more involved, can be readily derivedfor a systemwith unequalstructural unit molecularweights.

In addition to being a simpler methodfor obtaining the average properties such as Mwand Mncomparedto the Flory and similar approaches [31], the recursive approachalso moreeasily allows an evaluation of the effect of unequalreactivity and unequalstructural unit molecularweights on the averageproperties [27,28,33]. Nonlinear

Step-Growth Polymerization

Polyraerizatlon

of A I (f > ~)

Thereaction betweensimilar f-functional moleculesis the simplest case of nonlinear polymerization.Anexampleis the etherification of pentaerythritol [27]: C(CH2OH)4-~ Branched polyether -1- H20 (5.176) Ignoring for the present the effects of any condensationproducts, we can schematically represent the polymerization of Af (f = 4) A A A A

(5.177) A

AA~ A~ A

Let the systemreact until somefraction p of the A groups have reacted; p is given by -NAt - NA P-N.~o

400

Chapter 5

where Nit denotes the number of moles of A groups and the indicates the value at zero time. Let us pick an A group at as A~ in Eq. (5.177), and ask: What is the weight, W,~ut, looking out from its parent molecule in the direction chosen at random, W,~ut is a random variable. Thus, W/~U t

= { 0

zero subscript random, labeled Iattached to A

~ ? Since A’ is

if A’does not react W/~n if A~doesreact (with ~, say)

(5.178)

where W~,n is the .weight attached to A~ looking along _~2, into Ares parent molecule. By Eq. (5.160), E (W,~ut) = E (W/~Ut I A does not react)

P (A does not react)

+ E (w~ut I A reacts) = 0(1-

p)

+ E(WiAn)p

P (A reacts) (5.179)

E (w~n), the expected weight on any A loo~g into its parent molecule, will be the molecular weight of AI plus the sum of the expected weights on each of the remaining f -- 1 arms which is just E (W~ut) for each arm. Thus, E(W~ n)

"t = )M~,

+ (f-

(5.180)

1)E(W~

and the recursive nature of the branched molecule leads us back to the starting point. ~e weight-average molecular weight is obtained by pic~ng an AI or an A group at random and then finding the expected weight of the molecule of which it is a part; that is, ~w : E(W~ n) + E(Wf ut) (5.181) n) ut ) Solving Eqs. (5.179) and (5.180) for E andE (W and s ubst ituting in Eq. (5.181) yields l+p -~w = MA, 1 -- p(f -- 1)

(5.1s2)

or

Xw

~w(P)_ M~,(O)

1 + 1 - p(f

1)

(5.1s3)

which is in agreement with Flow’s result [cf. Eq. (5.153)] derived by muchlonger process involving size distributions [1]. It may be noted that solutions to Eqs. (5.179) and (5.180) exist when p(f -- 1) < 1. If p(f -- 1) > 1, then Mwdiverges and the

Condensation (Step-Growth)Polymerization

401

system forms a gel or infinite network. Thecritical conversionPc for gel formation in the polymerization of Af (f > 2) is thus 1 (5.184) Pe - f-1 Since only monomer AS(f > 2) is present, the branching coefficient (or), defined earlier, is simplyp and the critical condition for gel formationis thus given by 1 (5.1S5) ~e --

f-1

This is in agreementwith the result of Flory’s gel point theoD, [cf. Eq. (5.129)], described previously. In a situation wheremorethan one multifunctional (f > 2) species is present, cee would be computedusing average f. If a condensation product is involved, as in Eq. (5.176) above, one needs to subtract out Me, the molecular weight of the condensate. In polyetherification, Me= 18. Stockmayer[19] suggested that networkswith condensation products could simply be treated by replacing MA~with MA s -- fMe/2 in the relations derived without condensation. This approach is better than ignoring Meentirely but it neglects the unreacted ends of the moleculesand can only be strictly valid as p approaches1. Polymerization

ofA/ (f > :~)

+

An example of Af + B2 system, where f > 2, is urethane formation from pentaerythritol and 1,6-hexanediisocyanate.ConsiderNAsmoles of ffunctional A-type monomer reacting with NB2moles of bifunctional B-type monomershownschematically (for f = 4): A A

.A

A A +

S--S

B’¢2 J

~

E¢’3

A

A $~ I

(5.186)

G]aapte~ 5

402

Let the system react until some fraction PA of the A groups and some fraction PB of the B groups have reacted. If the reaction takes place only between A and B groups, then the number of A’s reacted must equal the numberof B’s reacted, that is, PA (fNA,) or

Pa

Again pick an A group at °ut attached the weight, W the direction 1~ ? Since

= PB (2NB~) --

~

PA

= rpn

(5.]87)

random, labeled as At in Eq. (5.186). What to Ar looking out from its parent molecule, in tA is chosen at random,

if A’does not react W~,Ut = { 0 (5.188) VlfiBn, if Ardoesreact (with ~) Therefore, according to the law of total probability for expectations [Eq. (5.160)], E (W/~ut) n) ---- PA E (Wt~ (5.]89) In a similar way as before we can write expected weights following the arrows in Eq. (5.186) until the repetitive nature of the structure leads back to the starting position: E (W~ n)

at = )M~ + E (Wt~

(5.190)

E°ut) n) = rpA E

(5.19])

E (W~ n) = MA, + (f -- 1) (W ut) (5. 192) Let WA]be the total molecular weight of the molecule to which a r~domly chosen Ay belongs and WB2the weight for a randomly chosen B2. ~en, E(WA,)

= E(W~ n)

+ E(W2ut)

= MAt + rE(W2 or) = MB~ + 2E (We ut)

(5.193)

E(WB~) = E(W~n) + E(W~ut) (5.194) Solving Eqs. (5.189)-(5.192) and substituting into Eqs. (5.193) and (5.194) gives [MB~ +~rpnMn,] (5.195)

=

+

+ pn(f

=

-1)M~

-

To find the weight-average molecular wei~t, we pick a unit of m~s at random and determine the expected weight of the molecule of which it is a part [by another appfication of Eq. (5.160)]: ~ = ~ (W~,)

~,

+ ~ (W~)

~

(~.~7)

403

Condensation (S~ep-Crowfll) Polymerization where WAr and WB2are the weight fractions defined by

of AI and B2, respectively,

NAt MA~ w~,~ = NA~ MAt + NB2 MB2

(5.198)

WB~ = 1 --

(5.199)

wa~

Substituting into Eq. (5.197) and simplifyingyields (2r/f)(1 Mw

~

2

2

+ rpn)M~, + 4rpnM~,M~ [1 + ( f - 1 )r p~n]M~ (2rMn,/f + MB~)[1 -- r(y- 1)pl]

(5.9,00) This result is the same as that which Stockmayer [19] obtained by tortuous combinatorial arguments and manipulation of distribution functions. If some of the starting species are oligomers, with a distribution of molecular weight, average molecular weights must be used in Eq. (5.200) [33].

Problem5.33 Without intramolecular

reactions, ~ can always be calculated from stoichiometry. Derive a general expression to calculate M~for the general system ~A~ + ~Bj at the extent of reaction PA of the A groups. Answer: At the extent of reaction PA, M~is just the total mass, mr, over the number (mole) of molecules, N, present after reaction, i.e.,

= Denoting the initial number (mole) and the molecular weight of the monomers by N and M, respectively, with appropriate subscripts, rat

= Y~’~MA, NA, + ~_,MBjNB~

N, defined above, is just the number(mole) of molecules present initially, No, less the number of new bonds formed, Nb (since with the formation of one bond the number of molecules decreases by 1). Further, since the reaction is only between A and B, the number of bonds formed equals the number of either A or B groups reacted. Thus,

i

M~

=

E~ M,~ Na, + Ej MB~NB~

Polymerization

ore Ai + E Bj

In practice, Ai (i > 2) is often mixedwith A2to control the chain length between the branch points and small amounts of A1 can be present as impurities. Similarly, Bj can be a mixture of B-typespecies with different functionalities. Consider such a mixture of NA,moles of A/andNB~moles of Bj. Assuming that A groupsreact only with B groups, this general system can be represented schematically, following Stockmayer’snotation [19], by

(5.2o

The expected weight along ~ is of the same form as for the AI + B~ system treated above but nowwemust consider all the possible Bj’s with whichan A can react. Equation(5.160) is therefore generalized to the form [271 ut) -- E (W~Ut[ A does not react) P (A does not react) E (W~ + E E (W.~ut J A reacts with Bj) P (A reacts with Bj)

- 0(1 PA) = pn~nB~E~

i’~ (W~

(5.202)

wherenB~= mole fraction of all B groups that are on Bj molecules and is given by j N~, (5.203) n~ - Ej j N~ 2 In direction -----* there will be a relation of the sameformas Eq. (5.160) for each Bj molecule E (W]~) = M~ + (j - 1) E (W~’’t) (5.204)

405

Condensation (Step-Growth) Polymerization The expected

weights

3 4 ~ and ~ are derived

along

similarly

to Eqs.

(5.219) and (5.221): E (W]~ ut)

where TtA,

---

PB E//’n‘ i

E (W~?)

(5.205)

- 1) E "t)

E (w~.n.)

= Mn‘ + (i

(5. 206)

mole fraction

of all A groups that are on Ai molecules and

is given by ~Ai

and PB is related

i Nn‘ EiiN&

~

(5.207)

to PA as in Eq. (5.204): Ei=l i Nn‘

PS --

Solving the system of Eqs. (5.202) E (W/~ ut)

--’~

and (5.204)-(5.206)

paMb +

(5.208)

Pa = rpa yields

paps(fs - 1)

1 - PAPs (fA- 1)(fspBM~ + PAPS (fA -- 1)M~ E (Wt~ ut)

--

I -PAPB (fA --

X)(fB

where fA and fB are weight average functionalities respectively,

--

(5.209) (5.21o)

of Ai and Bj molecules,

defined by

fh -- E~i2 Nn‘ _ E inn,

(5.211)

fB

(5.212)

Eii

and M~ and Mb are

--

given

Nn‘

EjJ2Ns¢

i

= ~_,jnB~

by

M= = ~ Mn‘ nn‘

(5.213)

i

Mb = E Ms, ns, J Let WA, be the total chosen Ai belongs, similarly

(5.214)

molecular weight of the molecule to which a randomly

and WB~the weight for a randomly chosen Bj. Then,

to Eq. (5.193) and (5.194),

ut E )(Wn‘) = Mn‘ + i E (W~ E (Ws~) = Ms~ + ut E )(WI~

(5.215) (5.216)

Ghapter5

406 To find M~o,Eq. (5.214) is generalized M---~

= ~E(WA,)WA, q’- ~E(WBI)WBj (5.217) i j where WA~and wBj are weight fractions of Ai and B j, respectively. WA~ is given by MA~ NA~ WA, = Ei M~,, N~ + Ej MB~ (5.218) and wBj is defined similarly. Substituting and rearranging we obtain pBrn~a + pAm~b psma + pAmb

Mw

PAPB [PA(fn- 1)M~ Pt ~(fB- 1) M: + 2M,~Mb] (pBma q- pAmb)[ 1 -- PAPB(fA 1) (fB -(5.219) where rna

= EMA, N&/EiNA , -= EMA, nA,/i i i i rn~ = Y-~ M~, NN / }-~ i g~,, = i i i and mb and rn~ are analogously defined for the Bj’s. Equation (5.219) is a general relation which covers nearly all nonlinear stepwise polymerizations. Equations (5.182) and (5.200) are just special cases of Eq. (5.219). The extent of reaction at the gel point can obtained from Eq. (5.219) for M~,-~ oc; thus (PAPB)ge!

= (fa-

1 1)(fB-

Substituting for PB from Eq. (5.208) and representing Pc, we obtain 1 1/2 Pc : {r(fA1)(fB-

(5.220) (PA)ge! simply (5.221)

Problem 5.34 Calculate the gel-point conversions for the systems cited in Problem5.27 using the recursive approach for comparisonwith the corresponding values calculated according to Flory-Stockmayertheory. Answer:. (a) System: A3(1 mole) + A4(1 mole) + B2(4 Eq.

(5.211):

fa

32xl -t- 42xl = 3xl + 4×1 = 3.571

40?

Condensation(Step-Growth) Polymerization 22x2 2X2 = 2 3xl + 4xl r = 2x4

Eq. (5.212):

fB --

Eq. (5.208):

= 0.875 1

Eq. (5.221):

u~a)g~,= vo = {o.s75(3.571- 1) (2 - 1)}1/2 [of. Problem5.27(a):

(/gA)gel

0.667

0. 676]

(b) System: A(2 moles) + A2(3 moles) + A3(3 moles) + Bl(2 moles) mole) + B4(3 moles) From Eqs. (5.211), (5.212), and (5.208): fA = 2.412, f13 = 3.375, = 1.062 Then from Eq. (5.221): (PA)gel Pc= 0.5 30. [cf . Pro blem 5.27(b): (PA)gel = 0.531] (c) System: A1(4 moles) + A2(30 moles) + Aa(2 moles) + A~(1 B1(5 moles) + B2(25 moles) + B,(5 moles) Similar calculations as above give: fA = 2.227, fB = 2.467, r -- 1.000, and (PA)gel = Pc = 0.745 [cf. Problem 5.27(c): (PA)gel = 0.756]. Problem5.35 Consider the polymerization system (c) in Problem5.34. Calculate the weight-average molecular weight at 50%conversion of the A groups, given that all the monomers in the system have equal molecular weights of 100.

Answer. FromEqs. (5.218) and (5.219):

m’a = m~ =

,

100 (4 + 30 + 2 + 1) (1x4+2x30+3x2+5×l) = 49.33 (100) 2 (4 + 30 + 2 + 1) = 4933.8 (1x4+2x30+3x2+Sxl) 100(5 + 25 +5) = 46.67 1×5+2×25+4×5)

(loo)2 (5 + 25 + 5) = 4666.7

m~ = (l×5+2x25+4xS)

From Problem 5.34(c): v = 1.00. fa = 2.227, fB = 2.467, P~ = vpA = lx0.50 = 0.50, andMa = M~ = 100. Substituting these values in Eq. (5.219) yields: ~,~ = 417

Also,

408

Post-gel

Chapter 5

Properties

The recursive method of Macoskoand Miller [27] has been described earlier for calculating molecular weight averages up to the gel point in nonlinear polymerization.A similar recursive method[34] can also be used beyondthe gel point, particularly for calculating weight-fraction solubles (sol) andcross-link density. Toillustrate the principles, weconsiderfirst the simple homopolymerization,that is, reaction betweensimilar f-functional monomersA.f and then a more commonstepwise copolymerization, such as reaction of Af with B2. Polymerization of A I Weschematically represent the stepwise homopolymerizationof Af by A A

A I

A~’~2, 1

(5.222)

A~I A Let polymerization proceed until somefraction p of the P~s have reacted. Picking an A group at random(say, ~ i n Eq. ( 5.222) we a sk: Whatis the probability that following ~ (i.e., looking out from the molecule)leads to a finite or dangling chain rather than to the infinite network, i.e., to the walls of the container? Let F~ut be the event that 1 ~ is the start of a finite chain, then from Eq. (5.160) it follows that P (F~ ut)

-- e

(F.~"t I

A reacts)

P (A reacts)

+ P (F~"t A does not r eact) P (Adoes not reac = P(F)~")p

l( 1-p) =

pP (F)~")+ 1(5.223)

where _b"~n is the event that ~ in Eq. (5.222) is the start of a finite chain. For Art to lead to a finite chain all of the other arms of ASmust

Condensation (~ep-~row~h) Pol~meriz~ion

40~

be finite. Thus, P (F~, n = P (F,~Ut) 1-1

(5.224)

and, as with the weight-average molecular weight, the simple repetitive nature of this simple branched molecule leads us back to the starting situation. CombiningEqs. (5.223) and (5.224) to solve P (F~t) yield s

/9

P (F~ut) "f-1 --

_P (Fh~ ut) -- ]9 + 1 : 0

(5.225)

n): and for P (F~ (5.226) [p P (FiAn) + ] __ p]f-1 .-- l~(Fin ) Physically, when P (F~ut) = 1 the system has not yet gelled. In order to drive post-gel relations we therefore desire roots of Eqs. (5.225) and (5.226) between 0 and 1. Since P (F,~ ut) = 1 will always be a root of Eq. (5.225), and not of interest, we can factor it out, reducing Eq. (5.225)

’ - 1 =0 p Z~’ (F£~t) i=0

(5.22z)

which can be readily solved for a given value of f. Thus,

ut) = (1 -p)/p for f = 3, P(F~ (5.228) ut) 1/2for f = 4, P (F/~ = (lip- 3/4) 1/2 (5.229) Roots between 0 and 1 for higherf are, however, easy to find numerically.

Problem5.36 Since at the gel point the infinite network begins to form, the probability of a finite chain should provide an independentmeansfor determining the gel point. Starting with Eq. (5.225) for the simple A/ (f > 2) homopolymerization, derive a relation betweenthe critical extent of reaction Pc for gelation and the monomerfunctionality f. Answen If P (Fh~ut) = 1, then it is impossible to find an A on an infinite chain, that is, infinite networkhas not started to form. Thusto find the gel point one has to characterize the situation for whichthere is no solution of Eq. (5.225) between and 1. Factoring Eq. (5.225) yields

410

Chapter5

Thus it suffices to determine when g(x) = p E{~o i - I = 0

(P5.36.2) has no roots between 0 and 1. Because g(x) is monotonically increasing for between 0 and 1, and g(0) = -1 and g(1) p(f - 1)- 1 , Eq.(P5. 36.2) has no root between 0 and 1 if and only if p(f - 1) - 1 < 0. Thus, if p <_ 1/(f - 1), P(F~ ut) = 1 and if V > 1/(f- 1), P(F~ ut) < 1 and an infinite networkhas started forming.Thusthe critical extent of reaction Pc whichmust be exceededfor the formation of infinite networksto becomepossl"ole is given by (P5.36.3) Pc = 1/(f - 1) This is identical with Eq. (P5.24.7) derived fromthe Flory theory of gelation.

Weight fraction solubles and cross-link density Up to the gel point all molecules are finite and the weight fraction of solubles, ws, is thus unity. Beyondthe gel point molecules are rapidly incorporated into the network and we decreases rapidly. For the homopolymerization of Af (f > 2), a randomly chosen Af molecule will be part of the sol if all f of its arms lead out to finite chains. Thus,

=P

(5.230)

Using Eq. (5.228) for f = 3 gives

= (1 - p)3/p3

(5.231)

which agrees with Eq. (5.158) derived by Flory for the homopolymerization of trifunctional monomers (or = p). Note that these relations neglect the formation of condensation products. If a condensation product forms during polymerization, such as water in Eq. (5.222), the above analysis must be modified to accommodate this phenomenon(cf. Exercise 5.28). An important network property is the cross-link density or concentration of effective junction points in the infinite network. An A.f unit chosen at random will act as an effective junction point if three or more of its arms lead out to the infinite network, or to the container walls. (If only one arm is infinite, this A.f unit will be just hanging from the network; if, on the other hand, two arms are infinite, Ay forms part of a chain connecting two effective junction points, but it itself is not an effective junction point. Consider tetrafunctional A4 as an example of AS. The probability that A4 is an effective crosslink of degree 4 is just P (X4,4) = [1 - P (F/~Ut)] 4 (5.232)

411

Condensation(Step-Growth)Polymerization

However, A4 can also be an effective cross-link of degree 3 and the probability that exactly three of the four arms are infinite is P (X3,4)

~--

(43

)

P (F~ut)[1

- P (F~Ut)] 3 (5.233)

In general, the probability that an AI~ unit will be an effective crosslink of degree m is then P(XmJi)

= ( fi l P(F.~Ut)fi-m[1~ ]m

P(F~Ut)]

m (5.234)

The concentration of effective ne~ork junctions or cross-link densi~ is just the initial concentration of the appropriate Af~ species, [Af~]o, times the probabi~ P (Xm,f~) su~ed over fi = m to the highest functionali~ [X~] = Z [AI,]oP

(X~j,)

(5.235)

The total cross-link densi~ IX] is just the sum of the individual [Xm]’S from m = 3 to A. Since m p goes to 1, P(F~ ut) becomes zero, in the limit of complete reaction theoretically [Xm] = [Ay~]o. The cross-li~ densi~ is an important parameter as it ~n be related to the concentration of effective ne~ork chains and hence to shear modulus of the cross-finked polymer [29,34]. Problem5.37 Consider the polyether network formation by the stepwise polymerization of pentaerythritol. Usingthe recursive methodof direct computation, determine the following network properties as a function of the extent of reaction: (a) weight-average molecular weight, (b) weight fraction of solubles, and (c) cross-link densities. Neglectthe effect of condensationproducts on these properties.

(a)MA,(C5H,204) -1/3. FromEq. (5.182):

-M~, =

f -- 4. FromEq.(P5.36.3), Pc --1/(4- I)

(136 - 3p

Figure 5.15 showsa plot of M~,vs. p. (b) CombiningEqs. (5.248) and (5.249) for f

w.=[(1/v- 3/4)1/2 for p > Pc. Figure 5.15 showsa plot of w, vs. p for p > 1/3.

412

Chapter

(c) Substituting Eq. (5.229) in Eqs. (5.230) and (5.233) ’l PCX4,,t) = [3/2 - Clip- 3/4)’P]

for p > Pc. Figure 5.15 shows a plot of P(X4,4) and P(X3,t) vs. p for p > 1/3. The actual densities, X4and X3, are, however,given by

x, = [As]oe(X,,,) X3 = [ASIoP where [As]o = 1/(136 g tool -1) = 7.35×10-3 tool g-1.

Polymerization

olaf

+ ]~

°ut As with molecular weight averages, discussed earlier, calculation of P (FA ) can be extended to a general system of Ai’s reacting with Bj’s. For illustration, however, consider a simple case of AI reacting with B2, schematically represented by

A

A +

B--B

~

(5.236)

I

B°3

A°~4 I AB~ A~ Looking out from A.f along 1.~ and using Eq. (5.160) gives [cf. Eq. (5.223)]: P (FA °ut) = Pa P (F~ in) + 1 - PA (5.237) where PA is the fraction of A groups andpB the fraction of B groups which have reacted. These are related by [el. Eq. (5.187)] r~

= "r’A

(5.238)

Condensation (S~ep-Crow~h) Pol~merizs~ion

12

4~3

f

10 ~

8

P (X4’4)

0.8

~6

0.~ ~

I:~ 0.~ 2

~

0.2 ;~ ~,,"~ 0.4

~

I ~"--~ 0.6

Extent of reaction,

I 0.8

1

O 1.0

p

Figure5.15 Calculated properties for polyether network formation by the stepwise polymerization of pentaerythritol

(Problem 5.37).

wherer is the initial moleratio of Agroups to B groups. Theprc~bability of Br leading

to a finite

chain in ~ is just

P (FB i’)

is,

Looking out from B2 along e(FB °"t)

= pBe(F2

u)

the same as Brr in -~, that

= P (FB °"t)

3 -~ and using + 1-pB

(5.239)

Eq..(5.160).gives

= rPAe(Fai

u)

+ 1-

rpA (5.240)

and finally following4 ~ we obtain P (Fa in) y= 1P (Fn°Ut)

(5.241)

thus leading us back to the starting situation of Eq. (5.237). Combining equations to solve for P (FA°ut) yields rp~

P (F~°Ut)

-~-1

- P (F~ °"t)

-

rp~

+ 1 = 0

(5.242)

Since P (FA°ut) = 1 will always be a root of Eq. (5.242), and not interest as it signifies no gelling, we can factor it out and solve the remaining

414

Chapter5

equation for roots between 0 and 1. This yields for for

f = 3, P (F~ ut) f = 4, P(F~ ut)

= (1 - rp2A) / rp2A = (1/rp2A3/4) 1/2

--

1/2

(5.243) (5.244)

For higher f the roots are easy to find numerically. Wecan readily extend Eq. (5.230) for sol fraction to a mixture of and B2 by weighting each species by its mass fraction in the mixture. Thus, "~ + wB~P(F~ut) 2 ws = wA, P(f~,’t) (5.245) Further generalization to a system of Ei Ai + Ej Bj (i, j = 1, 2, 3, ...) is straightforward [291: -~ ws = Ew~P(F~Ut) i + ~.wB, P(F~Ut) j (5.246) i 3 For crosslink density, Eqs. (5.230)-(5.235) are, however, applicable to + B2 system since only Ay~ (fl > 2) can act as an effective junction points in the infinite network.

Problem 5.38 Calculate the sol fraction and the degree of cross-linking in the urethane, networks formedby stepwise polymerization of 2-hydroxymethyl-2ethyl-l,3-propanediol and 1,6-hexamethylenediisocyanate to 90%conversion of the hydroxylgroups. Comparethese properties for two urethane systems with (a) r = 1 and (b) r = 0.75, wherer is the moleratio of hydroxyl to isocyanate groups. Answer: Let OH--= A, NCO---- B, and the system represented by A3+ B2 ---* network. Substituting Eq. (5.241) in Eq. (5.240) yields P (Ft~ ut) = rpA P (F~ut) I-’ + 1 - rp,~ (P5.38.1) For f = 3, Eq. (5.245) thus becomes w,=WA,P(F,~Ut) 3 q- WB,[rpAP(F,~Ut)2 2-I- 1 -- rpA] MA, (CeH~O~) = 134, (a) r = 1,

MB, (CsH~N~Oz) --

A3:B2 mole ratio = 2:3, p~ = 0.90 2x1M w~ = 2x134 + 3x168 = 0.347 3 x 168 wB~ = 2x134 + 3x168 = 0.653 From Eq. (5.137), 1 = 0.707 1/9 ~ P~ = [1 + 1]

(P5.38.2)

415

Condensation (Step-Growth) Polymerization Since PA > Pc, equations for post-gel properties

can be used.

From Eq, (5.243), P (F~ut) = (1 - 0.9) 2 / (0.9) 2 = 0.2346 From Eq. (P5.38.2), 2 2ws = (0.347)(0.2346) 3 + (0.653)[(0.9)(0.2346) ÷ 1 - 0.9] = 0.02 So at 90%reaction the finite species have essentially disappeared. From Eq. (5.234), = [1 - P(F~ut)] 3 = (1 - 0.2346) 3 = 0.448

P(X3,3)

From Eq. (5.235), [X3] = (0.347) [1 / (134 g = 1.16 X 10-a mol (b)

r = 0.75,

tool-l)]

A3 : B2 mole ratio

1 x 134 WA~ = lX134 + 2X 168 2 × 168 wB= = 1×134 + 2× 168

(0.448)

-- 1:2,

PA = 0.90

= 0.285 = 0.715

From Eq. (5.137), (0.75

1 + 0.75)1/2

0.816

Since PA > Pc, equations for post-gel properties are applicable. From Eq. (5.243), P(F~ ut) =(1-

0.75x0.92)/(0.75×0.92)

= 0.646

From Eq. (P5.38.2), ws = (0.285)(0.646) = 0.17

3 + (0.653)[(0.75)(0.9)(0.646)

2 2+ 1 - 0.9]

So even at 90% reaction there are nearly 17%solubles in the polymer. From Eq. (5.234), P (X3,3) = (1 - 0.~46) 3 = 0.044 From Eq. (5.235), [X3] ---- (0.285)[1 / (134 g raol-~)] (0.044) = 0.9 -4 tool Both the sol fraction and the cross-link density are thus highly sensitive to the value of r, the former increasing and the latter decreasing rapidly with the increase in the proportion of the b/functional component.

416 DENDRITIC

~hap~er5 POLYMERS

Nearly five decades ago, Flory [1] theorized about synthesizing condensation polymers from multifunctional monomers. These polymers were predicted to have a broad molecular weight distribution and to be nonentangled and noncrystalline due to their highly branched structure. Since such polymers would provide materials with poor mechanical properties, they were considered to be less interesting, and Flory at that time did not consider it worthwhile to pursue this line of research. However, a little more than 30 years later, the first papers on synthesis of dendritic polymers (dendron, Greek for "tree") appeared [35,36], revealing a number of very unique and different properties of these polymers, comparedto their linear analogs. For instance, at high molecular weights the dendritic polymers were found to be globular and, in contrast to linear polymers, they behaved more like molecular micelles [37]. The descriptors starburst, dendrimers, arborols, cauliflower, cascade, and hyperbranched used for such polymers all describe specific geometric forms of structure. Following the first papers of Tomalia et al. [35] and Newkomeet al. [361 in 1985 dealing with dendrimers, a large number of dendrimers have been presented in the literature ranging from polyamidoamine [38], aromatic polyethers [39] and polyesters [40], aliphatic polyethers [41] and polyesters [42], polypropyleneimine [43], polyphenylene [44] to polysilane [45]. Copolymers of linear blocks with dendrimer segments (dendrons) and block copolymers of different dendrons have been described. Dendrimers, as shown in the generalized form in Fig. 5.16(a), are obtained when each ray in a star molecule is terminated by an f-functional branching from which (f -- 1) rays of the same length again emanate. next generation is created when these f -- 1 rays are again terminated by the branching units from which again rays originate, etc. In recent years, the chemistry of preparing dendrimers has becomevery successful, although the synthesis of perfect monodisperse dendrimers is time-consuming and painfully cumbersome[47,48]. Another serious drawbackis the space filling due to which it has not been possible to prepare more than five generations. Either the reaction to a higher generation stops completely or, as it happens in practice, the outermost shells develop imperfections. Though perfect monodisperse dendrimers have very interesting material properties, for use as engineering materials they are far too complicated and costly to produce. This was soon realized by several researchers at DuPont Experimental Station working on dendritic polymers as rheology control agents and as spherical multifunctional initiators. The need to obtain the material rapidly and in large quantities, forced them to develop a route for a one-step synthesis of dendritic polymers. These polymers were, however, polydisperse and had defects in the form of linear segments between branch

4~7

Condensation(5~ep-Grow~h)Polymerization

Dendritic potymers A A~/A

-’~ o.

A A

A

Dendritic..j~ 0~.. A A=~.A Linear unit ~ A~ A Terminatunit ~ A (a) Denclrimer

(b) Hyperbranchedpotymer

Figure 5.16 Schematic representation of dendritic polymers comprising dendrimers and hyperbranchedpolymers(’g’ level indicates generation number). points but they were highly branched dendritic molecules. Kimand Webster [49,50] named them hyperbranched polymers. (The structures are called hyperbranched, since due to the chemical constraint a very high branching density becomespossible without gelation.) Ever since, a wide variety of hyperbranched polymers have been synthesized and reported in the literature [51]. While in a perfectly branched dendrimer derived from A~Bmonomer only one type of repeat unit can be distinguished apart from the terminal units carrying the chain ends [see Fig. 5.16(a)], a hyperbranched polymer AzB(at high conversionof B) will have three different types of repeat units as illustrated in Fig. 5.16(b); these are dendritic units (fully incorporated AzB monomers), terminal units having two A groups unreacted, and linear units having one A group unreacted. The linear segments are generally described as defects. Since the hyperbranched polymers are allowed to contain some linearly incorporated AzBmonomers,the synthesis of such polymers, unlike that of dendrimers, does not require the use of protection/deprotection steps. The most commonsynthesis route thus follows a one-pot procedure

418

Chapter 5

where AzB monomersare condensed in the presence of a catalyst. Such one-step polycondensations result in highly branched polymers even though they are not as idealized as the generation-wise constructed dendrimers. Tedious and repetitive synthetic procedures for dendrimers [36] result in expensive polymers and limit their availability. On the other hand, the one-step process allows production of hyperbranched polymers on a large scale, giving them an advantage over dendrimers in applications requiring large amounts of material, although the properties of hyperbranched polymers are only intermediate between those of dendrimers and linear polymers. Hyperbranched polymers are also unique in that their properties are easily tailored by changing the nature of the end groups. For some areas, such as coating resins and tougheners in epoxy resins, hyperbranched polymers are envisaged to play an important role; Various other applications of these novel materials have been suggested. One reason for the emerging interest in hyperbranched polymers is their different properties compared to conventional linear polymers. Polycondensation

of A~B Monomers

Stockmayer[52] was a pioneer in investigating polycondensations leading to branched polymers and was closely followed by Flory [1] who also described the polycondensation reaction of AzBmonomersfrom a theoretical point of view. The treatment was simplified by assuming that (i) the only allowed reaction is between an A group and a B group, (ii) no intramolecular condensation reactions occur, and (iii) the reactivity of a functional group is independent of molecular size. Flory predicted on the basis of such assumptions that condensation of AzB monomerswould give highly branched molecules without network formation and having a multitude of end groups of the same type (Fig. 5.17). If z molecules of monomerAzBare coupled together, the resulting zmeric species will contain only a single B group at one end and (f- 2)z + A groups at other ends, where f(= x + 1) is the total numberof functional groups of the monomer.Consider, for simplicity, an AI_IB monomerwith f = 3. The probability that an arbitrarily chosen A group has reacted is PA and equals the fraction of A groups reacted, The fraction of B groups reacted PB is related to PAby

PA= PB/(Y -- 1) A branching coefficient, o~, is defined as the probability that a given functionality group on a branch unit is connected to another branch unit; o~ thus equals PA- An expression for the branching coefficient is obtained

419

Condensation(Step-Growth)Polymerization

A

A

A

Figure 5.17 Formation of a condensation polymer based on A,~B monomeras proposedby Flory [1]. if PB in Eq. (5.247) is replaced with the conversion p. Thus, a = p/(f - 1) (5.248) ¯ The total numberof molecules at the extent of conversion p is No (1 -p), where No is the total number of monomerunits initially present. Hence the number-average degree of polymerization, Xn, can be derived as 1

1

xn- 1-p- 1- ~(f- 1)

(5.249)

and the weight-average degree of polymerization as

[1 - ~2(f_I)] From Eqs. (5.249) and (5.250), Xw/Xn, of the system is Xw/Xn = [1 - a2(f

(5.250)

the molecular weight distribution,

1 -~(f- -1)]1) _

(1 - cep)

1 -p

(5.251)

It is seen from Eq. (2.251) that as the conversion is driven towards completion, i.e. p ---+ 1, the molecular weight distribution increases dramatically. Theoretically, polycondensation of A2Bmonomersshould thus form an infinite molecule at extremely high conversions, though in practice, this is seldom observed. Intramolecular reactions, for example, reduce the molecular weight and molecular weight distribution. Flory concluded that condensation of AxBmonomers would give randomly branched molecules

420

Chapter 5

without network formation [1]. However, the occurrence of unwanted reactions (an A group reacting with an A group, for instance) will eventually give rise to an infinite network. Intramolecular reactions, on the other hand, reduce the molecular weight and molecular weight distribution. Synthetic

Approaches

A wide variety of A2Bmonomers,such as 2,2-dimethyl propionic aci.d, 3,5diacetoxybenzoicacid, 3,5-b/s(trimethylsiloxy)benzoyl chloride, 3,5-dihydroxybenzoic acid, and 5-acetoxyisophthalic acid have been used for the synthesis of hyperbranched polymers. A majority of the polymers are synthesized via step-wise polymerizations where AzB monomers are bulk-polymerized in the presence of a suitable catalyst, typically an acid or a transesterification agent. To achieve satisfactory conversion, the low-molecular-weigh~condensation product (e.g. H20, CHzOH)formed during the reaction has to removed. This is most often accomplished by passing an inert gas or by reducing the pressure in the reaction vessel. A drawback of polycondensation polymers is that they are prone to depolymerization by hydrolysis, which might restrict their use.

¯ B

.,,~B B*

* ¯

Hyperbranched potymer

Figure 5.18 Schematic representation of self-condensing vinyl polymerization used for the synthesis of hyperbranchedpolymers(* represents a reactive site whichcan initiate polymerization). (After Ref. 53.)

Condensation(Step-Growth) Polymerization

42~

A "second generation" of hyperbranched polymers appeared only a few years ago when Fr~chet et al [53] reported the use of self-condensing vinyl polymerization (SCVP) to prepare hyperbranched polymers by carbocationic systems (Fig. 5.18). They carried out SCVPof 3-(1-chloroethyl)ethenylbenzene. The reaction involves a vinyl monomerof AB*type in which B* is a group capable of initiating the polymerization of vinyl groups. The chain initiation is the addition of an activated B* group to the vinyl group of another monomerforming a dimer with two active sites and one double bond. Both the initiating center, B*, and the newly created propagating center (shown by * in Fig. 5.18) can react with the vinyl group of another molecule (monomeror polymer) in the same way. Similar procedures based on vinyl monomersbut using radical polymerization were demonstrated by Hawker et al. [54] and Matyjaszewski et al. [55]. Hawker and coworkers [54] applied the concept to TEMPO (2,2,6,6-tetramethyl-l-piperidinyloxy, free radical)-mediated "living" free-radical polymerization of hyperbranched polystyrenes. Matyjaszewskiet al. [55] developed atom transfer radical polymerization (ATRP)techniques to obtain hyperbranched polystyrenes. Since then, a number of different approaches, based on vinyl monomers and various initiating systems, have been explored to yield hyperbranched polymers such as poly(4-acetylstyrene), poly(vinyl ether), polyacrylates, polymethacrylates. A technique of forming hyperbranched polymers using a "graft-on-graft" concept was first described by Gauthier and Mrller [56] whoperformed sequentially several functionalization (such as chloromethylation) and anionic grafting (based on coupling reaction involving chloromethylated polystyrene and living polystyryl anions) reactions on a linear polystyrene. The "grafton-graft" procedure results in polymers with a treelike structure, in analogy to Cascade-type molecules. In this case, however, the grafted branches are distributed randomly along the chains serving as backbones, rather than strictly at their end. Because the building blocks in this case are polymers rather than small molecules, the molecular weight of the polymer grows muchmore rapidly and fewer reaction steps are necessary to prepare high polymers. Gauthier and Mrller [56], for example, obtained molecular weights up to (3-4)× 106 after three chloromethylation and grafting cycles, polydispersity of the polymersincreasing slightly for successive generations. Applications Amongthe various applications suggested for hyperbranched polymers are surface modification, additives, tougheners for epoxy-basedcomposites, coatings, and medicines. It has been demonstrated that h, vdrophobic, fluorinated, hyperbranched poly(acrylic acid) films can passivate and block electrochemical reactions metal surfaces thus preventing surface corrosion [57]. Hyperbranchedfilms can be synthesized on self-assembling monolayers on the metal surface via sequential grafting reactions to obtain thick and homogeneousfilms.

422

Chapter5

The lack of mechanical strength makes hyperbranched polymers more suitable as additives in thermoplast applications. Hyperbranchedpolyphenylenes, for example, have been shownto act successfully as rheology modifiers when processing linear thermoplastics. A small amountadded to polystyrene thus results in reduced melt viscosity [58]. The use of epoxidized hyperbranched polyesters as toughening additives in carbon-fiber reinforced epoxy composites has been demonstrated [59]. The use of hyperbranched polymers as the base for various coating resins has been described in the literature. For example, a comparative study [60] between an alkyd resin based on a hyperbranched aliphatic polyester and a conventional (less branched) high-solid alkyd showed that the former had a substantially lower viscosity and much shorter drying time than the conventional resin of comparable molecular weight. An important application of dendritic polymers being explored in medicine is in advanced drug delivery systems. However, most applications within this field, described, in the literature, deal with dendrimers and not with hyperbranched polymers. In a study on the effect of dendrimer size when used inside the humanbody, it was found [61] that large dendrimers (lV[w ca. 87,000) were excreted into the urine within two days, whereas smaller dendrimers (Mwca. 5,000) accumulated mostly in the liver, kidney and spleen with no urine excretion. Hyperbranched polymers, being mostly polydisperse, are thus unsuitable in vivo applications. A special feature of dendritic polymers is the possibility to combine an interior structure with one polarity, with a shell (end groups) having another polarity, e.g., a hydrophobic inner structure and hydrophilic end groups. Thus hyperbranched polyphenylenes with (anionic) carboxylate end groups have been described [49], where the carboxylate end groups make the polymer water soluble while the hydrophobic interior hosts nonpolar guest molecules. In another example [62], hyperbranched aromatic poly(ether ketone)s having acid end groups have been used to solubilize hydrophobic molecules in water. In such a case, a critical micellar concentration (CMC) is not observed and instead there occurs a steady increase in solubility .of the hydrophobic compoundwith polymer concentration. Such dendritic polymers have been described as unimolecular micelles. In a recent review [63], the guest-host possibility is described for various dendritic polymers considered suitable for medical applications such as drug delivery. During recent years a number of papers have been presented where dendritic polymers have been designed for special applications. For example, a hyperbranched polyester containing 3,6-di-acceptor-substituted carbazole chromophoreswas suggested for use in nonlinear optics applications [64]. The hyperbranched polymer in this case was prepared from 3,6-diformyl-9-(ll-hydroxyundecyl)carbazole via Knoevenagel condensation in

Condensation(Step-Growth) Polymerization

4~3

one-pot reaction. Concluding

Remarks

Variation of macromoleculararchitecture has been increasingly used during the last few decades as a means for developing new polymer materials with special properties. One area of such activity concerns dendritic macromolecules. Thus, numerous polymers with highly branched structures have been synthesized and characterized. Dendritic polymers, comprising dendrimers and hyperbranched polymers are derived from A~:B monomers,i.e., monomershaving one B-type functionality and 2 or more A-type functionalities, so that each repeat unit in the resulting polymers has a potential branching point. The difference between dendrimers and hyperbranched polymers is that while the former are well defined, layer-wise constructed macromoleculeswith a branching point in each repeat unit, the latter contain some linearly incorporated or not fully reacted monomersin the polymer backbone. Hyperbranched polymers, which thus have less well-defined architecture than dendrimers, still have properties that differ greatly from conventional linear or moderately branched polymers and they are also less tedious to make than dendrimers, allowing them to be produced on a large scale and at a reasonablecost. While the synthesis of dendrimers requires the use of a series of timeconsuming and elaborate steps, the synthesis of hyperbranched polymers can be simplified to a one-pot procedure based on classical condensation reactions. These reactions are made either in bulk or in solution where AzB monomers are condensed by themselves or in combination with a By core monomer. The use of a By core monomeraffords better control over the molecular weight and polydispersity of a hyperbranched polymer. Initially, hyperbranched polymers were mainly condensation polymers such as polyesters and polyethers since the required monomerswere the most readily available. The methodof self-condensing vinyl polymerization using vinyl-functional monomers,introduced later, has greatly increased the number of possible monomersthat can be used for this type of polymers. One structural variation which has been widely employed for hyperbranched polymers is modification of the end-group type. Hyperbranched polymers contain numerous end groups and these have been varied over a wide range from highly polar groups such as carboxylic acids to nonpolar end groups. The end groups can also be reactive entities such as acrylates, giving a cross-linkable hyperbranched polymer. Hyperbranched polymers differ greatly from linear or moderately branched polymers. For example, the solubility is muchhigher for hyperbranchedpolymers but not as high as for dendrimers. Hyperbranched polymers normally exhibit an amorphous, nonentangled behavior, i.e., a Newtonian behavior in the melt. The nonentangled state also makes hyperbranched polymers rather brittle. Several thermoset resin materials have been described where the hyperbranched polymer exhibits a low resin viscosity, thereby reducing the need for solvents to attain the application viscosity. At the same time,

rapid curing and good film properties are obtained, offering advantages in coating applications. Hyperbranched polymers allow a wide range of variation in properties, which depend on several parameters, the most important however being the backbone and end-group structure in combination. The glass transition temperature, for example, can be shifted 100°C simply by changing the polarity of the end groups, while keeping the backbone structure unchanged. Attachment of reactive end groups in various amounts also leads to thermoset structures where the Tg and cross-link density can be greatly varied for the same hyperbranched polymer. Though most hyperbranched polymers are considered to be amorphous, some examples of crystalline and liquid crystalline hyperbranched polymers have been described in the literature. The possibility of crystallinity has further expanded the application potential of these polymers. In some cases, in application as toughening additives, the polarity of the hyperbranched polymer relative to the thermosetting matrix resin can be adjusted to give a reaction-induced phase separation in composites, resulting in a dramatic increase in toughness but still retaining the overall good mechanical properties (such as high modulus) of the system. This has been demonstrated for hyperbranched aliphatic polyesters added to epoxy/amine thermoset system. Use as toughening additives for composites represents a successful application of hyperbranched polymers. Though numerous applications have been suggested for hyperbranched polymers, few have reached the stage of full commercialexploitation. Hyperbranched polymers is, however, a young and growing area in the domain of macromolecules. The special properties of these polymers are now receiving greater attention and a number of interesting applications of the hyperbranched polymers are expected to bring them to the market place. The future thus looks bright for these novel materials.

Condensation (Sfep-Crow~h ) Polymerization

42g

REFERENCES 1. P. J. Flory, Principles of Polymer Chemistry, Cornell University Press, Ithaca, N.Y. (1953). 2. I. Vancso-Szmercsanyi and E. Makay-Bodi, Eur. Polym. J., 5, 145, 155 (1969). 3. D. H. Solomon, J. Macromol. Sci. Rev. Macromol. Chem., Cl(1), 179 (1967). 4. D. H. Solomon, "Polyesterification," Chap. 1 in Step-Growth Polymerizations (D. H. Solomon, ed.), Marcel Dekker, NewYork (1972). 5. S. D. Hamann, D. H. Solomon, and J. D. Swift, J. Macromol. Sci-Chem., A2(1), 153 (1968). 6. A. A. Frost and R. G. Pearson, Kinetics and Mechanism, 2nd ed., Chaps. 2, 3, and 9, Wiley, NewYork (1961). 7. C. C. Lin and P. C. Yu, J. Polym. Sci., Polym. Chem. Ed., 16, 1005, 1797 (1978). 8. J. H. Saunders and E Dobinson, "The Kinetics of Polycondensation Reactions," Chap. 7 in ComprehensiveChemical Kinetics, vol. 15, (C. H. Bamford and C. E H. Tipper, eds.), American Elsevier, NewYork (1976). 9. D. B. Jacobs and J’. Zimmerman,"Preparation of 6,6-Nylon and Related Polyamides," Chap. 12 in Polymerization Processes (C. E. Schildknecht, ed. with I. Skeist), Wiley-Interscience, NewYork (1977). 10. M. Katz, "Preparation of Linear Saturated Polyesters," Chap. 13 in Polymerization Processes (C. E. Schildknecht, ed. with I. Skeist), Wileyolnterscience, New York (1977). 11. P. E. M. Allen and C. R. Patrick, Kinetics and Mechanismof Polymerization Reactions, Chap. 5, Halsted Press (Wiley), NewYork (1974). 12. H. Swada, Thermodynamics o[ polymerization, Chap. 6, Marcel Dekker, New York (1976). 13. C. A. Campbell, E. E Elton, and E. G. Bobalek, J. Polym. Sci., 14, 1025 (1970). 14. O. Levenspiel, Chemical Reaction Engineering, 2nd ed., Wiley, NewYork (1972). 15. P. J. Flory, J. Am. Chem. Soc., 63, 3083, 3091, 3096 (1941); Chem.Rev., 39, 137 (1946). 16. G. J. Howard, "The Molecular Weight Distribution of Condensation Polymers," vol. 1, pp. 185-231 in Progress in High Polymers (J. C. Robb and W. Peaker, eds.), Iliffe Books, London(1961). 17. J. R. Schaefgen and P. J. Flory, J. Am. Chem. Soc., 70, 2709 (1948). 18. G. V. Schulz, Z. Physik. Chem., B43, 25 (1939). 19. W. H. Stockrnayer, J. Polym. Sci., 9, 69 (1952); 11, 424 (1953); J. Chem. Phys., 11, 45 (1943). 20. E. G. Bobalek, E. R. Moore, S. S. Levy, and C. C. Lee, J. AppL Polym. Sci., 8, 625 (1964).

426

Chapter 5

21. H. Mark and A. V. Tobolsky, Physical Chemistry of High Polymer Systems, Interscience, NewYork (1950). 22. D. S. Argyropoulos, R. M. Berry and H. I. Bolkar, Makromol. Chem., 188, 1985 (1987). 23. M. Gordon and M. Judd, Nature (London), 234, 96 (1971). 24. W. Burchard, Polymer, 20, 589 (1979). 25. D. Durand and C. M. Bruneau, Makromol. Chem., 180, 2947 (1979). 26. T. M. Orlova, S. S. A. Pavlova, and L. V. Dubrovina, J. Polym. Sci., Polym. Chem. Ed., 17, 2209 (1979). 27. C. W. Macoskoand D. R. Miller, Macromolecules, 9, 199 (1976). 28. L. Lopez-Serrano, J. M. Castro, C. W. Macoskoand M. Tirrell, Polymer, 21, 263 (1980). 29. D. R. Miller and C. W. Macosko, Macromolecules, 9, 206 (1976); 11, 656 (1976). 30. D. R. Miller and C. W. Macosko, Macrmolecules, 13, 1063 (1980). 31. L. C. Case, J. Polym. Sci., 29, 455 (1958). 32. E. Ozizmir and G. Odian, J. Polym. Sci., Polym. Chem. Ed., 18, 2281 (1980). 33. K. D. Ziegel, A. W. Fogel and R. Pariser, Macromolecules, 5, 95 (1972). 34. N. R. Langley and K. E. Polmanteer, J. Polym. Sci., Part A-l, 12, 1023 (1974); N. R. Langley, Macromolecules, 1, 348 (1968). 35. D. A. Tomalia, H. Baker, J. Dewald, M. Hall, G. Kallos, J. R. Martin, J. Ryder and P. Smith, Polymer J., 17, 117 (1985). 36. G. P,. Newkome,Z. ¥ao, G. P,. Baker and V. K. Gupta, J. Org. Chem., 50, 2004 (1985). 37. E Zeng and S. C. Zimmerman, Chem. Rev., 97, 1681 (1997). 38. D. A. Tomalia, D. M. Hedstrand and M. S. Ferritto, Macromolecules, 24, 1435 (1991). 39. C. J. Hawker and J. M. J. Fr6chet, J. Am. Chem. Soc., 112, 7638 (1990). 40. T. M. Miller, E. W. Kwoek and T. X. Neenan, Macromolecules, 25, 143 (1992). 41. A. Buyle Padias, H. K. Hall, D. A. Tomalia and J. R. McConnell, J. Org. Chem., 52, 5305 (1987). 42. I. Ihre, A. Hutt and E. Soderlind, J. Am. Chem. Soc. 27, 6388 (1996). 43. E. M. M. de Brabander-van den Berg and E. W. Meijer, Angew. Chem. Int. Ea.EngL, 32, 308(1993). 44. T. M. Miller, T. X. Neenan, R. Zayas and H. E. Bair, J. Am. Chem. Soc., 114, 1018 (1992). 45. A. W. van der Made and P. W. N. M. van Leeuwen, J. Chem. Soc., Chem. Commun., 1400 (1992). 46. I. Gitsov and J. M. J. Fr6chet, Macromolecules, 26, 6536 (1993). 47. K. L. Wooley, C. J. Hawkers and J. M. J. Fr6chet, Angew. Chem. lnt. Ed., 33, 82 (1994). 48. J. M. J. Fr6chet, Science, 263, 1710 (1994). 49. Y. H. Kim and O. W. Webster, J. Am. Chem. Soc., 112, 4592 (1990).

Condensation (S~ep-Grow~h) Polymerization

427

50. Y. H. Kim and O. W. Webster, Macromolecules, 25, 5561 (1992). 51. A. Hult, M. Johansson and E. Malmstr6m, Adv. Polym. Sci., 143, 1 (1999). 52. W. H. Stockmayer, J. Chem. Phys., 11, 45 (1943); J. Chem. Phys., 12, 125 (1944); J. Chem. Phys., 18, 58 (1950). 53. J. M. J. Fr6chet, M. Henmi, I. Gitsov, S. Aoshima, M. Leduc, R. B. Grubbs, Science, 269, 1080 (1995). 54. C. J. Hawker, J. M. J. Fr6chet, R. B. Grubbs and J. Dao, J. Am. Chem. Soc., 117, 10,763 (1995). 55. S. G. Gaynor, S. Z. Edelman and K. Matyjaszewski, Macromolecules, 29, 1079 (1996). 56. M. Gautiaier and M. M611er, Macromolecules, 24, 4548 (1991). 57. M. L. Bruening, Y. Zhou, .G. Aguilar, R. Agee, D. E. Bergbreiter and R. M. Crooks, Langmuir, 13, 770 (1997). 58. Y. H. Kim and O. W. Webster, Macromolecules, 25, 5561 (1992). 59. L. Boogh, B. Pettersen, S. Japon, J-A. Manson, Proceedings of 28th International Conference on Composite Materials, Whistler, Canada, vol. 4, p. 389 (1995). 60. B. Pettersen and K. Sorensen, Proceedingsof the 21st Waterborne, Higher Solids & Powder Coatings Symposium, NewOrleans, Louisiana, p. 753 (1994). 61. J. Roberts, M. Bhalgat and R. Zera, J. Biomed. Res., 30, 53 (1996). 62. C. J. Hawker and E Chu, Macromolecules, 29, 4370 (1996). 63. K. Uhric, Trends in Polym. Sci., 5(12), 388 (1997). 64. Y. Zhang, L. Wang, T. Wadaand H. Sasabe, Macromol. Chem. Phys., 197, 667 (1996).

EXERCISES 5.1. A mixture of R(COOH)2 and R’(OH)2 in the mole ratio of 1:1.1 was used studying the kinetics of polyesterification. The mixture having R(COOH)2 concentration of 3.808 mol/kg was initially polymerized under mild conditions to about 80% conversion of the carboxylic acid groups. The mixture was then further polymerized in the presence of an added strong acid catalyst at a sufficiently high temperature to facilitate removal of the water of condensation by passing a stream of nitrogen.gas. The polymerization in the second stage yielded the following data:

428

Chapter 5 Time, min 0 5 10 20 30 40 60 90

[COOH], mol/kg 1.71 1.47 1.28 1.00 0.80 0.66 0.47 0.30

Determine the polyesterification [Ans. 0.013 kg tool -1 min-1.]

rate constant for the monomersystem.

5.2. (a) Whatwill be the expected degree of polymerization and extent of reaction if the polymerization between equimolar amounts of a diamine and diacid which has an equilibrium constant of 200 is carried out in a dosed system without the removal of the by-product water ? (b) To what level must the water concentration be lowered in order to obtain a degree of polymerization of 100, if the initial concentration of carboxyl groups is 2M? [Ans. (a) 15.1; (b) 4.04x10-2 M.] 5.3. A linear polyester is to be synthesized from phthalic anhydride and propylen¢ glycol. While the glycol is of high purity, the available phthalic anhydride sample is impure, conta/ning 5% (w/w) phthalic acid. In what proportions should the impure anhydride and glycol sample be reacted to obtain maximum molecular weight ? [Arts. 1.644 : 1.0] 5.4. Polymeric decamethylene adipate of X,, -- 190 and having only hydroxyl end groups is to be prepared by reacting a small excess of the glycol with the dibasic acid until the condensation was substantially complete. Calculate the mole percent excess glycol that is to be used. [Ans. 10.61 mole%] 5.5. Poly(decamethylene adipate) obtained in Exercise 5.4 was subjected to alcoholysis with 5%of its own weight of (a) ethylene glycol and (b) n-octyl alcohol in the presence of p-toluene sulfonic acid catalyst till there was no further change in viscosity. Estimate the limiting X,~ of the mixture. Assume equal reactivity of functional groups. [Ans. (a) 9; (b) 17.6] 5.6. A sample of phthalic anhydride which contains 5% (w/w) phthalic acid impurity is to be polymerized with propylene glycol. What would be the limiting degree of polymerization if ’stoichiometric’ polymerization were carried out without taking note of the impurity ? [Arts. Xn = 370.]

Condensation (5~ep-Grow~h) Pol~meriz~ion 5.7.

A simplified alkyd recipe is made of phthalic anhydride (1.4 mol), oleic acid (0.9 mol), glycerol (1.35 mol), and ethylene glycol (0.5 mol). (a) Calculate number average degree of polymerization when the esterification reaction is complete. (b) An operator making this mixture forgets to add the ethylene glycol. If he tries to run the reaction to completion, at what extent of conversion will he notice the fact of this omission ? [Arts. (a) X,~ = 9.2; (b) p = 0.9865.]

5.8. In the polycondensation of HzN(CH2)6NH~and HOOC(CH2)4COOH, equimolar quantities, to form nylon-6,6 what is the molecular weight of the species that has the largest yield by weight at 99%conversion ? [Arts. M, = 11,300.] 5.9. In the polymerization of H~N(CH2)10COOH to form nylon-11, fraction of the reaction mixture has the structure

what weight

-[-NH-(CH2) 10-CO-] 10owhen 99%of the functional groups have reacted 9. [Arts. 3.7 x 10-3.] 5.10. Poly(ethylenoxy benzoate) is produced by the following reaction: n HOCH2CH20-Ph-COOCH3 ~-(-OCH~CH20-Ph-CO-),

+ nCH3OH

(where Ph = p-phenylene). At 75% extent of reaction, what is the weight average degree of’polymerization (a) of the reaction mixture and (b) the polymer ? [Ans.(a) 7.0; (b) 5.11. When an equilibrium step-growth polymerization is 99% complete, what fraction of the reaction mixture is still monomer(a) on mole basis and (b) on weight basis ? [Am. (a) 0.01; (b) 0.0001.] 5.12. In the polymerization of H2N(CH2)10COOH to form nylon-ll, what is the molecular weight of the species that has the largest weight fraction in the reaction mixture at 99% conversion ? [Am. 18,300.1 5.13. Consider synthesis of nylon-6,10 from stoichiometric amounts of hexamethylene diamine, HzN(CH2)6NH~, and sebacic acid, HOO(CHz)sCOOH. 98% conversion of carboxylic acid groups, determine (a) M~ and M~ the product, and (b) the sum of the mole fractions and that of the weight fractions of species containing 1 through 6 (inclusive) monomerunits. [Am. tmsrm (a) 7,050 and 13,960; Co) 0.1141 and 0.0079.] 5.14. In the polymerization of decamethylene glycol and acidic acid in equimolar proportions at ll0°C in the presence of an externall} added strong acid catalyst, the degree of polymerization showed an increase of 40 in

430

Chapter 300 min, the reaction being carried out with continuous removal of water of condensation. (a) Determine the rate constant k’ as defined -d[COOH]/dt = k’[COOH][OH]. (b) Suppose in the removal of water from the start of reaction 2%of the initially charged glycol were lost by steam distillation, but no adipic acid. What maximumdegree of polymerization could be achieved ? [Ans. (a) 2.2×10-a s-l; (b) 99.]

5.15. In a synthesis of nylon-6,6 from hexamethylenediamine and adipic acid with 1 tool% excess of the latter, the water of condensation is removed by evacuation. At 99% conversion of the limiting reactant determine (a) and M~of the reaction mixture, and (b) cumulative mole fractions and weight fractions, separately, of species with two carboxylic acid end groups, those with two amine end groups, and those with one carboxylic acid and one amine end groups, having degrees of polymerization S~ < 10. What would be the corresponding values if the reaction mixture were stoichiometrically balanced ? Commenton the results. [Ans. Nonstoichiometric case: (a) M,~ = 7572, M~o = 15042. (b) Molefractions: 0.0622, 0.0157, 0.0616; weight fractions: 0.0045, 0.0011, 0.0054. Stoichiometric case: (a) M,~= 11300, M~= 22487. (b) Mole fractions: 0.0239, 0.0239, 0.0478; weight fractions: 0.0012, 0.0012, 0.00~28.] 5.16. In Exercise 5.15, estimate at 99%conversion of the limiting reactant the mole fractions of species which have (a) -COOHend groups, (b) -NH2 end groups, and (c) both -COOHand -NH2 end groups. [Ans. (a) 0.4459; (b) 0.1125; (c) 0.4415.] 5.17. A nonstoichiometric mixture consists of 1 tool of glycerol and 5 moles of phthalic acid. Can gelation occur in this system ? [Ans. No; Xn of mixture at p = 1 is 2.] 5.18. Calculate the extent of reaction at which gelation occurs for the following mixtures : (a) phthalic anhydride and glycerol in the molar ratio 1.500:0.980; (b) phthalic anhydride, glycerol, 1.500 : 0.990 : 0.002; and

and ethylene glycol in the molar ratio

(c) phthalic anhydride, glycerol, 1.500: 0.500: 0.700.

and ethylene glycol in the molar ratio

Comparethe gel points calculated from the Carothers equation with those using the statistical approach of Flory. [Ans. (a) Pc (Carothers) = 0.8435, Pc (Flory) = 0.7143; (b) Pc (Carothers) 0.838, Pc (Flory) = 0.7105; (c) Pe (Carothers) = 0.931, Pc (Flory) = 0.8259.] 5.19. A polymerization system made of Af (f = 3) and B-B is to be reacted until gelation. What should be the stoichiometric proportions of Af and B-B if

Condensation (Step-Growth) Polymerization

431

it is desired to effect gelation of the system when90%of the A groups have reacted ? [Arts. 0.588: 1 (Carothers); 0.411 : 1 (Flory).] 5.20. Computethe time to reach the gel point for the acid catalyzed system 2A/ Also calculate X,~ (f = 3) -b- A-A-4- 3.SB-B, where cok~ -- 4x10-4 -1. s at the gel point. [Ans. 1.67 h; 7.7] 5.21. The network polyurethane obtained by reacting a diisocyanate R(N=C=O)2 with pentaerythritol C(CH2OH)4 contains, according to elemental analysis, 0.2% (wAy) nitrogen and has a density of 1.05 g/cma. Determine (a) polymer chain segment density in mol/cm3, and (b) molar mass of chain segments between branch points. [Ans. (a) 7.5 × -~ to ol er a-3; (b ) 14,000 g tool-1.] 5.22. Calculate the statistical gel point conversions of A and B groups in the system consisting initially of A (4 moles), A-A(51 moles), A3(2 moles), (3 moles), B (2moles), B-B (50 moles), B3 (3 moles), and B~ (3 [Ans. A groups: 0.7799; B groups: 0.7643.] 5.23. A polyesterification batch consists of tricarballylic acid (1 mol), adipic acid (4 moles), and diethylene glycol (5.5 moles). Calculate (a) the gel conversion and (b). the weight fractions of polymeric mixture containing, respectively, 1, 3, and 5 chains per molecule at this conversion. [Ans. (a) 0.8864; (b) = 0.2 5, w~ = 0.1 25, ~o~= 0. 078.] 5.24. Equimolar amounts of trimethylol propane and tricarballylic acid were reacted till 70%esterification took place. Assumingequal reactivity of functional groups, calculate the weight fractions of sol and gel in the polymeric mixture, and the weight fraction of unreacted monomerin the sol. [Arts. ws = 0.037; wg = 0.963; wl (sol) = 0.422.] 5.25. Use Eq. (5.221) derived by the recursive approach to calculate the gel point conversions of A and B groups for the following step-growth polymerization systems and compare with the corresponding values calculated according to the gel point theory of Flory: (a) A4 (2 moles) + B-B (4 mols); (b) (3 moles) + A4 (1 mol) + B-B (4 moles); (c) A4 (1 mol) + and (d) A (1tool) + Aa (2 moles) + B4 (2 [Arts. (a) PA = 0.577 (cf. Flory 0.577), /913 = 0.577; (b) = 0.6 67 (of. Flo ry 0.667), PB = 0.833; (e) = 0.3 33 (el . Flo ry 0.3 33), PB = 0.3 33; (d) 0.471 (cf. Flory 0.467), PB= 0.412.] 5.26. Recalculate the gel point conversion of the polymerization system of Exercise 5.22 by the simpler recursive approach and compare the results. Also calculate the Mwof the polymer formed at 50% conversion of the A groups. Assume that the molecular weights of all the monomeis in the polymerization system are equal and have a value of 100. I/Ins. PA = 0.771 (cf. Flory 0.779); M~, = 399.]

432

Chapfer 5

5.27. A chemist formulated an alkyd recipe as Phthalic anhydride C6H4(CO)201.5 tool Tricarballylic acid CH2(COOH)CH(COOH)CH~(COOH) 1.2 Pentaerythritol C(CH2OH)41.0 tool Can the system be reacted to complete conversion without gelation ? If not, what is the extent of conversion of the acid functionality at the gel point calculated from (a) the Carothers equation and from the statistical approaches of (b) Flory-Stockmayer and (e) Macosko-Miller [Arts. (a) 0.69; (b) 0.494; (c) 0.497.] 5.28. Although the relations for probability of a finite chain and cross-link probability derived by the recursive approach remain unchanged on taking into account the formation of condensation byrproducts, the latter do affect the calculated value of sol fraction. Modify Eq. (5.231) for sol fraction w~, homopolymerization of A~, (f = 3), taking into account the formation condensation by-product which is removed from the reaction mixture. [,,Ins.

L ~--~-,~-~-~ J where-~f’c is the molar massof condensation by-product.Notethat for J~c the correction factor cqnals1 andthe aboveequationreducesto Eq. (~.2~]). 5.29. ~oly~rethane foamwas madeby stepwise ~olymefization of methylenediphe~yl isoeyanate CH2(C~¢NCO)2 and 2-hydro~methyl-2-ethyl-L3-propanediol the mole ratio of 1 : 0.7. Calenlatethe weightfraction solubles andthe total crosslink density in the network if the e~tent of reactio~ of the isocyanate is 90%. [Ans. ~ = 0.03; [~1 = 0.7~ x 10-s tool ~-~.] 5.30. A stoichiomet~c mixture of A¢ and B2 type monomersin which only the reaction of A with B is possible waspolymefized to 80%conversion of the A groups. Calculate (a) the sol fraction and(b) the ratio of c~oss-links degree 3 and degree 4 in the network polymer. [~olecula~ weights: ~ns. (~) 0.0¢: (b) 2.~8:

p~ope~ies o~ these polymers ~o~ld be expected i~ vie~ o~ ~eir st~et~l br~ched ~olyme~sa~d the ~e~etio~s i~volv~d i~ s~ch

433

Condensation (S~ep-Growdl) Polymerization

APPENDIX

5.1

SUMMATIONS OFTEN NEEDED IN CALCULATION DISTRIBUTION OF POLYMERS

5.1.

1

ZpZ_ 1 = ==1

I -- p’ p being a constant:

OF MOLECULAR

0

Derivation Z p~-I py~

~- 1 + p -{- p2 _{_ ......

+ p=-~

= p + p2 + p3 + ......

+ p,~

By subtracting, (1

As n --~ co, p~ --~ O; it then follows Z p=-i _ 1 1 - p ==i

1 (1 - p)2’ p being a constant: 0 < p <

5.2.

Derivation ~-~xp ~-~ = 1 +2p+3p 2 + ...... I9 Z ~gPZ-1 -~ P q" 2P2 -F 3p3 -F ......

’~-~ +rip ~ -F rtp

By subtracting (1-p)

Z~p~-i

= l+p+p2

+p3

.. .....

Taking into account the previous result, with p’~ --~ 0 1

=

(A5.1.2)

434 Chap~e

Der/vation

--- 1 + 4p + 9p~+ 16p~+ ......

’~-~ + n~p

= p+4p~÷gp~+16p~+ ......

’~ +n~p

By subtracting,

1 + 3p + 5p2 + 7p~+....... ~ n2P

whence

==I = p H- 3p2 + Spa "k 7p4 -t- ....... By subtracting these ~o equa~ons,

z=l

......

~2pn+l

whence

v(1- v)~ Z

= p+2p2+2p3+2p~+ ......

By subtracting again,

~ n ~ ~, pn+~ ~ 0; it follows

Similarly, it c~ be shownthat

+

Chapter

6

Radical

Chain Polymerization

INTRODUCTION In the previouschapter, the synthesis of polymersby step polymerizationand the kinetics of the process were considered. Weturn our attention nowto chain-growthpolymerizations.Thereader shouldrecall that the features.that distinguish step-growth and chain-growth polymerizations are summarized in Table 5.1. A large numberof different class of unsaturated monomers, such as ethylene (CH2---CH2,the simplest olefin), a-olefins (CH2=CHR, where R is an alkyl group), vinyl compounds(CH2=CHX, where X C1, Br, I, alkoxy, CN,COOH,COOR,C6H5,etc., atoms or groups), and conjugated diolefins (>C--C-C=C()readily undergo addition or chaingrowth or simply chain polymerization. Polymerization of such monomers will be discussed in this and several of the subsequentchapters. Chain-growth polymerization,as well as all other typical chain reactions, are fast reactions typified by three normallydistinguishableprocesses,viz., (i) initiation of the chain,. (ii) propagationor growthof the chain, and(iii) termination of the chain. (A fourth process, chain transfer, mayalso be involved.) Theinitiation is usually a direct consequenceof generationof a highly active species R*by dissociation or degradationof somemonomer molecules (M)under the influence of such physical agencies as heat, light, radiation etc., or as a consequenceof dissociation or decompositionof somechemical additives commonly knownas initiators (I): I ~ R* (6.1) Thereactive species R*maybe a free radical, cation, or anion, whichadds itself, if conditions are favorable, to the monomer moleculeby openingthe ~" bondto form a fresh reactive center (radical, cation, or anion center), dependingon the nature of R*. The newreactive center adds to another 435

436

Chapter 6

monomermolecule, M, and the process is repeated in quick succession resulting in addition of many more molecules to the same growing chain at its reactive center to carry on the chain propagation process: R* + M ----+

R-M* ~ R-MM* +riM

,

R_(M)n+IM.

(6.2)

The chain growth is terminated at some stage by annihilation of the reactive center by one or more convenient and appropriate mechanism which depends largely on the type of reactive center (radical, cation or anion), nature of the monomerM, and the overall chemical environment and condition of reaction. In the present chapter, the basic principles of chain polymerizations in which the reactive centers are free radicals will be considered in detail, focusing on the polymerization reactions in which only one monomeris involved. Copolymerizations involving more than one monomerare considered separately in ChaPter 7. Chain-growth polymerizations in which the active centers are ionic are reviewed in Chapter 8.

OVERALL SCHEME POLYMERIZATION

OF

RADICAL

CHAIN

Radical chain polymerization, as noted above, is a chain r,eaction consisting of a sequence of three steps-initiation, propagation, and termination. The initiation step is considered to involve two reactions. The first is the production of free radicals. There are many ways to accomplish this, but the most commonmethod involves the use of a thermolabile compound, called an initiator (or catalyst), which decomposesto yield free radicals. The usual case is the homolyticdissociation of an initiator I to yield a pair of radicals R"

i 2R.

(6.3)

where kd is the rate constant for initiator dissociation. Its magnitude is usually of the order of 10-4--10-6 s-1. (The radical R" is often referred to as an initiator radical or aprimaryradical). The second part of the initiation step involves the addition of the radical R" to a monomermolecule as in

R"

H H I + CH2=C ~ R-CH2--C" I X X

(6.4)

437

Radical Chain Polymerization

Reaction (6.4) maybe abbreviated and generalized R" 4-

M--~M~"

where Mstands for monomerand MI" actually represents RM’, that is, a monomer-ended radical containing, one monomerunit and an end group R. The rate constant for the reaction is kl. Initiation is followed by chain propagation, whichconsists of the growth of MI" by the successive addition of large numbers of monomermolecules. Each addition creates a new radical which has the same identity as the previous one, except that it is larger by one monomerunit. The successive additions may be represented by MI" 4- M k-~P M2"

(6.6a)

MZ" + M--~M3"

(6.6b)

M3" 4- MR--~’->P M4", ....

(6.6c)

or, in general terms, Mn" 4-

(6.6d)

M--~Mn+I"

where kp is the rate constant for propagation. The value of kv for most monomersis in the range 102-104 L/moPs. This is a large c, onstant, much larger than those usually encountered in step polymerization (see Table 5.2). Growth of the chain to macromolecular proportions thus takes place very rapidly. At somepoint, the propagation reaction is terminated due to annihilation of the radical center of the propagating chain. The annihilation of the radical centers occurs by bimolecular reaction between radicals. Tworadicals react with each other by combination (also knownas coupling), H

H

H H

vwwCH2-- C~. + ¯ C-CH2vvwv, ~ vwwCH2-- ~-- C~-CH2vww (6.7) X X X X or, by disproportionation in which a hydrogen radical that is beta to one radical center is transferred to another radical center, resulting in the formation of two polymer molecules-one saturated and one unsaturated (at the end unit):

438

Chapter 6 H

~vwCH2--C-+ X

H

H

-C-CH2vww~ X

v~vCH2--C-H X

H + C=CHwvw X

(6.8) These two different terms by

modes of termination can be represented in general

Mn"

4-

Mn" 4-

Mm" ~ Mn+m Mrn"

~ Mn 4-

(6.9) Mm

(6.10)

Termination can also occur by a combination of coupling and disproportionation. However, since both the reactions result in the formation of dead polymer molecule(s) one can also express the termination step kt Mn" 4- Mm° ~ dead

polymer

(6.11)

where the particular modeof termination is not specified and the overall termination rate constant let is given by kt = ktc q- ktd

(6.12)

Typical termination rate constants are in the range of 106-108 L/tools or orders of magnitude greater than propagation rate constants. The much greater value of kt (whether ktc or ktd) compared to kp does not, however, prevent high molecular weight polymer formation because the concentration of radical species is very small (low value of kd) and because the polymerization rate is dependent on only the one-half power of kt (see p. 443). It is apparent from the above schemeof radical chain polymerization that the initiator fragment originating in the form of radical R" are incorporated as end groups R [see Eq. (6.4)] in the polymer structure, sealing one both ends of the polymer chains. A distinctive feature of radical chain polymerization, in sharp contrast to condensation or step-growth polymerization (cf. Table 5.1), that a partially polymerized system of the former would practically consist of full grown polymer molecules of high molecular weight and the unreacted monomer; no species in an intermediate stage of growth can be isolated. The fact is that the polymer molecules formed in the early stages of the reaction are usually comparable in chain length or molecular weight to those formed at a muchlater stage of the process. It is largely a consequence of the very large difference between kp and kd values that the time needed for the full growth of a polymer molecule is an insignificantly

Radical Chain Polymerization

439

small fraction of the time needed for a measurable overall conversion of the monomer.

RATE EXPRESSION POLYMERIZATION

FOR

RADICAL

CHAIN

Equations (6.3) through (6.12) constitute the detailed mechanism free-radical initiated chain polymerization. According to this scheme, the initiator I first decomposesinto a pair of primary (free) radicals, R’, as Eq. (6.3), the rate of radical generation being then given d[R’]/dt

= 2kd[I]

(6.13)

since for each incidence of decomposition of an initiator molecule, two radicals R" are produced. Each R" then attacks a monomermolecule M to produce a chain radical MI" (which has an R end group, not shown), as in Eq. (6.5). These two steps are characterized by the rate constant kd for the decomposition of the initiator and the rate constant ki for the chain initiation reaction, respectively. Of the two steps, the decomposition of the initiator, whichis muchslower than the initiation reaction, is the rate controlling step. The rate of chain initiation, .R/, will thus be the same as the rate of radical generation given by Eq. (6.13), if there is no wastage the primary radicals R" by side reactions. For practical purposes, however, the rate of initiation is expressed by

wherethe factor f is the initiator efficiency or the efficiency of initiation, representing the fraction of primary radicals (R’), which actually contributes to chain initiation as given by Eq. (6.5). The rate of propagation is the sum of many individual propagation steps [Eqs. (6.6a)-(6.6c)]. It is, however, assumedthat the propagation constant kv is independent of the size or the number of monomerunits in the propagating radical. One can therefore express the rate of propagation by Rp = kv [M] [M’]

(6.15)

where [M] is the monomerconcentration and [M’] is the total concentration of all chain radicals, that is, all radicals of size MVand larger. The termination rates ]~t correspon~ling to the different modes of termination are 2.Rtc = 2~c [M’]

(6.16)

440

Chapfer 6

from Eq. (6.9), R,dfrom Eq, (6.10),

2 2ktd[M’]

(6.17)

and 2~t

: 2~[M’] 2 2: 2(k,

tc q- ktd)[M’]

(6.18)

from Eqs. (6.11) and (6.12). The factor of 2 is used in the above expressions for the simple reason that for each incidence of a termination reaction, two chain radicals disappear from the system. It is important to note that as radicals are generated in pairs, they are also destroyed in pairs. In order to obtain a kinetic expression for the overall rate of polymerization, it is necessary to assume that both kp and kt are independent of the size of the radical. [This assumption is inherent in Eqs. (6.15) through (6.18).] It may be recalled that the same type of assumption was also employed in deriving the kinetics of step polymerization (Chapter 5). There is, however, ample experimental evidence which indicates that although radical reactivity depends on molecular size, the effect of the size vanishes after the dimer or trimer [1].

Problem6.1 According to simple collision theory, the rate constant for propagation may be written as

where ~p is the steric factor, a~ is the cross section of the collision, # is the reduced mass of the colliding pair, Ep is the activation energy for propagation, is the gas constant, and T is the absolute temperature. Showthat the assumption made in free-radical polymerization kinetics that kp is independent of the size or degree of polymerization of propagating radicals can be rationalized in terms Eq. (P6.1.1). Commenton the similar assumption also made for Answer: Since the only effective collisions in propagation will be those of monomerM with the growing end of the propagating radical Mn’, the product ~p~rp should be roughly independent of the degree of polymerization of the radical. Because the chemical nature of the reactive end of the radical is independent of the degree of polymerization, E~ should also be independent of radical size. The factor that remains to be considered is the reduced mass # which will be defined by /z

=

MM,. MM MM.. -k

(P6.1.2)

MM

where MM,. and MM are the masses of the propagating radical and monomer, respectively. Since MM,. >> MM,except in the initial stage, Eq. (P6.1.2) may be written as /z ~ MM (P6.1.3)

Radical Chain Polymerization

441

It maytherefore be concluded that kp should be a constant that is independent of radical size (except in the initial stages of propagation). For mostterminationreactions, the rate constant is determinedby the collision frequencyof radicals (diffusion controlled reactions), and the reduced massdoes not becomeindependentof radical size as in Eq. (P6.1.3). Hencethe assumption that kt is independentof size is usually less valid. Neverthelessthis assumption must be madein order to obtain tractable results. Monomerdisappears by the initiation reaction [Eq. (6.5)] as well by the propagation reactions [Eqs. (6.6a)-(6.6c)]. rate of monomer disappearance, which is synonymouswith the rate of polymerization, is thus given by

dt

_

+

(6.19)

However, the number of monomermolecules reacting in the initiation step [Eq. (6.4)] is insignificant comparedwith those consumedin the propagation step [Eqs. (6.6a)-(6.6c)] for a process producing high molecular weight polymer. Thus, to a very dose approximation, the former can be neglected and the polymerization rate is given simply by the rate of propagation:

diM]-dt

/~ = v[M][M ’] k

(6.20)

This equation is not directly useful for evaluating Rp as it contains a term for the concentration of radicals. Radical concentration~ are difficult to measure quantitatively, since they are very low (,,~ 10- mol/L), and is therefore desirable to eliminate [M’] from Eq. (6.20). In order to this, the steady-state assumption is made that the concentration radicals increases initially, but quickly reaches a constant steady-state value. This can be explained as follows. Whena free-radical polymerization is started, the number of radicals in the system will" increase from zero as the initiator begins to decompose according to Eq. (6.2). The rate of termination reaction will also increase from zero in the beginning because the rates of these reactions are proportional to the square of the total concentration of radicals in the system [Eqs. (6.16)-(6.18)]. Eventually the rate of radical generation be balanced by the rate at which radicals undergo mutual annihilation, and the concentration of radicals in the system will reach a steady value. It can be shownthat in typical polymerizations this steady state is reached very early in the reaction. The assumption that the rate of initiation equals the rate of termination is called the "steady-state assumption." It is equivalent to the following two statements:

Ri = Rt at steady state and

d[M’]/dt

= 0 at

steady

(6.21) state

(6.22)

442

Chapter6

The theoretical validity of the steady-state assumption has been discussed [2] and its experimental validity has been shown in many polymerizations.

Problem6.2 Experimentally, it is found that, except in the very earliest

(and generally negligible) stages of the reaction, the loss of monomeris accounted for quantitatively by the appearance of the polymeric product. Justify on this basis the steady-state approximation that all free radicals present in a polymerizing system are at steady-state concentrations. Answer: Let [M]0 be the initial concentration of the monomer. Monomermolecules that have reacted must be contained either in the propagating radicals or in the polymer (i.e., product molecules). Therefore, the stoichiometry requires that IM]o = [M] + Z ~[P~] + Z ~[ram’] Differentiating

Eq. (P6.2.1) with respect to time and rearranging,

According to experimental observation, dt

-

Therefore,

~n(~)

=

This means that dt

dt

Equation (P6.2.3) represents the steady-state

approximation.

Since the steady state is reached soon after polymerization starts, we can assume without significant error that it applies to the whole course of the polymerization. Substituting /~t from Eq. (6.18) into Eq. (6.21) obtains t~ = 2kt Rearrangement

2[M’]

(6.23)

of Eq. (6.23)

[M’]= (

1/2

(6.24)

tLadical ChainPolymerization

443

and substitution into Eq. (6.20) yields

dt - E = [M] E for the rate of polymerization. Whe~initiation takes place by thermal decomposition of initiator, i.e., Eq. (6.3), substitution for R,/ from Eq. (6.14) gives d[M] d$ --

R,p

(fkg [I]) 1/2 = kp[M]k~j

(6.26)

Equations (6.25) and (6.26) have the significant conclusion that the of polymerization depends directly on the monomerconcentration and on the square root of the rate of initiation. Thusdoubling the rate of initiation or initiator concentration does not double the polymerization rate, but the polymerization rate is increased only by the factor V/~. This behavior is a consequence of the bimolecular termination reaction between radicals. It is further evident from Eqs. (6.25) and (6.26) that the polymerizability a monomerin a free radical polymerization is related to the ratio kp/~/2 rather than to kp alone. This ratio will appear frequently in the relations we develop for radical polymerization. Integrated Rate of Polymerization Expression It should be noted that Eq. (6.26) represents the instantaneous rate polymerization corresponding to [M] and [I] values at any instant. Since these values change with conversion, Eq, (6.26) must be integrated over period of time to determine the overall extent of polymerization. If the initiator decomposesin a unimolecular reaction [cf. Eq. (6.3)], the correspondingrate expression is first order in initiator: -- d[II/dt

= kd[I]

(6.27)

Integration of Eq. (6.27) between [I]0 at t = 0 and [I] at t gives -k~t [I] = [I]oe

(6.28)

444

Chapter 6

Table 6.1 Half-Lives of Initiators Initiator 50°C Azobisisobutyronitrile 74 h Benzoyl peroxide Acetyl peroxide 158 h Lauryl peroxide 47.7 h t-Butyl peracetate Cumylperoxide t-Butyl peroxide Source:Datafrom Refs. 3 and 4.

Half-life at 70°C 100°C 4.8 h 7.2 min 7.3 h 19.8 min 8.1 h 3.5 h 12.5 h

130°C

18 rain 1.7 h 6.4 h

218 h

[For first order reactions, it is often convenient to integrate between[I]0 at t = 0 and [I] = [I]0/2 at t = tl/2, the half-life of the initiator, to obtain t,/e

= (ln2)/ka

.(6.29)

The half-life is thus independent of the initial concentration. It is a convenient criterion for initiator acfivities. Table 6.1 lists the half-lives for several commoninitiators [3,4] at various temperatures.] Substituting Eq. (6.28) into Eq. (6.26) one obtains

[M]-On integrating

-~ (fkd[I]o)

between [M]oatt

- In [M] _ 2 (-~)

1/2

(e -kd)

dt

(6.30)

= Oand [M]att: (@~°)1/~(1-

e-~et/2)

(6.31)

The extent of monomerconversion, p, is defined as

So 1 --p

(6.32) [M]o = [M]/

Therefore, Eq. (6.31) mayalso be written -ln(1-

p)= 2 (~/23

(f[I]°~-x/2(1-

e-kdt/2)

This gives the amount of polymer produced (in terms of the moles of monomerconverted) in time t at agiven temperature. It is also useful for

445

Radical Chain Polymerization

Table 6.2 Reaction Parameters in Radical Chain Polymerization Quantity /{i

Units mol L-I -1 s -1 s

kd

[I] [M’] P~ [M] kp Rt /q

General range of values 10-8 -1° - 10 -6 10-4

_ 10

-1 mol L

10-2 -4 - 10

-1 tool L

10 -7 -9 - 10

mol L-1 -1 8 -1 mol L -I -1 L tool s -1 mol L -1 s -1 L mo1-1 s 12 kp / k: (L mo1-1 $-1)1/2

10-4 -6 - 10 10-1 - 10 102 4 - 10 10-8 -l - °10 106 - 108 10-2 -- 1

determining the time needed to reach different extents of conversion for actual polymerization systems where both [M] and [I] decrease with time. Table 6.2 shows the general range of values of the various concentrations, rates, and rate constants pertaining to the above kinetic scheme. These values are typical of radical chain polymerizations.

Problem6.3 The decomposition of benzoyl peroxide is characterized by a halflife of 7.3 h at 70°C and an activation energy of 29.7 kcal/mol. What concentration (mol/L) of this peroxide is needed to convert 50%of the original charge of a vinyl monomer to polymer in 6 hours at 60°C ? (Data: f = 0.4; k~/k~ -2 = 1.04x10 L/m61-s at 60°C.) Answer: FromEq. (6.29), at 70°C, In 2

In 2 (7.3 x 3600 s)

= 2.638 x 10-~ -1 s

To calculate ka at 60°C, In

r. ] [(2.638 x :tO -~ ~-1)1

From the given value of E~, (29.7 x 103 cal mo1-1) Therefore,

(1.987 cal mo1-1 °K-I) (kd)~0 o = 7.128 x 10-~ -1 s

= 1.495 x 104 °K

446

Chapter 6

Using Eq. (6.33), - ln0.5

for 50% conversion at 60°C,

= 2(1.04

× 10 (7.128 × 10-6 s-l)’/2 [1 - e -(7"12s

(0.4

[I]0)

1/2

× 10-6 s-1)(6

X

× 3600 s)/2]

tool L [I]0 = 3.75 × 10-2 -1

Problem6.4 For a new monomer 50% conversion is obtained in 500 rain when polymerized in homogeneoussolution with a thermal initiator. Predict the time for 50% conversion in another run at the same temperature but with four-fold initial initiator concentration. Answer: Approximating

(1 - e-kat/2)

by karl2, Eq. (6.33) may be written

-ln(1-p) Taking ratio of equations for the runs (same conversion, same rate constants), 1 -- [~1°~(~ t~ t2 = (500

min)/v~

[~102 / [~10,1 = 4

’

= 250 rain

Problem6.5 If a 5% solution of a monomerA containing l0 -4 mol/L of peroxide P is polymerized at 70°C, 40% of the original monomercharge is converted to polymer in 1 h. Howlong will it take to polymerize 90%of the original monomer charge in a solution containing (initially) 10%A and -~ mol/L of peroxide P ? Answer: Approximating (1 -- e -~:~t/2)

-ln(1-p)

by k~t/2,

Eq. (6.33)

becomes

=

where K is a lumped constant, same for both runs. Taking ratio to eliminate If, t = 0.1(ln 0.1/ In 0.6) = 0.45

447

RadicalChai~Polymerization A Note on Termination

Rate Constant

Researchers in the United Kingdomand the United States whopioneered muchof our knowledgeof free-radical polymerizationsused different conventions for termination rate constants. All United States texts and data compilations adhere to the Americansystem while United Kingdomtexts rely on the British method. Thoughthe convention is seldom explicitly specified, it can be usually inferred fromthe particular context. In the British convention, ktc and ktd are defined by --diM’i/at It followsthen that

= ktc[M’] 2 + 2ktdiM’]

d[polymer]/dt---

~]¢,tc[M’] 2 2+ ~td[M’]

(6.34)

(6.35)

since a pair of radicals give rise to onepolymermoleculeif the termination occurs by combination,but two polymermoleculesif the termination is by disproportionation. Onthe other hand, the Americanconvention, whichis followed in this text, uses the followingdefinitions: --d[M’]/dt

=_ 2ktc[M’] 2 -+- 2ktd[M’] 2

(6.36)

and hence, d[polymer]/dt --- ktc[M’] 2 + 2~a[M’] 2 (6.37) The rate constants quoted according to the Americanconventionwill thus be exactly half those measuredby the British system. However,the same conclusions about polymerization rates are reached wheneither usage is adopted because a compensatoryfactor of 2 is present in the kinetic equations that use Americanrate constants and not in those corresponding to the British system. Hence,each conventionis unobjectionable if used consistently but the two cannot be mixed. EXPERIMENTAL DILATOMETRY

DETERMINATION

OF P~:

As Eq. (6.26) shows, polymerization rates depend on the monomerand initiator concentrations. Since these concentrations are knownmost accurately at zero time, it is useful to workwith initial rates of polymerization, as given by o = _ (d[M]~ = lim [M]o- [M] (6.38) k, dt ]0 at-~0 ~ Unlessotherwisespecified all rates mentionedin the followingsections will be initial rates and the superscript and subscript on ~ and (diM]/dr), respectively, in Eq. (6.38) will be dropped.

448

Glhapter6

It is easy to determine the rate of polymerization since, according to Eq. (6.38), it is merely necessary to measure the monomerconcentration [M] a function of reaction time t andthen determine the initial slope of a plot of [M] versus t. Moreover,since such plots are usually sufficiently linear up to about 10-15% monomerdepletion, only one or two experimental points are often adequate. The most obvious and direct way of measuring [M] is to stop the reaction at some predetermined time and then isolate the polymer from the reaction mixture (e.g., by sudden chilling of the reaction mixture followed by drying) in either of two ways: precipitation of the polymer by addition of a nonsolvent or distillation of the monomer.The amount of polymer formed, and hence the amount of monomer consumed at this reaction time, may be determined simply by weighing. The amountof monomerremaining may be determined directly by weighing or calculated from the monomerconsumed. The technique is, however, time consuming and requires the preparation of a new reaction mixture for each experimental point on the concentration-time plot. It is muchfaster and more convenient to measure some physical property of the reaction ~nixture that changes as the polymerization proceeds and that may be related to the concentration of the monomer. Although a numberof suitable physical methodsof analysis are available, dilatometry is most often used in the measurement of polymerization rates. Because the density of a polymer is usually greater than the monomerfrom which it was formed (see Table 6.3), the rate of addition polymerization can be followed by observing the contraction in volume of a fixed weight of monomeras it is polymerized. The sensitivity of the change in volumewith conversion can be increased significantly if the shrinkage in volume is observed in a tube of very narrow diameter. Such a device is called a dilatometer. It can be constructed with a reservoir to contain a sufficient volumeof liquid (to make the total volume change occurring during conversion to be meaningful) and fitted with a capillary tube to make the volume change accompanying the conversion of monomerto polymer readily measurable. The dilatometer maybe constructed in a variety of ways. The principle of operation of a dilatometer maybe seen from the simple apparatus shown Table 6.3 Densities of SomeMonomersand Polymers Density (g/cm3) at 25°C Monomer Monomer Polymer Volume Acrylonitrile 0.800 1.17 Methylacrylate 0.952 1.223 0.940 1.179 Methyl methacrylate Styrene 0.905 1.062 Vinyl acetate 0.934 1.191 1.406 Vinyl chloride 0.919 Source: Datafrom Ref. 5.

change (%) 31.0 22.1 20.6 14.5 21.6 34.4

Rad./cal ChainPolymerization

449

Figure 6.1 Sketch of a dilatometer.

in Fig. 6.1. The total volume of the dilatometer (B to D) is predetermined (e.g., by the difference in weight when empty and when filled with water at a fixed temperature). The monomeror a solution of the monomerin solvent are introduced, along with an initiator, into the apparatus through filling tube A until the liquid is drawnwell up into the capillary tube D and the stopcock B is then closed. The dilatometer is placed in a thermostatic oil bath maintained at the desired reaction temperature and is immersedin the oil so that the capillary tube protrudes from which the volumeshrinkage is to be recorded. As the volumeof liquid increases by thermal expansion, the liquid will rise, fill the capillary tube, and overflow. The excess solution can be removed by wiping the top of the capillary tube with a piece of filter paper. As soon as the solution has come to a thermal equilibrium, the volume of solution will decrease. The change in height of the liquid in the capillary is measured periodically with a cathetometer (a rigidly mounted, vertically sliding telescope) or from the scale on the dilatometer. The rate of poly-

450

Chapter 6

merization can then be determined of the liquid.

from the rate of change in the height

Problem6.6 Show how the rate of a free-radical mined from the measured changes in the capillary dilatometer (Fig. 6.1).

polymerization can be deterliquid height (Ah) in a simple

Answer: Referring to Fig. 6.1, total volume of the dilatometer or the reaction system is V = VBC q- ~rr2h

(P6.6.1)

where VBCis the volume contained in the space from the stopcock B through the bulb C to the entrance to the capillary tube D, r is the radius of the capillary, and h is the length of the capillary or the height of the liquid in the capillary above the entrance. It follows from Eq. (P5.5.1) that the change in volume of the reaction system is proportional to the change in h, as in AV = Y o - V = 7rr2Ah where V0 is the initial volume and V is the volume at time t, given by

(P6.6.2)

V = w,~,~ + w~p + ws~s (P6.6.3) Here w,~, wp, and ws are the weights and v,~, vp, and ~, are the partial specific volumes of monomer, polymer, and solvent, respectively. (For polymerization of the bulk monomer, ws = 0.) To a very good approximation, o (P6.6.4) Wp

Wm

~

--

Wm

0 is the initial weight of monomer. where w m Substituting Eq. (P6.6.4) into Eq. (P6.5.3) and rearranging wm =

V wm o-v~ -- ws~s

(P6.6.5)

Since no polymer is present initially,

the initial volumeis given by (P6.6.6) ~ + ws~ Vo = wm° ,~ Assuming that all the monomeris converted to polymer after a sufficiently long time (t = cx~) when there is no further decrease in h, the final volume (P6.6.7) V~ = w,.~ o + w~~s Equations (P6.6.6) and (P6.6.7) from Eq. (P6.6.5) to yield V -

Voo o

may be used to eliminate

~,~ - ~p (P6.6.8)

On a weight fraction basis, the fractional yield of the reaction, y, may be written as

Y = w,,

-

w,~

(P6.6.9)

Radical Chain Polymerization

451

Then combining with Eq. (P6.6.8) and (P6.6.2), Vo - V = Ah(t_~_) (P6.6.10) Vo - V~ Ah(cc) Utilizing the definition of the initial rate of polymerization [Eq. (5.38)], one may then write (e6.6.11) y [M]o Ah(t) [M]o t Ah(t = cx~) y

=

Since Ah(t = c~) can be calculated (see Problem 6.7), P~ can be determined from the observed decrease in liquid height (Ah) in the capillary of a dilatometer at time t.

Problem6.7 A dilatometer (total volume 49.0 cm3, capillary radius 0.1 cm) was filled with a freshly distilled sample of styrene containing 0.1% (by wt) of 2,2’azo-bis-isobutyronitrile. The dilatometer was placed in an oil bath at 70q-l°C and was immersed in the oil so that the capillary tube protruded. Whenthe volume of the solution began to decrease after coming to thermal equilibrium, the fall in height of the liquid (Ah) in the capillary tube was determined periodically from the scale on the dilatometer. This yielded the following data: Time (s) 300 600 1800 3600 5400 7200

Ah (cm) 0.83 1.68 5.03 10.05 15.10 20.13

Determine the kinetic parameter kp/k~/2 at 70°C, given that f = 0.60 and ka = 4.0x 10-s s-1 at 70°C. [Density (g/cmz) at 70°C: styrene 0.860; polystyrene 1.046] Answer: Initial

volume of styrene (V0) = 49.0 3 at 70°C.

Wt of polystyrene at 100%conversion = Initial gcm-z) = 42.14 g

wt of styrene = (49.0 cm~)(0.860

V~ = (42.14 g) / (1.046 -3) = 4 0. 287 cmz W0 - V~ = (49.0

- 40.287)

or 8.713 3

3) = 277.22 (8.713 cm 2 Ah(t = c~) = (3.143)(0.1

[M]o = (1000

cmaL-1)(0.860 (104 g mo1-1)

gcm -a)"

The yield of polymer (y) can be calculated

cm

-1 8.27moiL from Eq. (P6.6.10)

and the poly-

452

Chapter 6

merization rate (~) from Eq. (P6.6.11) using the observed values of Ah. gives: Time(s) 300 600 1800 3600 5400 7200

% Conversion (y x 100) 0.30 0.62 1.84 3.69 5.55 7.39

/~ x l0 s (tool -1 s -1) 8.40 8.51 8.49 8.48 8.50 8.49

For low degrees of conversion (generally below10%)both [I] and [M], and hence /~, may be assumedconstant. The average value of/~ obtained from the above data is 8.49x10-5 mol L-1 -1. s [I]o = (1 g L-1)/(164 g mol -1) 6.1 -3 mol L-1 FromEq. (6.26), using initial conditions,

[M]0 (Ykd [t]o)) (8.49 × 10-s moll-t -1) s (8.27 molL 0.027 mol

METHODS

OF INITIATION

Equation (6.25) for rate of polymerization is general in that the reaction for the production of radicals [e.g., Eq. (6.3)] is not specified and the reaction rate is simply shownas /?4. A variety of initiator systems can be used and radicals can be produced from them by a variety of thermal, photochemical, and redox methods [3-5]. (The term catalyst is often used synonymouslywith initiator. This is incorrect in the classical sense, since initiator fragments appear as end units [cf. Eq. (6.4)] and the initiator is thus consumed. The use of the term catalyst may, however, be condoned since very large numbers of monomer molecules are converted to polymer for each initiator molecule which is consumed.) Polymerization can also be induced by supplying the initiation energy through irradiation with visible and ultraviolet light, high-energy or ionizing radiation, or by the passage of an electric current. The conversion of a monomerto a polymer will occur through the" normal propagation, termination, and transfer reactions as in free-radical and ionic chain polymerizations (see Chapter 8). Only the initiation processes will be different.

/Lad/ca/Chain Polymerization Thermal Decomposition

453

of Initiators

The thermal scission of a compoundis the most commonmeans of generating radicals to initiate polymerization. The numberof different types of compounds which can be used as thermal initiators is rather limited. Compounds with bond dissociation energies in the range 100-170 kJ/mol are usually suitable. (Others with higher or lower dissociation energies will dissociate too slowly or too rapidly to be useful.) The major class of compoundswith bond dissociation in this range contain the O-Operoxide linkage. There are numerous varieties of compoundsof this type and some are listed in Table 6.4.

Table 6.4 SomePeroxide Initiators for Radical Polymerizations Type Diacyl

Example peroxides

(~ C-O-O-C--~

Temperature (°C) for tl/2 of 10 h 73

0 0 Dibenzoyl peroxide (CH3)3 C-C-O-O--C (CH3) 3 II II 0 0 Diisobutyryl peroxide - CH CH 3 3 I I Dialkyl peroxides

21

115 CH3 CH 3 Dicumylperoxide

Peroxyesters

CH3-C-O-OC (CH3) 3 II O t-Butyl peracetate CH 30

Dialkyl peroxydicarbonates

O CH 3 II II I H-C-O- C-O-O- C-O-C-H I I C2H5 C2H5 Di-sec-butylperoxydicarbonate

102

45

454

Chapter 6

Thermal decomposition is ideally a unimolecular reaction with a firstorder rate constant, kd, which is related to the half-life of the initiator, tl/2, by Eq. (6.29). For academic studies it is convenient to select initiator whoseconcentration will not change significantly during the course of an experiment so that instantaneous kinetic expressions, such as Eq. (6.26), maybe applicable. From experience it seems that an initiator with a tu2 of about 10 h at the particular reaction temperature is a good choice in this regard. This corresponds to a lea of 2×10-5 s -1 from Eq. (6.29). For the peroxide initiators listed in Table 6.4 the required reaction temperatures for 10 h half-life (t~/2) are also shown. It should be noted, however, that the temperature-half-life relations given in Table 6.4 mayvary with reaction conditions, because some peroxides are subject to accelerated decompositions by specific promoters and are also affected by solvents or monomers in the system. Aside from peroxides, the main other class of compounds used extensively as catalysts are the azo compounds. By far the most important memberof this class of initiators is 2,2’-azobisisobutyro-nitrile (AIBN) which generate radicals by the decomposition reaction: CH3

~H3

CH3-C-N--N-C-CH3 CN CN

CH3 -~ 2CH3-C" + CN

N2

(6.39)

The activation energies for decomposition of azo compoundsare similar to those of peroxides and they undergo facile dissociation like the latter, although the azo initiators do not contain a weak bond like the O-O linkage. In spite of the high dissociation energy (,-~ 290 kJ/mol) of the C-N bond, docomposition of AIBN proceeds because the nitrogen which is formed [Eq. (6.39)] is a very stable gas and has a very high enthalpy formation. This initiator has a 10-h t~/2 at-64°C. Azo compoundsare preferred for scientific investigations because the choice of reaction conditions has muchless effect on the behavior of azo initiators and their kg values do not vary with the particular polymerization system as muchas those of peroxides. Initiator

Efiiciency

Initiators are not used efficiently in free-radical polymerizations. This becomes apparent when a material balance is performed on the amount of initiator that is decomposed during a polymerization and compared with the amount that initiates polymerization and thus becomes a part of the polymer formed. The values of f for most initiators lie in the range of 0.3-0.8. The major cause of this low f is wastage of primary radicals in "cage" reactions. When

Radical ChainPolymerization

455

an initiator molecule decomposesforming primary radicals, the radicals are each other’s nearest neighbors for about 10-l° s. During this short interval the primary radicals are surrounded by a "cage" of solvent and monomer molecules through which they must diffuse to escape each other. While in the cage, the radicals can be expected to be colliding on the average once every 10-13 s (the vibrational frequency of a diatomic molecule at reaction temperatures being approximately 1013s-1) and so any reaction can take place. To illustrate, the decomposition of benzoyl peroxide could lead to the following reactions: ~bCOO-OOCq~ ~ (2q~COO’)cage (q~COO’)cage (¢CO0")cage @CO0" + M ~COO" -- ~"

+ M

(2q~COO’)cage ~ (q~COO~ + COs)cage + M ~ ¢COOM" ~ ¢C00" ~ ¢COOM" + COz

(6.40) (6.41) (6.42) (6.43) (6.44) (6.45)

Equation (6.40) represents the primary step of initiator decomposition into two radicals, which are held within the solvent cage (with solvent and/or monomermolecules comprising the cage wall). The radicals in the solvent cage mayundergo recombination [i.e., reverse of Eq. (6.40)], reaction with each other [Eq. (6.41)], reaction with a monomermolecule contained the wall of the cage [Eq. (6.42)], or diffusion out of the solvent cage [Eq. (6.43)]. Once outside the solvent cage, the radicals may react with monomer[Eq. (6.44)] or decompose according to Eq. (6.45)] to yield a radical which may react with monomer. Once one or the other radical leaves the cage, it is extremely unlikely that the pair will encounter each other again. Recombination of the primary radicals [the reverse of Eq. (6.40)] has no effect on the initiator efficiency. Initiation of the polymerization occurs by reactions (6.42), (6.44), and (6.46). The initiator efficiency is decreased by the reaction indicated by Eq. (6.41) since the reaction products are stable and cannot give rise to radicals. This reaction is usually muchmore significant than any other in decreasing the value of f. Since the rate constants for radical-radical reactions are in the range 107 L/mol-s and higher, and the concentration of radicals in the solvent cage is high (,-~ 10 mol/L), there is a reasonable probability that the radical pair in the solvent cage will react as indicated by Eq. (6.41). All initiators suffer cage wastage reactions. For AIBN,for example, f is only 0.6 at 60°C because of these reactions.

Problem6.8 Consider the following schemeof reactions for free-radical chain polymerizationinitiated by thermal homolysisof initiator.with cage effect [6]:

456

~hapter6

I

~ (2R.)

(2R-)

(P6.8.1)

-~

(P6.8.2)

(R-) + ~ --~ (R’)

(P6.8.4)

~

R" + R. R"

+ M k.~

M-

+

M ~

M"

+

M’

~

Q’

(P6.8.5)

M"

(P6.8.6)

M"

(P6.8.7)

V

(P6.8.8)

where (R’) indicates a primary radical within a cage provided by the medium, is the stable product of reaction between a pair of primary radicals within a cage, M" denotes chain radicals having an end group R (derived from initiator), and P a dead polymer molecule; other species are as defined earlier. Reactions (P6.8.2), (P6.8.3), and (P6.8.4) are the alternatives available to a primary radical within cage, namely, reaction with its primary partner, reaction with a monomermolecule contained in the wall of the cage, and diffusion out of the cage. Reactions (P6.8.5) and (P6.8.6) are analogous to (P6.8.2) and (P6.8.3), but refer to the main body the solution, that is, they are reactions which may occur only after (P6.8.4). Neglecting Chain transfer, derive an expression for the rate of polymerization showing the cage effect. Answer: It should be noted that the rate of reaction (P6.8.2), unlike reaction (P6.8.5), first order with respect to [(R’)], since the primary radicals are necessarily formed in pairs in independent cages. It is also knownthat for all practical values of [M], reaction (P6.8.5) is negligible. The net rate of formation of (R’) is thus given d[(R-)]

- 2kd[I] - /¢, [(R.)]- ks [(R.)][M]kn[(R.)]

For a steady state concentration

[(R-)]

of (R.), d[(R’)]/dt

= 0. Therefore,

2ka[I] k~ + kD + k~[M]

(P6.8.9)

If reaction (P5.8.6) is fast, the rate of initiation will be given ~ = kD[(R’)] + k~[(R’)][M] = [(R.)] (ko + k,

(P6.8.10)

Combination of Eqs. (P6.8.9), (P5.8.10), and (5.25) (P6.S.11) -k~ + kD + k,[M] [IIV~[MI ( ~ ) x/2 kD+ k ~ [M] Comparison of Eq. (P6.8.11) with Eq. (6.26) shows thatthe polymerization is reduced by the cage effect. R~ = kp

457

Radical ChainPolymerization For k v >> k~[M]

and kD >> kr,

Eq. (P6.8.11) reduces to Eq. (6.26)

ideal kinetics and negligible cage effect. The initiator efficiency is not an exclusive property of the initiator alone, but it varies to different extents depending on the prevailing condition of polymerization, including the identities of monomerand solvent used. The viscosity of the reaction mediumhas an effect on f. Withincreasing viscosity, the lifetimes of radicals in the solvent cage are increased, leading to greater extents of radical-radical reactions within the solvent cage and hence to a lower value of f. In some cases, f varies with solvent due to the solvent reacting with (scavenging) radicals before the latter can initiate polymerization. The change of initiator efficiency with solvent mayin a few instances be due to solvation of radicals by solvents. The initiator efficiency for any particular initiator mayalso vary depending on the monomerwhich is polymerized. For example, the value of f for AIBNranges from 0.6 to 1.0 in the polymerizations of methyl methacrylate, vinyl acetate, styrene, vinyl chloride, and acrylonitrile, increasing in that order [9]. This is a consequence of the differences in the rates with which radicals add to the different monomers. Experimental

Determination

of f

Several methods can be used for the evaluation of the initiator efficiency. One method depends on the direct analysis of initiator fragments as end groups in the polymer formed compared to the amount of initiator consumed. The use of isotopically labeled initiators such as 14C-labeled benzoyl peroxide and other related peroxides, and 14C-labeled AIBNand asS-labeled potassium persulfate provide appropriate sensitive methods for determining the number of initiator fragments trapped as end groups in the resulting polymers [10,11]. A second method involves measurement and comparison of both polymer production and initiator decomposition. Since reaction environment has a significant effect on initiator decompositionreaction, it is necessary that the measurement of initiator decomposition is done during an actual polymerization. The decomposition of AIBNcan be followed relatively easily by monitoring the evolution of nitrogen. Determination of the number-average molecular weight, and hence the number of polymer molecules formed in a given time, and comparing with the number of radicals produced over the same period (estimated from initiator decomposition) allows a determination of f. This method, however, requires a knowledge of whether termination occurs by coupling or by disproportionation, since the former results in two initiator fragments per polymer molecule and the latter in only one. A third method involves counting of radicals in a system with radical scavengers which stop chain growth rapidly and effectively. The stable free

458

Chap~e~ 6

radical diphenylpicrylhydrazyl (DPPH)has been widely used for this purpose [10,11]. The DPPHradical (purple or deep violet), obtained by oxidation of diphenylpicrylhydrazine with PbO2, reacts with other radicals to form a non-radical adduct (light yellow or colorless): 02N

02 N

NOz(6.47) O2N

~ O2N

The reaction can be easily followed spectrophotometrically because of the color change associated with the reaction. Other radical scavengers have been used in similar counting of radicals [14]. The approach of determining f using radical scavengers is, however, not very useful as the reaction between the scavengers and radicals is often not quantitative. A fourth and probably the most useful method of determination of initiator efficiency is based on the dead-end effect in polymerization technique which is treated in a later section. This technique allows treatment of kinetic data obtained under dead-end conditions to evaluate both the rate constant for initiator decomposition (kd) and the initiator efficiency (f) under experimental conditions. Redox Initiation Manyoxidation-reduction reactions produce free radicals that can be used to initiate polymerization. This type of initiation is referred to as redox initiation or redox catalysis. A prime advantage of redox initiation is that radical production occurs at reasonable rates over a very wide range of temperatures, including 0-50°C and even lower, depending on the particular redox system. This allows a greater freedom of choice of the polymerization temperature than is possible with the thermal dissociation of initiators. This redox initiation can be arranged to proceed quickly under mild reaction conditions and is particularly useful for low and ambient temperature radical polymerizations. A wide range of redox systems, including both inorganic and organic components, either wholly or in part, maybe employed for this purpose. A general redox reaction described by A- B + X ----+ A" + Be + Xe (6.48) can proceed with any molecule AB, provided the reducing agent X is strong enough to split the A-B bond. For practical purposes, the A-B bond in redox systems must be relatively weak and this limits the choice of such materials. Someof the commonredox systems are described below.

459

Radical Chain Polymerization

(a) Peroxides in combination with a reducing agent are a commonsource of radicals, for example, the reaction of hydrogen peroxide with ferrous ion: HOOH + Fe z+ --~ HO" + OH- + Fe 3+ (6.49) Ferrous ion also promotes the decomposition of a variety of other compounds including various types of organic peroxides [4]: ROOR + Fe 2+ 3+ ~ RO" + RO- + Fe (6.50) 2+ 3+ RCOOH -+- Fe ~ RO" q- OH- q- Fe (6.51) +, 2+, 2+, 2+, 3+ Other reductants such as Cu Co Cr V and Ti can be employed in place of ferrous ion in manyinstances. Most of these redox systems are aqueous or emulsion systems. In organic media amines can be used as the reductant [15,16] for redox initiation with acyl peroxides, an examplebeing the combination of benzoyl peroxide and an N,N-dialkyl-aniline. O

O

R

q~-C-O-O-C-q~

+ q~-N-R

O ~

O

4-c-o- + -o-c-4 +

R ~.-N’+ R

Initiation is only by the q~COO"radical because the amino cation radical is not an effective initiator as shownby the absence of nitrogen in the polymer. The rate of initiation by this redox system will thus be given by R./ = k d [peroxide] [amine] (6.53) (Note the absence of factor 2 in this expression.) Substitution of R/in Eq. (6.25) gives t~ = 1/2 kp [M] (kcl[per°xide]2kt [amine])"

(6.54)

as the equation for the rate of polymerization with peroxide-amine redox initiation. Similar procedure may be followed for other redox systems. The decomposition rate constant kd for pure benzoyl peroxide in styrene polymerizations is 1.33x10 -4 s -I at 90°C, while that for the benzoyl peroxide-N,N-diethylaniline redox system is 1.25x 10-2 L/mol-s at 60°C and 2.29x10-3 L/mol-s at 30°C [15]. The redox system thus has a muchlarger decomposition rate. Peroxide decomposition is also accelerated in the presence of transition metal ion complexes such as copper (II) acetylacetonate and ammoniumsalts [17,18]. (b) The combination of a variety of inorganic reductants and oxidants can be used to initiate radical polymerization [4,19], for example, redox systems with persulfates as the oxidant:

460

Chapter 6

-o s-o-o-so + -O S-O-O-SO;+

+ SO- + Fe (6.55) + SO- + S,O(6.56/

2+. Other redox systems include reductants such as HSO~,SO~-, and Fe (c) A combination of inorganic oxidant and organic reductant initiates polymerization, usually but not always by oxidation of the organic 4+, component, for example [20], the oxidation of an alcohol by Ce Ce 4+ -~- RCH2- OH ~ Ce 3+ n t- H+ + RCHOH (6.57) or by V5+, Cr6+, or Mn3+ [21]. Other redox pairs include oxidation of aldehydes and ketones by Ce4+ and V5+ [22], oxidation of thiol compounds such as thioureas, thioglycolic acid, and 2-mercaptoethanol by Fe3+, 4+, Ce BROW-, and $2082- [23,24], and oxidation of oxalic, malonic, and citric acids by permanganate and Mn3+ [25]. Someredox polymerization involve termination by reaction between the propagating radicals and a component of the redox system. Thus in the alcohol-Ce4+ system [Eq. (6.57)], termination occurs according Mn" + Ce 4+ ~ Ce 3+ + H+ + dead

polymer

(6.58)

at high eerie ion concentrations. The propagating radical loses a hydrogen to form a dead polymer molecule with an olefinic end group. The rates of initiation [Eq. (6.57)] and termination (neglecting the usual bimolecular termination mechanism) are given by R4 = kd ICe4+] [alcohol] nt ---- /~ [Ce4+1 [M’] By making the usual steady-state assumption (i.e., the polymerization rate as

(6.59) (6.60) Ri = Rt), one obtains

kd kp [M] [alcohol]

(6.61)

Someredox systems involve direct electron transfer between oxidant and reductant, while others involve the intermediate formation of oxidantreductant complexes. For those redox initiations involving the equil~rium formation of intermediate complexesthat lead to radical formation, derivation of the kinetics follows in a straight forward manner (see Problem 6.9). Problem6.9 The free-radical polymerization of acrylonitrile initiated by the redox system Mna+-cyclohexanol(CH) was investigated [22] in aqueous sulfuric acid in the temperaturerange 30-45°C.The following reaction mechanisminvolving

461

Radical Chain Polymerization

3+ and the alcohol, whose decomposition the formation of a complex between Mn yields the initiating free radical with the polymer chain being terminated by the mutual combination of growing chains was suggested: K ~ complex (a) 3+ + CH k~

(b)

complex

R" + Mn2+ +

(c)

R" + 3+ -- ~ Mn 2+ +

(d)

R"

+ M -~

(e)

M~" + M ~ M,~+I"

(f)

M~" + M,~"

H+

products

MI"

~ polymer

Derive suitable rate expressions for evaluation of rate parameters from initial rates of polymerization.

Rate of Mn3+ disappearance Applying the steady-state

principle to the pfima~ radial R’, z+] k.K [Mn [CH] ko [Mna+] + hi [M] For reaction (a), [complex] K [Mn3+1 [CH]

(P6.9.1)

(P6.9.2)

3+ For total Mn [Mna+]tot

[Mn3+] + [complex] [Mn~+] (1 + K [CHI)

(P6.9.3)

Therefore, d [Mna+]tot dt

a+] d [complex] d [Mn dt dt k,- K [Mna+] ~+] [CH] + ko [R’] [Mn = k,.

ko k~ K [Mn3+]2 [CH] K [Mn3+ [CH] + 3+]

ko[Mn + k, [M]

If ko [Mn~+] >> hi[M], Eq. (P5.9.4) reduces to the form d [Mn~+]tot 3+] [CH] 2k,. K[Mn dt 2k,- g [Mn3+]tot [CH] 1 + K [CH] = 2k~ [Mn3+]tot

(

1 [CI-~

1 K)

(P6.9.4)

462

Chapter 6

The values of K and kr can be computed by plotting (- d [Mn3+]tot / dr) versus a+. From the intercept and the ratio 1/[CH] at constant concentration of total Mn intercept/slope of the plot the rate constant for the unimolecular decomposition of the complex (k~) and its formation constant (K) can be calculated. Rate of polymerization Applying the steady-state principle to the growing chains, i.e., of reactions (d) and (f),

equating the rates

~ k~ JR’] [M] = k, [M’] ~-"

[M’] " (ki JR’] ’/2 [M]) k,

(P6.9.6)

Substituting for [R’] in Eq. (P5.9.5) from Eq. (P5.9.1) and then substituting [M.] ~n Rp = kp [M-] [M], yields p~ = k~, k~/2 (k~ K [Mn3+] [CH])1/2 ~/~ [M]

k:/2

+ [M])1/2

If k0 [Mn3+] >> k~ [M], Eq. (P5.9.7)

(P6.9.7)

can be approximated to the form

I~ = k~ k~/~ (k~ K [CH])~/~ ~/~ [M] The com~osk¢ ~ns[am k~ (k~ / ko ~/~ can be obtained ~om ~h¢ slope of the plot of ~M]~ / R~) versus -~/[CH], if k~ and g are ~own from ~q. (P5.~.5). ~or fitfin~ ¢xpe~mcntal ra~e dam ~q. (P5.11.7) can b¢ conveniently rea~anged to the form

This equation can be used to evaluate rate parameters using the iNtial rate of polymeNzation (~) data that may be obtained ~om the iNtial slope of the conversion vs. time plots. Sin~ k~ and K are obtained from Eq. (P6.11.5), the value of the mmposite constant k~ (k~/k, ~)~1~ can be obtained ~om the inter~pt of a plot of ([M]/~)~ against 1/[Mn ~+] at mnstant ~ncentration of the monomer[M] and the reducing agent [CH]. ~e slope gives the parameter k~/k~/~ while the quotient of intercept and slope yields k,/k~.

Photochemical Initiation Photoinitiation of vinyl polymerization offers significant practical advantages. An obvious advantage is the avoidance of chemical contamination by initiator residues. Moreover, there is a marked convenience to photochemical reactions that appeals to many researchers. Photoinitiation and

Radical Chain Polymerization

463

polymerization can be spatially directed (i.e., confined to particular regions) and turned on or off simply by turning the light source on or off. Moreover, the initiation rates can be controlled by a combination of light intensity, source of radicals, and temperature. Extensive and expanding use is being madeof these advantages in the printing and coating industries. Somemonomersundergo direct photoinitiation and free-radical chain polymerization when exposed to ultraviolet or visible light. For other monomers,a photosensitizer must be added to the system. Photosensitizers are compoundsthat absorb ultraviolet or visible light and then dissociate into free radicals or transfer energy directly to the monomer. Direct

Photoinitiation

Two different types of direct photoinitiation can be recognized. In the first, absorption of light photons (quanta) yields an electronically excited monomer molecule M* : M +

hv

~

M*

(6.62)

which subsequently decomposes to give radical fragments: M* ~ R" + R" (6.63) capable of initiating polymerization of the monomer. Examples of monomersin this category include alkyl vinyl ketones and vinyl bromide both of which dissociate whenirradiated with ultraviolet light by the following reactions: O

O

R-C-CH=CH~ -----> O R-C"

~

R"

CH2=CHBr --~

+

R-C"

+ "CH=CH2

(6.64)

CO

Br"

(6.65) + CH2=CH"

(6.66)

The resultant monoradicals add to the monomerand radical chain polymerization takes place. The second type of initiation mechanism is exemplified by the photopolymerization of styrene or methyl methacrylate. Absorption of light in this case does not result in decomposition of monomermolecules. Instead, it has been suggested [26] that radicals are produced by a complexinitiation mechanism,illustrated for styrene by the following equations: + hv’ Ph-CH=CH2

~

(PhCH=CH2)*

__~PhCH=CH2 t*ph~H-~H2

(6.67) (6.68)

464

Chapter 6

Ph~H -- ~H2 4Ph~HCH2CH2~HPh

PhCH -- CH2 ~ Ph~HCH2CH~HPh (6.69) 4- PhCH : CH2 ~ CH3~HPh 4PhCH = CHCH~HPh

(6.70)

According to this mechanism, the absorption of light produces an excitdd singlet state of the monomerwhich may either fluoresce [Eq. (6.67)] be converted to an excited (and long-lived) triplet st.ate [Eq. (6.68)]. latter maybe regarded as a diradical, that is, oCH2-C(H)X.Attack on the monomerby this diradical ultimately yields two monoradicals [Eqs. (6.69) and (6.70)], which, in turn, initiate polymerization. For both types of direct photoinitiation the rates are proportional to the light intensity and to the extinction coefficient of the monomer(see later). Photosensltization Even if direct light absorption as above does not occur, polymerization can still be initiated if photosensitizers are present that produce free radicals when they absorb ultraviolet or visible light. The same substances that are used for thermal initiation are often used for photosensitization. For example, azo compoundsand peroxides are photosensitizers, and the photoinitiation reaction is the sameas is the thermal initiation process, described earlier in this chapter. However, the photoinitiation can take place at much lower temperatures than in the thermal initiation by the same initiators. Moreover, manyiniti£tors can be used as photosensitizers even though they do not dissociate thermally at convenient rates or temperatures to be useful as thermal initiators. For example, azoisopropane does not dissociate sufficiently rapidly below 180°C to be useful thermal initiator. However,it photodissociates even at low temperatures when irradiated with near-ultraviolet light: (CH3)2CHN

= NCH(CH3)2

+ h~ (3000~

< A < 4000~)

2(CH3)2~H 4- N2 (6.71) Other examples of such photosensitizers include carbonyl compoundssuch as ketones, both aliphatic and aromatic. Aromatic ketones are, however, more useful in commercial practice, since their absorptions occur at longer wavelengthand their efficiency of initiation (quantumyield) is higher. Benzophenone and acetophenone and their derivatives are the most commonly encountered aromatic ketones. Ketones undergo homolysis by one or both (often simultaneously) of two processes- fragmentation and hydrogen abstraction. Fragmentation involves O O O

¢-c-¢’

+ "¢’

(6.72)

while hydrogen abstraction occurs only in the presence of a hydrogen donor (RH):

RadicalChainPolymerization 0 II

¢-c-¢’

hu

465

0 [I RH C (¢--¢’)* ------>

OH

I

¢-C-¢’ + R"

(6.73)

Amineswith abstractable c~-hydrogensare the mostefficient and extensively used hydrogendonors; less efficient donorsinclude alcohols and ethers. The hydrogenabstraction is generally moreefficient than fragmentation and occurs at longer wavelengths(lower energy photons). The relative amounts of the fragmentationand hydrogenabstraction reactions vary dependingon the type of initiator and on the hydrogendonor employed. Similarly photolytic reactions occur with benzoin (I), benzoin ethers (II), benzil (III), and benzil ketals (IV). These and related compounds are extensively used as photoinitiators in commercialpractice, usually in combinationwith hydrogendonors. 0 OH II I

0 OR II I

(I)

(II)

0 0 II II

0 OR II I

(Ill)

I OR (IV)

Dye-sensitized photopolymerizations are of interest in that the spectral range of photoinitiation can be extendedinto the visible region. A variety of dyes such as methyleneblue, thionine, tluorescein, and eosin undergo excitation and the excited dye can interact with an appropriate substanceto produceradicals which, in turn, can initiate polymerizationof the substrate monomer.Dye-sensitized photopolymerizations often comprise redox systems involving electron or hydrogentransfer betweenthe excited dye and the other substance. Anexampleis the methyleneblue-p-toluenesulfinate ion system. Thep-toluenesulfinate ion is oxidized by the irradiated dye to the correspondingradical whichinitiates polymerizationof the monomer:

CH3--~~SO~

~

CH3---~SO

2"

(6.74)

Somedye-sensitized systems appear to proceed by energy transfer from the excited dye D* to another compoundC, D* + C ~ D + C*

(6.75)

466

Chapter 6

Radicals then form by decomposition of the excited state C, C* ~ radicals

(6.76) The net outcomeis that, whereas C* cannot be produced by direct irradiation of C with light of frequency t~, the excitation of C takes place because D* is able to transfer energy to C at a different frequency, ut, which can be absorbed by C. Rate

of

Photopolymerization

In photochemistry, a mole of light quanta is called an Einstein. A mole of light quanta of frequency ~ or wavelength A has energy Nhvh~, or NAvhC/A,where NAyis Avogadro’s number, h is Planck’s constant, and c is the speed of light. The rate of photochemical initiation maythen be expressed as P~ = 2~I~ (6.77) simply by replacing led [I] of Eq. (6.14) with Ia, the intensity of absorbed light in moles (Einsteins) of light quanta per liter per second, and replacing f with d~ for photochemical polymerization. The factor of 2 in Eq. (6.77) is used to indicate that two radicals are produced per molecule undergoing photolysis. The factor of 2 is not used for those initiating systems that yield only one radical instead of two. Referred to as the quantumyield for initiation, ~ is synonymouswith f in that both describe the efficiency of radicals in initiating polymerization. Thus, the maximumvalue of ¢~ is 1 for all photoinitiating systems. An expression for the photopolymerization rate is obtained by combining Eqs. (6.25) and (6.77) to yield P~p = kp [M] (9

I~ / ~)1/2

(6.78)

Problem 6.10 The polymerization of methyl methacrylate (1 mol per liter of solution) is carried out using a photosensitizer and 3130 ~ light from a mercury arc lamp. Direct measurementby actinometry showsthat light is absorbed by the systemat the rate of 1.2x105ergs/L-s. If the quantumyield for chain initiation in this systemis 0.50, calculate the rates of initiation and polymerization.[k~,/k~/2 at 60°C = 0.102 L1/2 mo1-1/2-1/2] S Answer: Energy of 1 photon (hc/A) = (6.63 x -27 er g s) (3 x 1 01 (3.13 x 10-5 cm) -12= 6.35 x 10 erg

467

Radical Chain Polymerization

Energyof 1 mol of light quanta -12 -1) erg quantum = (6.02 x 10~z quanta tool -1) (6.35 10 = 3.82 × 1012 ergs mo1-1 I, = (1.2 × 105 ergs L-ls-1)/(3.82 × 1012ergs tool -1) = 3.14 x 10-8 -1 tool L-is Ri = 2~ar, ---- 2 x 0.5(3.14 × 10-s molL-is-1) ----- 3.14 x 10-s -1 tool L-is 1/~ /~ = (k,/k~/~)

[M] (R,/2)

= (O.102Ll/2mo,-l’2s -1/2) (1.0mol -I) ( 3.14× 10-8mo1 L-is-l) = 1.28 x 10-5 -1 mol L-is Assuming,for simplicity, that the incident light intensity does not measurably vary with thickness of the reaction system, I, will be given by

io = do [A] b

(6.79)

where lo is the incident light intensity, A is the species that undergoes photoexcitation, e is the molar absorptivity (extinction coefficient) of A at the particular frequency of radiation absorbed, and b is the thickness of the reaction system being irradiated. (e usually has units of liter/molcm.) Combination of Eq. (6.77) with Eq. (6.79) yields ~ = 2+ao [A] b (6.80) This can be substituted into the general equation (6.25) for P~o to give P~ ---

/¢p [M]( ~, fl~,Zo -~[Al b ) 1/2

(6.81)

Equation (6.81) indicates that for photopolymerizations P~ is first order in [M], ½-order in light intensity and ½-order in [hi for the case where there is negligible attenuation of the light intensity in traversing the reaction vessel. Whenthe photoexcitation involves monomer,i.e., A is M, Eq. (6.81) becomes 1~ = kp[M] 3/2

(~elo

b~ 1/2

(6.82)

that is, the dependence of Rp on [M] becomes 3/2-order. Abnormal orders in [M] have been found under certain circumstances. An example is the photopolymefization of methyl methacrylate with ketone photoinitiators [27]. For this system the dependence of Rp on [M] drops from first-order [Eq. (6.81)] to ½-order as [M] increases, leading to quenching by the monomer. In simple terms, a quencher (which can be A or M or solvent) undergoes energy transfer with the photoexcited species to dissipate the excitation energy. Quenchingcan be observed if the concentration of A or Mis too high. It leads to a lower than expected dependence of Pqo on the component of the system that acts as a quencher.

The use of Eq. (6.82) assumes that there is negligible attenuation the light intensity in passing through the reaction vessel. This wiIl be nearly true ~only whenthe light absorption is quite low or very thin reaction vessels are employed. For most polymerizations, however, the light absorption will not be negligible and hence the incident light intensity will vary with thickness. From Lambert-Beer’s law, -~ I [A]b = Io e

(6.83)

where ~r is the light intensity at a distance b into the reaction vessel. The light intensity absorbed, Ia, is then given by

= ±o- = ±0 where b now reprrsents the total thickness of the reaction vessel. An expression for the polymerization rate can be obtained by combining Eq. (6.25) with Eqs. (6.77) and (6.84) to

1/2 R~ = kp[M] [.~I°

(1 - ~,e-~[Alb)]

¯

(6.S~)

The light intensities delivered by various light sources are usually known in units such as kcal/s, kJ/s or erg/s, and it is necessary to convert them into the appropriate units of moles (Einsteins) of light quanta per liter per second before use in the above equations. Measurement

of Absorbed

Light

One may avoid the use of Eq. (6.85) and, instead, use Eq. (6.78) [with Ia defined by Eq. (6.84)] by directly measuring /’a in the particular polymerization system (see Problem 6.10). Measurementof light intensity, generally referred to as actinometry, may be performed with chemical, thermal, and electrical actinometers. The thermal and electrical actinometers are generally more convenient to use than chemical actinometers and include semiconductor photodetectors, photomultipliers, and thermocouptes which operate on the principle of converting photon energy to either electrical or thermal energy. The absorbed light intensity Ia is measured by placing an actinometer directly behind the reaction vessel and measuring the differences in light intensity whenthe vesgel is emptycomparedto when it holds the reaction system.

tL~d~calChainPolymerization Initiation

by Ionizing

469

Radiation

High-energy or ionizing radiation can induce chain reaction polymerization of a pure monomer (or a solution of the monomer). Alpha particles, beta rays (electrons), gammarays, or high-velocity particles from particle accelerator can all be used. (Ionizing radiations have particle or photon energies in the range 10 keV-100 meV compared to 1-6 eV for visible-ultraviolet photons.) Practically any monomerthat polymerizes by a free-radical mechanism can be polymerized in this way. Though rapid polymerization of monomers and the cross-linking of polymers by highenergy radiation have been considered as possible manufacturing processes, radiation-induced polymerization has achieved far less commercial success than photochemical polymerization. The reason is the higher costs and safety problems of ionizing radiation sources compared to photochemical sources. Gammaradiation is the most convenient type of high-energy radiation for initiating polymerization because its high penetrating power affords uniform irradiation of the system. Because gammarays are absorbed to the same extent by solids as by liquids, solid monomerscan be polymerized readily. This allows polymerization of manymonomersat low temperatures. The chemical effects of the different types of radiation on molecules are qualitatively the same, though there are quantitative differences. Molecular excitation mayoccur with the subsequent formation of radicals in the same manner as in photolysis, but ionization of a compoundby ejection of an electron is more probable because of the higher energies of these radiations. (For this reason such radiations are termed ionizing radiations.) Thus a compoundC, on excitation, may yield a radical-cation C+" by ejecting an electron, C + radiation -~ C+" + e(6.86) The radical-cation can propagate at the radical and positive centers depending on reaction conditions. It mayalso dissociate to form separate radical and cationic species: C+" ~ A+ q- B" (6.87) The initially ejected electron maybe attracted to the cation A+ with the formation of another radical: A+ q- e- ----* A"

(6.88)

The ejected electron may also be captured by the component C forming an anion C- which may or may not be ex6ited depending on the energy of the electron, and may further dissociate producing radicals and ions:

470

Chapter 6

C + e-

~

C-

-[-B-

~ A"

B- --~

C-

B" -[-e-

(6.89)

(6.90) (6.91)

The high-energy irradiation of either a pure monomeror a solution of a monomergenerates both free radicals and ions. In light of the above discussion, these seem to result more from the gross "damage" sustained by the monomerthan from the selective induction of specific chemical reactions. Becauseof this lack of selectivity, it is not possible to calculate a meaningful quantumyield for radical-induced reactions. Instead, the energy yield is described by a G-value, which is defined by (say, for substance A), G(:/:A)

number of molecules of A formed or consumed xl00 numberof electron-volts of energy absorbed

(6.92)

Since the high-energy irradiation generates both free radicals and ions, either free-radical or ionic chain polymerizations may be induced, the preponderance of one or the other mechanism being dependent on the nature of the monomer,as well as on the temperature, and purity of the reagents. Somemonomerssuch as acrylic esters, vinyl esters, and vinyl fluoride, polymerize only by free-radical mechanisms,while others such as styrene, acrylonitrile, isoprene, apparently polymerize by both radical and ionic types of chain mechanisms.Isobutylene, however, polymerizes only by a cationic chain mechanism.It is usually only at low temperatures that the ionic species are stable enough to initiate polymerization. At ambient or higher temperatures, the ionic species are usually not stable and dissociate to yield radicals. Initiation mayalso be carried out using initiators or other compounds which are prone to undergo decomposition on irradiation. Three distinct phases may be identified in a radiation-induced polymerization. In the first phase, the interaction of radiation with monomer, -16 solvent molecules or other compoundsoccurs within a brief interval of 10 to 10-15 s, resulting in the formation of electronically excited molecules, ions, and electrons. In the second phase, which takes place some 10-14 to 10-1° s after the initial interaction, these products dissociate or react with monomerto yield a set of initiating free radicals and ions. In the third phase occupying the time regime of 10-1° to 10-1 s, these free radicals and ions undergo the normal elementary reactions of initiation, propagation, transfer, and termination characteristic of chain polymerization. The irradiation of a pure monomer gives rise to the simplest initiation reactions, because only one chemical species absorbs the radiation. However,

Radical Chain Polymerization

471

solutions undergo more complicated initial reactions because both the solute and the solvent can absorb radiation energy to produce ions and free radicals. As a useful approximation, one can assume that the fraction of the total energy absorbed by a particular species in solution is proportional to the electron fraction of that species. The electron fraction is defined as the number of electrons contained in that species divided by the total number of electrons in the whole solution.

Problem6.11 If a 0.1 Msolution of styrene in carbon tetrachloride is irradiated to initiate polymerization,whatfraction of the total energyabsorbedby the system will be absorbed by the solvent ? Whattype of species will be the predominant 3] initiating species ? [Styrene density -- 0.905 g/cm3;CC14density = 1.594 g/cm Answer: Basis: 1000 cm~ solution Vol. of styrene (CsHs) ---- (0.1 mol)(104 g mo1-1)/(0.905 -3) ~ = 11.49 cm Assumingadditivity of volumes, Amountof CC14 = (1000cm3 - 11.49cm3) (1.594 gcm-a) -1 / )(154 g tool = 10.2 tool [CCl4]

=

-a 10.2 tool L

The numbersof electrons in styrene and CC14 molecules are 56 and 74, respectively. Therefore, the electron fraction of CC14 in solution is 74 [eel4] 74 [CCl4].q- 56 [CsH8] 74 x 10.2 = = 0.993 74×10.2 + 56×0.1 Thus, 99.3%of the total energy absorbed by the system will be absorbed by the solvent, and hencemostof the initiating ions and free radicals will be derived from the solvent. Thestyrene polymerizationis this case will be initiated predominantly by species such as CCI~, CCI~+, CI’, CC13", etc. eCC h =

The kinetics of radiation-initiated polymerizations follow those of photolytic polymerization in a relatively straightforward manner. The rate of initiation is determinedby the intensity of irradiation and the concentration and radiation susceptibility of the compoundthat decomposesto yield the initiating species (ions and/or radicals). The final expression for the rate

472

Chapter6

polymerizationis, however,determinedby the exact details of the initiation, propagation, and termination steps. l~ree-Radical

Chain Initiation

Themostimportantcharacteristic of the initiation step in radiation-induced free-radical chain polymerizationis the yield of radicals generatedby the absorption of a given radiation dose. This characteristic is quantified by a G-value, defined as the numberof radicals producedfor 100 eV of energy absorbed. This is referred to as GM(R" ) or Gs(R"), dependingon whether the radiation energy is absorbedby the monomer or by the solvent. SuchGvalues have been determined for a numberof commonmonomersby radical scavengingtechniques. Sometypical values are shownin Table 6.5. Froma knowledge of such values the rate of formationof radiation-initiated radicals can be calculated in the samewayas wasdescribed earlier for thermal and photochemicalinitiation. Thus, for the radiation-induced polymerizationof a pure monomer, the rate of formationof initiating radicals is described by d[R’] aM(R’) (dQA~ d~ -- i00 \--~-] where(~A is the energyabsorbedper unit volumeof the reaction systemFor monomer solution, the correspondingexpression is

d[R.]_ dt

100

+

(dQA) 100 J \ dt

(6.94/

The energy absorption per unit volume, QA,and the rate of energy abTable6.5 G-Values(100-eVyields of initiating radicals) in RadiolyticFree-RadicalChainPolymerization Monomer or solvent Styrene Acrylonitrile Methylmethacrylate Vinylacetate Isobutylene n-Hexane Benzene Toluene

GM(R-) or ) Gs(R" (radicals/ 100-eV) 0.66 5.0 6.1 9.6 3.9 5.8 0.66 2.4

Radical Chain Polymerization

473

sorption per unit volume, dQA/dr, are related, respectively, to the dose and dose rates (the terms used by radiation chemists to express energy absorption per unit mass) by the density of the system. Radiation doses are often expressed in a unit called the rad, which is defined as 100 ergs/g. A typical dose rate from a 6°Co "),-ray source wouldbe 106 rads/h or 1 Mrad/h.

Problem 6.12 Consider the irradiation of (a) pure styrene (density = 0.905 g/cma) and (b) 1.0 Msolution of styrene in toluene (density = 0.871 g/era3) at 20°C with 7-rays and a dose rate of 1 Mrad/h. Calculate the rate of initiating radical formationin the twocases. Answer: (a)

dQ A dt

= (l×10 s radh -a)(10z

ergsg-a rad-1)(0.905

g -a)

(6.24 x 10la eV erg-a) (2.78 x 10-4 -1) h s la -a -1 = 1.57 × 10 eV crn s -- (0.66 x 10-2 radicals eV-a) (1.57 = 1.04 x 1034 radicals cm-3 -a s

× 10 la eV

cm-3 -1) s

(b) Assumingadditivity of volumesof styrene (Calls) and toluene (Crt-Is), obtains.by calculation: [CrHa] -- 8.34 mol -1 L Therefore,

d0h = (1 x 10~ tad h-~) (102 ergs g-1 rad-1) (0.871 g -~) dt

(6.24 × 10n eV erg-~) (2.78 x 10-4 -~) hs = 1.51 x 101~ -as eV -1 cm

The numbersof electrons in styrene and toluene are 56 and 50, respectively. Therefore, 56 (1.0 tool -a) ~C~H~= ’56(1.0 tool L-1) -t- 50(8.34 tool -a) =0. 12 eCTI-h = 1.0 -- 0.12 = 0.88 From Eq. (6.94), dt

-- [(0.12)(0.66 x lO-2radicais -1) -q-(0.88)(¢~.4 x (1.51 x 10’~ ~Vcrn-z -a) s TM -~ -~ = 3.31 × 10 radicals cm s

X

lO-~radicals eV-~)]

474

Chapter 6

Ionic

Chain Initiation

The G-values for the initial formation of ion pairs in liquids are in the range 3 to 4. Though this is comparable to the G-values found for free-radical formation, the efficiency of ionic chain initiation is much lower than that of free-radical chain initiation. This is because most of the gegenions (see Chapter 8) formed initially do not separate, but instead undergo mutual charge annihilation. The radiation yield of "free" ions, which .can initiate polymerization depends on the dielectric constant (e) of the medium, higher e promoting the yield of free ions. Thus, in hydrocarbons which have low dielectric constants (e-values 2 to 4), the G-value for "free" ions is only 0.1, whereas alcohols which have e-values in the range 20 to 40 show free-ion G-values of 0.6 to 1.5 and water with e = 78 has a free-ion G-value of about 2.5. Moreover, ionic chain polymerizations are especially sensitive to traces of impurities, and these impurities can exert a strong inhibiting effect.

Problem 6.13 It is observed that styrene that has been dried by distillation from sodium-potassium alloy polymerizes under irradiation about 200 times faster than styrene that has only been subjected to a single conventional distillation. Give a qualitative explanation for this difference. Answer: Ionic chain polymerizations are especially sensitive to traces of moisture and other impurities which can terminate ions. Water and other similar compoundsterminate ions by transferring a proton o~: negative fragment (see Chapter 8) For example, the cationic polymerization of styrene, the cationic chains are effectively terminated by proton transfer to water molecule:

"{" H20 "---’~ -I~Jn i~ CH2R ~-rCH?-CH Dr" IC+

R-~CH2 ~

O+ -~nCH =~

+H3

Styrene monomer,as normally purified by conventional distillation, has sufficient water present (ca 10-2-10-4 M) to prevent ionic polymerization and consequently only radical polymerization occurs. In an extremely dried sample of styrene (as obtained by distillation from sodium-potassium alloy) the polymerization, however, proceeds entirely by cationic propagation. Ionic polymerizations are generally much faster than free-radical polymerizations.

Radical Chain Polymerization

475

By analogy with free-radical polymerizations, Eqs. (6.93) and (6.94) can be used also to calculate the radiation-induced rate of formaIion of initiating positive and negative ions by replacing R" by R+ or R-. Pure Thermal Initiation Many monomers appear to undergo a spontaneous polymerization when heated without the addition of an initiator. In most cases, such polymerizations are due to initiation by the thermal or photolytic homolysis of impurities (including peroxides or hydroperoxides formed due to 02) present in the monomer. Thus, it is observed that most monomers, when exhaustively purified (and contained in exhaustively purified vessels) do not undergo a purely thermal, self-initiated polymerization in the dark. However, some monomershave been unequivocally shown to undergo self-initiated polymerization. Styrene and methyl methacrylate belong to this type of monomer,The rates of thermal, self-initiated polymerizations are much slower than the corresponding polymerizations initiated by the thermal homolysis of an initiator such as AIBN,but are far from negligible. For example, the self-initiated polymerization rates for bulk styrene at 60°C and bulk methyl methacrylate at 70°C are observed to be 1.98x10-~ mol/L-s [28] and 0.21x 10-6 mol/L-s [29]. The extent of self-initiated polymerization must therefore be taken into account in any polymerization study if it is conducted at temperatures at which self-initiation constitutes a significant part of the total initiation process. At very low initiator concentrations, thermal initiation makes an appreciable contribution to the polymerization rate for styrene. The self-initiation mechanismfor styrene polymerization has been established [30]. It involves the formation of a Diels-Alder dimer (V) of styrene followed by transfer of a hydrogen atom from the dimer to a styrene molecule : 2CH2=CH (6.95)

(v) CH3-~:H (6.96) Ph

Ph

476

Chapter 6

The Diels-Alder dimer has not been isolated confirmed by ultraviolet spectroscopy.

but its existence has been

Problem6.14 There is evidence [30] that thermal self-initiated polymerization of styrene maybe of about five-halves order. Showthat this is in agreementwith the established initiation mechanism involving a Diels-Alderdimer formation [Eqs. (6.95) and (6.96)]. Answer: The higher than second-order rate observed for thermal conversion of monomer indicates that Eq. (6.96) is the slow step. Representing the concentration Diels-Alder dimer (V) by [Ell and that of styrene by [M], R, l= k,[DI[M = ki K whereK is the equilibriumconstant for the reaction of Diels-Alderdimerformation [Eq. (6.95)] and ki is the rate constant for the initiation reaction [Eq. (6.96)]. Using steady-state approximation (gi k,[M’l 2 3= kiK[M] [M’] =

a/2 [MI

Substituting this into Eq. (6.20) gives

showingfive-halves order in monomer concentration.

The self-initiation mechanismfor methyl methacrylate appears to involve the formation of a biradical by reaction of two monomermolecules followed by hydrogen transfer from some species in the reaction system to convert the biradical to a monoradical [32]: CH.~ CH3 ~ ~ "C-CH2-CH2-C’I I H-C’CH2"CH2-CI"l (6.97) 2CH2~---.C. COOCH3 COOCH3 COOCH3 COOCH3 COOCH3 CH3

CHz

CH3

The possibility of the diradical itself undergoing propagation to form large chains is very small because a diradical must almost inevitably cyclize at someearly stage in its growth.

Rad/caJ Cha/n Polymerization

DEAD-END

477

POLYMERIZATION

Under normal situations where sufficient initiator concentration is present, the free-radical polymerization of a given monomer solution at a given temperature proceeds until equilibrium is reached at which point the monomer concentration decreases to the equilibrium or critical value [M]e (see later) corresponding to the temperature of polymerization. This represents the maximum possible conversion (Pc) of monomer to polymer at the given temperature and it may be attained if only the initiator molecules do not get depleted during the process to below a minimumrequired to sustain the polymerization. However, in the event of initiato~ concentration used being low and insutficient, leading to a large depletion or complete consumption of the initiator before maximum conversion of monomer to polymer is accomplished, it is quite likely to observe a limiting conversion poo which is less than Pc. This is known as the dead-end effect in radical polymerization. The effect is shown schematically in Fig. 6.2. However, if there is autoacceleration effect or gel effect (see later) leading to a sharp rise in rate of polymerization, viscosity of medium, and degree of polymerization, pure dead-end effect cannot be observed.

1.0

-

(Injection of freshinitiator)

....

o 0.2

o o

0 Time Figure6.2 Schematic representation of dead-end effect in radical polymerization showing a limiting conversion of monomerto polymer (due to initiator concentration used being low and insufficient). Injection of the initiator in adequate amounts in the system at time t] immediately causes formation of more polymers up to a maximumconversion corresponding to the equilibrium monomerconcentration

478

Chapter6

Consider a dead-end polymerization initiated of an initiator. Recalling Eq. (6.33)

\r~t

by the thermal homolysis

I t"~--d) (1 -- e. -kdt/2)

(6.33)

where p is the fractional extent of monomerconversion at time ~ and [I]0 is the initial concentration of the initiator, let p~ be the limiting conversion attained at long reaction times (~ ---~ oc). Equation (5.33) then becomes -ln(1 - p~) = 2kl~lf[I]°l

’l~f

\

(6.98)

Dividing Eq. (6.33) by Eq. (6.98), rearranging, and then taking logarithms of both sides leads to the useful expression In

1 1~(1 )

(6.99)

=-2

A plot of the left side of Eq. (6.99), which is equivalent to the expression in [(ln[M]~ -- In[M]) /ln[M]~ - In[M],)], versus time permits evaluation of kd (see Problem 6.15). Since kd is determined, f can obtained from either Eq. (6.26) or (6.98) if the ratio kp/k~/2" is known from other studies. The thermal dissociation rate constants and activation energy values for several commonlyused initiators are listed in Table 6.6.

Table 6.6 Thermal Decomposition of Initiators Initiator Acetyl peroxide 2,2’-Azobisisobutyronitrile Benzoyl peroxide t-Butyl hydroperoxide t-Butyl peroxide Cumylperoxide

5kd x 10 2.39 0.845 5.50 0.429 3.00 1,56

Theunits of kd are s-i; the units of Edare Ll/mol. Source:Data from Ref. 3.

T°C 70 60 85 155 130 115

Ed 136.0 123.4 124.3 170.7 146.9 170.3

479

Radical Cha/n Polymerization

Problem6.15 Isoprene was polymerized in bulk at several temperatures using AIBNat an initial concentration of 0.0488 mol/L in dead-end polymerization experiments [33]. In every case, the conversion increased with time until a limiting value was obtained beyond which no further polymerization was observed. No autoacceleration effect was observed in this system. The data of fractional degree of conversion (p) with time, including the limiting value of conversion (p~), determined at each temperature are shown in table below: 60°C 70°C Time (h) .p Time (h) 0.054 4 8 16 0.086 6 30 0.167 12 0.220 24 48 72 0.280 30 96 0.315 48 144 0.360 72 240 0.390 96 300 0.390 150 oz 0.390 cxz Source: Data from Ref. 33

p 0.055 0.102 0.180 0.235 0.273 0.310 0.310 0.325 0.325 0.325

80°C Time (h) 2 4 6 8 16 24 48 o~

0.100 0.172 0.202 0.240 0.290 0.305 0.305 0.305

Determine the kinetic parameters kd and (kp / kit/z) fi/2 for isoprene-AIBN system and the activation energy of the initiator dissociation. Answer: The left side of Eq. (6.99) is plotted against time t in Fig. 6_3 using the conversion-time data of the isoprene-AIBN system. The slope of the linear plot yields kd/2, and hence kd, of AIBNin isoprene. (k~, / k~t/~) fl/2 is then calculated from Eq. (6.98). This yields Temperature (°C) 60 70 80

-8. 8.54 x 10 -5 3.08 × 10 -5 9.84x 10

3.27 4.94 8.17

It is possible to calculate f from the above values of (k~/k~tl2)f 112 if k~,/k~ I~ is known from other studies. An Arrhenius plot of the kd values as -Inkd vs. 1/7" yields from the slope, Ed = 29.3 kcal mo1-1.

480

Chapter 6

60 °C

0

I

40

80

I

120

I

160

Time(h) Figure 6.3 Test of Eq. (6.99) for dead-endpolymerizationof isoprene-AIBN systemat different temperatures(Problem6.15). Mathematical treatments have been. developed to extend the deadend polymerization technique to other situations such as polymerizations which are not dead-end, i.e., polymerizations go to completion and also polymerizations involving appreciable induced decompositionof initiator [34,35]. DETERMINATION OF ABSOLUTE RATE CONSTANTS Threedifferent types of rate constantsare of concernin ideal polymerization kinetics describedby Eq. (6.26)--those for initiation (kd), propagation(kp), and termination (k,t). The use of polymerization data under steady-state conditions allows the evaluation of only kd (see Dead,endpolymerization). The ratio kp/k~/2 or k2p/kt can be obtained from Eq. (6.25) since /gp,/{/, and [M] are measurable. However,steady-state data do not allow

Radical ChainPolymerization

481

the evaluation of the individual kv and k~ values. It is necessary to employ non-steady-state conditions to determine these individual rate constants. Non-Steady-State

Kinetics

One of the best means of obtaining individual rate constants using nonsteady-state conditions is photochemical polymerization. The treatment discussed here is essentially that of Flory [31] and Walling [36]. Wehave seen previously that the assumption of steady-state enables one to equate Pv/ to Rt to obtain Eq. (6.24) for radical concentration and hence Eq. (6.25) for the rate of polymerization under steady-state conditions. However,at the very outset of the polymerization, the radical concentration is zero, and a finite time must be required before it reaches the steady-state level. During the non-steady interval, the rate of change of radical concentration is given by the difference of their rates of production and termination -- Ri -- Rt 2= P,4 - 2kt[M’] (6.100) dt In photoinitiated polymerization it is possible to commencethe generation of radicals abruptly by exposure of the polymerization cell to the light source, and the time required for temperature equilibration in an otherwise initiated polymerization can be avoided. The rate of photoinitiation is given by Eq. (6.77) and Eq. (6.100) then becomes d[M’]/dt Since for steady-state

= 2~5Ia

--

2kt[M’] 2

(6.101)

conditions, d [M’]/dt = 0, Eq. (6.101)leads

2e~ra = 9.~ [M.]~

(6.102)

where the subscript s denotes steady-state condition. CombiningEqs. (6.101) and (6.102) one obtains d[M’] dt This equation gives the rate of change of the radical concentration under a non-steady-state condition as the difference in the rates of termination under steady-state and non-steady-state conditions. It is convenientat this point to define a parameter~’s, called the average lifetime of a growingradical under steady-state conditions, as

2) - 2k, ([M’I~ -[~’l

(6.103)

No. of radicals present at steady state No. of radicals disappearingper unit time at steady state

[M-I, - 2k,[M.]~ -- 9.a[M’ls

(6.10~)

482

~hapte~6

Combination of Eqs. (6.105) with Eq. (6.20) "~s -As [M] and (Rp)s are measurable,

at steady-state

yields

kp [M]

by determining

(6.105) -rs one can obtain

from Eq. (6.105), and since kp/k t or kp/kt can be obtained ~om steadystate measurements, both ~p and kt can be evaluat¢d individu~ly. It is the objective of non-steady-state experiments to determine Ts for this pu~ose. Integration of Eq. (6.103) yields

,1

-+

where to is the integration

: constant

such that [M’] = 0 at t =

S~neet~h-’~ = (~/~)t.[(~+ ~) / (~ - ~)], ~om~,~n~ (6.104)

and (6.106)

one may write -1 tanh

(~.107)

Therefore, ~ _ [M’]

_ tanh

(~)~

[M-]~[

The ~ in these equations nation.

may be identified

[.(t

-

(6.~08)

t0)

with the duration

of the i~umi-

Problem6.16 Typical % values determined from photoinitiated

radical chain polymerization with intermittent illumination are in the range 0.1-10 s. Calculate from this the duration of the non-steady-state period and commenton the validity of steady-state approximation made in a typical polymerization study. Answer: [M’]o = 0 at t = 0. Therefore, From Eq. (6.108), [M’] = tanh(t)

to = 0.

For a steady-state condition, assuming, for example, 99.9999%attainment of the steady-state radical concentration, t/~’s = 7. Thus the time required for [M.] to reach its steady-state value is 70, 7, and 0.7 s, respectively, for ~-s values of 10, 1, and 0.1 s. Thus, in the typical polymerization stu.dy the steady-state assumption is valid after a couple of minutes at most.

Now suppose the polymerization

is being conducted

with intermittent

483

Radical Chain Polymerization

illumination, that is, with alternate light and dark periods. At the very beginning the radical concentration [M’] is zero and it builds up in the period of illumination till it reaches a steady-state value [M’]s as shownin Fig. 6.4(a). If the light source is switched on and off and held for long but equal time periods of light (t) and darkness (t~), the radical concentration will alternately build up from zero to [M’]s, the steady-state value, and decay from [M’]s to zero with the repetition of the sequence t and t ~ of illumination and darkness, respectively [Fig. 6.4(a)]. The intensity of illumination is I0 during t and zero during t ~ with short zones of transition in between. The radical concentration during illumination is essentially consis-

O

-= [~]s

t

t~/"

t

>/ .t

t

(a)

Figure 6.4 Schematicrepresentation of variation of chain radical concentration I-M-] over (a) cycles of long illumination period (t) and dark period ~) and (b) cycles of short (intermittent) illumination period (t) and dark period tent with I0, but radicals are present only half of the time in itermittent illumination and hence the average rate of polymerization (~v) observed corresponds to one-half of the rate (P~)s, to be observed for the same intensity on the basis of continuous illumination. Thus for slow blinking, P"v / (P~p)* = 1/2.

484

CJhapter6

Whenthe frequency of blinking is high giving very short but equal light and dark periods [Fig. 6.4(b)], then, even if one starts with [M"] = 0 at the beginning of the first light period, the radical build up [curve OAEin Fig. 6.4(b)] is interrupted by the interception of light before it reaches [M’]s (at point E). [M’] reaches a maximumvalue, say, [M’]I < [M’]s the end of the illumination period (t). The decay of radical concentration from the [M’]I value begins immediately thereafter as the dark period commences.Since the dark period ({) is short, radical concentration does not decay to zero, but drops to a minimumvalue, say, [M’]2 > 0, when it starts rising as. the illumination is on again. With frequent blinking, the concentration of radicals alternates between [M’]I and [M’]2 as the system passes through the end of a light period to the end of a dark period in successive cycles of illumination and darkness [Fig. 6.409)]. With progressive increase in the frequency of blinking the difference [M’]x and [M’]2 will be progressively small and in the limiting case of very fast blinking the radical concentration would reach a constant or plateau value [M’]e below the steady value [M’].,. This constant value [M’]dwould effectively be that which correspgnds to a continuous illumination of intensity Io/2, as only 50%of the irradiation is received by the system. It then follows from Eqs. (6.80) and (6.24) that [M’]c will be proportional to (L/2)1/~. Since the rate of polymerization is proportional to the concentration of radicals [cf. Eq. (6.20)], the average rate of polymerization (~p) for very fast blinking will therefore be proportional to (/0/2) 1/2. In comparison, for steady conditions under continuous illumination (i.e., no blinking), (P~o)s (x 1/2. According to analysis and consideration detailed above for photoinitiated polymerization, it may now be concluded that R.p or [M’] can be varied by varying the frequency of blinking. Under otherwise comparable conditions, the average rate (P’-v) at different flashing conditions with the same L related to the steady-state (P~o)s -and

(P~p)

--

1 2

(for slow blinking) (for very fast blinking)

(6.109) (6.110)

For unequal light and dark periods such that t r = rt, i.e., the dark period is ~" times longer than the period of illumination in all successive cycles (r = 1 for the systems considered above), the general expressions for the relative rates may be written as:

(_P~)~

--

1 (r

+

(for slow blinking)

(6.111)

RadicalChainPolymerization

485 1

- (,. + 1)v,

(for very fast blinking)

(6.112)

Considering, as an example, a case where the ratio r of the length t r of the dark period to the length t of a light period is 3, the relative rate equals 1/4__for slowblinking and 1/2 for very fast blinking. Thusthe averagerate, (Pqo) increases from1/4 to 1/2 of the steady-state rate as the cycle time (time for one light period and one dark period) decreases from a muchhigher value to a muchlower value in comparisonwith the average lifetime of a growingchain (~’s), or as the frequencyof blinking 1/(t -F rt) increases from a muchlower value to a muchhigher value in comparisonwith 1/~s. The mathematicaltreatment of intermittent illumination has been described [37]. As explained above, in relation to Fig. 6.4(b), the radical concentration, after a numberof cycles, oscillate uniformlywith a constant radical concentration [M’]I at the end of each light period of duration and a constant radical concentration [M’]2 at the end of each dark period of duration f~ = ft. Considering first light period, if [M’] = 0 at t = 0, then from Eq. (6.107), t0 = 0; but for [M’] = [M’]2 > 0 at the beginning of light period (t = 0), Eq. (6.107) yields -- to/rs

= tanh -1 ([M’]2/[M’]s)

(6.113)

and hence tanh -1 ([M’]/[M’]s) -- tanh -~ ([M’]2 /[M’]s) = t / ~-s (6.114) At the end of light period t, [M’] = [M’]I [see Fig. 6.4(b)] and Eq. (6.114) becomes ta,nh_ 1 ([M’]I~

_ t&nh-’ ([M’]2~ _ __~

\[M.],J

(6.115)

On the other hand, during the dark period radical decay occurs according to d[M’]/dr’ = - 2kt 2[M’] (6.116) which on integration,

with [M’] = [M’]I at t’ = 0, yields 1 1 -- 2kit’ (6.117) [M’j [M’]I Multiplying Eq. (6.117) through by [M’]s and combiningwith Eq. (6.103), r[M’]s

[M’]s

-t

(6.118)

486

C,]aapter 6

Since t ~ -- z’t and [M’]. = [M’]2 at the end of dark period t t, one obtains fromEq.(6.118), [M’]s

[M’ls

--

(6.119)

Equations (6.115) and (6.119) permit evaluation of the maximum minimumradical concentration ratios [M’]I /[M’]s and [M’]2 /[M’]s forgiven values of ~ (= e / t) and t / rs. The average radial concentration [M’] over a cycle of light and dark periods is given by

Here [M" ] in the first integral covering the peNodof illumination is ~ven by (6.114) and that in the second integal coveNngthe dark period is given by Eq. (6.118). Evaluation of the integrals yields the following expression

[ al:

(6.121) Thus, for a given ratio (r) of dark period to light period, values [M’]/[M’]s or (~) / (Pqo)s may be calculated for different assumed values of t/"r s. A semilog plot of relevant data for r = 3 is shown in Fig. 6.5. The plot shows that the average radical concentration falls from one-half of the steady-state value for fast blinking (low t / ~-s) to one-fourth of the same for slow blinking (large t/~’s), in full conformity with Eqs. (6.111) and (6.112). In order to experimentally determine the "rs value for a particular polymerization system, one interposes a rotating sector or disc in between the system and the source of light. The sector has a portion cut out, which determines the value of r. The steady-state polymerization rate (/~p).__s first measured without the sector present. Then the average rate (P~) measured with the sector present at different (increasing) speeds of sector rotation. The blinking frequency as well as/; and t’ are determined by the speed of sector rotation. The polymerization is conveniently followed by dilatometric technique (p. 329). Thus a numberof rate ratios (P~p) / (P~)s are obtained for a given sector ratio (r) by varying the sector speed and hence t. The rate ratios are then olotted against log t. Alternatively, one can also plot the data as (~) / (R~)o~ since this ratio is related

487

Radical Chain Polymerization

0.55

0.45

0.35-

0.25 0.01

I 0.1

I

I

I

1.0

10

100

1000

t/’r Figure 6.5 Semilog plot of [M’]/[M’]s versus t/% for r = 3. (Adapted from Ref. 39.)

Pt:o / (/~)s through Eq. (6.112). The theoretical curve (e.g., Fig. 6.5) the same r value is placed on top of the experimental curve and shifted on the abscissa until a best fit is obtained. The displacement of one curve relative to the other along the abscissa yields log’rs since the abscissa for the theoretical curve represents (log t - log’rs). (Several experimental variations of the rotating sector method have been used [40,41]. One of these uses a glass tubular reactor surrounded by a metal cylinder having narrow regularly spaced slots through which light shines. Monomeris pumped through the reactor and passes alternately through light and dark regions. The mathematical-treatment [41] is very similar to that described above for the rotating sector method.) Evaluation of % then allows calculation of absolute rate constants kp and ~, making use of Eq. (6.105) relating Ts with k v / kt and of equations and approaches discussed earlier leading to evaluation of the parameter

488

Chapter 6

kp2 / k.t. Table 6.7 lists the k/~ and k,t values and the corresponding activation energies for some common monomers.

CHAIN

LENGTH

AND

DEGREE

OF

POLYMERIZATION It is relevant at this point to examine if the chain length or degree of polymerization (~---~n) of a polymer product could be predicted or calculated from measured rate of polymerization with a knowledge of kinetic constants. Since the number-average degree of polymerization is given by the average number of monomer molecules consumed per polymer molecule, it becomes necessary first to define and obtain an expression for the kinetic chain length. Kinetic

Chain

Length

The kinetic chain length (v) of a radical chain polymerization is defined as the average number of monomer molecules consumed (i.e., polymerized) per each radical which initiates a polymer chain. This quantity will obviously be given by the ratio of the rate of propagation (Rp) to the rate

Table 6.7 Kinetic Parameters in Radical Chain Polymerization Monomer

k~,x10

-3

Ep

AexlO -z

ktxl0

-w

-9 Et

A~x10

Methyl acrylate

2.09

29.7

10

0.95

22.2

Methyl methacrylate

15

0.515

26.4

0.087

2.55

11.9

0.11

Styrene

0.165

26

0.45

6.0

8.0

0.058

Acrylonitrile

1.96

16.2

-

7.8

15.5

-

Vinyl chloride (50°C)

11.0

16

0.33

210

17.6

600

Vinyl acetate

2.30

18

3.2

2.9

21.9

3.7

.... 54.0

1.3

-

-

-

Tetrafluoroethylene (83°C)

9.10

17.4

Ethylene

0.242

18.4

-

1,3-Butadiene

0.100

24.3

12

-

kp and k,t values are for 60°C unless otherwise indicated and have the units of L mo1-1 s -1. Theunits of Ep are kJ/moi of polymerizingmonomer and those of Et are kJ/moi of propagatingradicals. Sourde: Data mostly from Ref. 3.

489

/Lad/ca/Cha/n Polymerization

initiation (P,,/) or to the rate of termination (Rt), sinc~ Pq = Rt under the steady-state conditions. Thus,

v=

= P /Rt

(6.122)

Combiningwith Eqs. (5.15) and (6.18) we

=

kp [M]

(6.123)

Eliminating the radical concentration term [M" ] with ~e help of ~q. (6.15),

-

2~

~

The ~netic chain length is thus inversely dependent on the radial concentration [Eq. (6.123)] or the polymerization rate [Eq. (6,124)]. This is great practi~l signifi~nce .~ it shows that any attempt to incr¢~e the rate of polyme~zation by increasing the radial concentration will be only at the expense of produdng smaller size polymer molecules. Equations (6.123) and (5.i24) are applicable for all ~ses of bimolecular te~ination i~espective of th~ exact mechanism (combination or dispropo~ionation) ~d also irrespectiw of the nature of the initiation process. Thus, for any monomer the ~netic chain length will be independent of whether the polyme~tion is initiated by thermal, redox, or photochemical means, or of the initiator used, if the [M" ] or ~ is the same. For polymerization initiated through radicals, generated by the~al decomposition of initiator, Eq. (6.25) may be combined with Eq. (6.124) to give an alternative expression for ~:

u = ~(fke~)~/~

[I]~/~

Thus a four-fold increase in initiator concentration wouldresult in halving the size of pol~er molecules, though the rate of polymerization [Eq. (6.26)] would be doubled by this change in ~itiator concentration. Problem6.17 In a benzoyl peroxide initiated polymerization of styrene it is desired to double the initial steady rate of polymerization without changingthe initial number-averagedegree of polymerization. Howcould this be achieved by changing only the monomer and initiator concentrations ? Answer: FromEq. (6.26), [M]2[I]~/2---- 2 [M]I[Illl/2 [Mb_/ [M],= 2 [I]ff 2 / [I]~/2

(r6.17.1)

490

Chapter 6

From Eq. (6.125), since there is no change in [M]2/ [I]~/2 = [M]I / [I]11/2 or

[M]2/ [M], ---- [I]~/2 / [I]]/2

From Eqs. (P6.17.1) and (P6.17.2),

[I]2/[I]1

2:1.

Therefore, [M]2/ [M]I= : 1 Mode

of

Termination

The number-average degree of polymerization number of monomer molecules contained in a to the kinetic chain length. If the propagating or combination [Eq. (6.7)], a dead polymer kinetic chain lengths, that is, D---~,~

DPn, defined as the average polymer molecule, is related radicals terminate by coupling molecule is composed of two

= 2v

However, if the termination occurs by disproportionation dead polymer molecule is composed of only one kinetic D--’ff,~

= v

(6.126) [Eq. (5.8)], chain and so,

the

(6.127)

The mode of termination is experimentally determined from the observation of the number of initiator fragments per molecule. This requires the analysis of the molecular weight or the DPn of a polymer sample as well as the total number of initiator fragments contained in the sample.

Problem6.18 For a radical chain polymerization with bimolecular termination, the polymer produced contains on the average 1.60 initiator fragments per polymer molecule. Calculate the relative extents of termination by disproportionation and by coupling, assuming that no chain transfer reactions occur. Derive first a general relation for this calculation. Answer: Let n = number of propagatingchains etc = fraction of propagating chainswhichundergotermination by coupling 1 -- etc = fraction of propagating chainswhichundergotermination by dispropo~ionation b = average number of initiator fragments per polymer molecule formed = et--2-~ 2 + n(1 Total number of initiator fragmentsTotal number of polymer molecules 2 n

Number.of polymer molecules b =

nero~2 + n(1 - e,c)

2 -

(P6.18.1)

491

RadicalChainPolymerization Fromthis, etc = (2b - 2)/b

(1-,,o)

(P6.18.2)

= (2- b)/b

(P6.18.3)

For the givenproblem, Fraction of coupling= (2× 1.50 - 2)/1.50 = 0.75 Fraction of disproportionation= (2 - 1.5)/1.5 = 0.25

If b is the average numberof initiator fragmentsper polymermolecule and etc is the fraction of propagatingchains whichundergotermination by coupling, the two are related [cf. Eq. (P6.18.1)] b = 2/(2

- ere)

(6.128)

The number-averagedegree of polymerization DP,~will then be related to the kinetic chain length u by DPn --

(2)

2 - etc

u

(6.129)

CombiningEqs. (6.124) and (6.129) one obtains DPn =

(2 - tc)

(6.130)

Althoughexperimental data are not available for all monomers,for most monomersthe termination of propagating chains appears to occur predominantlyor entirely by coupling. However,varying extents of disproportionation are observed depending on the monomerand the reaction conditions employed.For example, disproportionation increases whenthe propagating chain radical is sterieally hindered or has morefl-hydrogens available for transfer (see later). Thus, whereasstyrene, methylacrylate, and acrylonitrile undergoterminationalmostexclusively by coupling, methyl methacrylate undergoes termination by both coupling and disproportionation. Theextent of disproportionationincreases as the temperatureis raised, the effect being mostsignificant for sterieally hinderedradicals, In methyl methacrylate, for example,the extent of disproportionation increases from 67%at 25°C to 80%at 80°C[42]. Problem6.19 Usingcarbon-14labeled AIBN as an initiator a sampleof methyl methacrylateis polymerizedat 80°Cto an averagedegree of polymerizationof 1.5 x 103. TheAIBN has an activity (per tool) of 9.1×t counts per minute ina scintillation counter,ff 1.0 g of the poly(methyl methaerylate)showsan activity of 337 counts per minute, determineby appropriate calculation the modeof

492

Chapter 6

termination in methylmethacrylate at 80°C. Answer: Mer weight of methyl methacrylate = 100 Molesof mers in 1.0 g poly(methyl methacrylate) = -2 -1 1 mol of AIBN--= 9.1×107 counts min Therefore 337 counts/min ---- 337/(9.1 × 107) or 3.70 × -6 to ol AI BN --= 7.4 × 10-6 mol chain radicals, since I tool of AIBN gives rise to 2 molesof chain radicals (assuming100%initiator efficiency). Let x fraction of chain radicals terminate by coupling and (1 - x) fraction by disproportionation. Therefore 1 moi of chain radicals give rise to 0.5x + (1 z) or (1 - 0.5z) tool polymer molecules. Hence 10-2 tool D--~ = = 1.5x103 .6 (7.4 × 10 moi) (1 -- 0.5x) x = 0.2 So 20%of the chain radicals terminate by coupling and 80%by disproponionation.

Average

Lifetime

of Kinetic

Chains

The average lifetime (’r) of the kinetic chain is given by the ratio of the steady radical concentration to the steady-state rate of radical disappearance : T --

[M’]_ 1 2kt [M’] 2 2~ [M’] Substituting for [M’] from Eq. (6.123) yields

"r -

(6.131)

(6.132)

For termination by disproportionation [cf. Eq. (6.127)], "r = D---~n / kp [M]

(6.133)

and for termination by coupling or combination,

r = / 9.kp [M]

(6.134)

Problem6.20 Consider the polymerization of styrene in bulk at 60°C initiated by 1 x 10-3 Mbenzoylperoxide. The density of liquid styrene is 0.909 g/cm3 at the reaction temperature. Whatis the average radical lifetime and what is the steadystate radical concentration ? [Data at 60°C: kn (styrene) = 6.0×107L/mol-s; (benzoyl peroxide) = 7.1x10-6 s-x, f = 0.5] Answer: CombiningEq. (6.132) with Eq. (6.125),

RadicMChain Polymerization

493

1/. = 2(ykdkt = 2 [(0.5)(7.1 x -6 s- 1)(6.0 x r L tool-1 s-l) (1 x 10 -3tool L-l)] 1/2 -~ = 0.923 s ~- = 1.1s FromEqs. (6.24) and (5,14), 1/2

[M-]=

r.(o-~)(7.1x lo-~~-’)(1~-~~o l = 7.7x10 -~ -1 tool L

CHAIN

TRANSFER

In manypolymerization systems, the polymer molecular weight is observed to be lower than predicted on the basis of Eqs. (6.126) and (6.127). effect is due to the premature termination of a growing polymer chain by transfer of its radical center to other species, present in the reaction mixture. These are termed chain transfer reactions and may be depicted as Mn" + XA ~ MnX + A" (6.135) where XA may be monomer, initiator, solvent, polymer, or any other substance present in the reaction mixture, and X is the atom or species transferred; ~tr is the chain transfer rate constant. The new radical A’, which results from chain transfer, can reinitiate polymerization by the reaction A" q- M "~ M" (6.136) where kr is the rate constant for addition of monomerto A" leading to chain reinitiation by the process

where kp is the normal prbpagation rate constant. The chain transfer to monomeris negligible for most monomers, but may be significant for some monomers, for example, vinyl acetate, vinyl chloride, and o~-methyl substituted vinyl monomers, e.g., propylene and methyl methacr]late : CH3

I

+

COOCH3

CH3 I ktr,M

CH3

COOCHs

COOCH3

---+

I

.CH2

+

I

7

COOCH~

494

Chapter 6

/gtr, Mis the rate constant for chain transfer to monomer.In general, the rate of transfer to monomeris given by

where

Rtr,M = ktr, M [M] [M’] (6.139) Manyperoxides used as initiators have significant chain ~ransfer reactions. Dialkyl and diacyl peroxides undergo chain transfer due to breakage of the O--O bond, e.g. H

0

0

wwvCH2-C. + R-C-O-O-C-R X

H

0

-~ vwvvCH~-C-O-C-R X

0 + R-C-O"

(6.140)

where ~r,I is the rate constant for chain transfer to initiator. The hydroperoxides are usually the strongest transfer agents amongthe initiators. In general, the rate of chain transfer to initiator is given by /~tr,I = k, tr,I[I] [M’] (6.141) Chain transfer to initiator was earlier referred to as induced initiator decomposition. In some laboratory polymerizations, the solvent itself acts as the chain transfer agent. For example, the chain transfer reaction for vinyl polymerization in CC14can be represented by H

H

vvvwCH2-C" + CCI4 ----> wvwCH2-C.CI + "CCI 3 I X X

(6.142)

wherektr,Sis therateconstant forchaintransfer to solvent. Solvents are normally, notusedin industrial free-radical polymerizations, foreconomic reasons, andthechaintransfer agentsin thesereactions areingredients thatareaddeddeliberately to limitthemolecular weight ofthepolymer. In general, therateof chaintransfer to solvents andaddedchaintransfer agents is givenby atr,s= ~r,sIS] [M’]

(6.143)

Sincechaintransfer stopsgrowing chain, it always results ina lowermolecularweightthanwouldoccurin itsabsence. Theeffectof chaintransfer on therateof polymerization varies, however, anddepends on therelative ratesof thetransfer [Eq.(6.135)] andreinitiation [Eq.(6.136)] compared to thatof thenormalpropagation reaction [Eq.(6.6d)]. Several possible situations thatmaybe encountered aresummarized in Table6.8.Theseare allinstances ofchaintransfer, buttheyareusually givendifferent names, as shown,depending on theneteffects on polymerization, rateandmolecular weight.

Radical Chain Polymerization

495

Table 6.8 Effect of Chain Transfer on Polymerization Rate and PolymerMolecular Weight Process name Normal chain transfer Telomerization Retardation Degradative chain transfer Inhibition

Characteristics

kp >>ktr, kr --- kp kp < ktr

Effect on polymeri- Effect on Polymer zation rate molecular weight None None Decrease

k~ << ktr, kr < k~ Large decrease kp << ktr, kr << kp No polymerization

Decrease Large decrease Decrease Large decrease -

If the new radical (A’) formed by the transfer reaction is reactive enough to add to a monomermolecule, and thus initiate a new chain, within about the same time period as that required for addition of a monomerended radical to a monomer molecule, then kr ~ kp and the rate of polymerization /~p is not altered. If, in addition, kp >>ktr, the polymer molecular weight is reduced but the product is still macromolecular. This is the case of normal chain transfer. If kp << ~r and kr "~ kv, there will be a large numberof transfer reactions compared to propagation reactions, and only a low-molecular-weight polymer will be made. This process, called telomerization, is illustrated by the radical reaction of ethylene and CC14which yields waxy products of the general structure CI(CH2CH2)nCC13with n _< 12. The chain transfer and reinitiation reactions in this case are vvwvCH2CH2" -I- CC14 k’~tr vwwCH2 -- CH2-- CI q- "CC13 ¯ CC13 -l-

CH2 = CH2 "-~ C13C - CH2 -- CH2"

(6.144) (6.145)

If the reactivity of the radical formed by chain transfer is lower than that of propagating chain radicals (kr < kv), the rate of polymerization will be reduced along with the’ polymer molecular weight. Such cases are examplesof retardation. If the reduction in polymerization rate is so severe as to make it effectively nil, it is a case of inhibition. Retardation and inhibition are described more elaborately in a later section. Degradative chain transfer results if chain transfer is predominant (ktr >> kv) and the new radical formed by this reaction reinitiates poorly (kr < kv), with the result that there is a large decrease both in polymerization rate and molecular weight.

496

G~pter 6

Problem 6.21 A vinyl monomerof molecular weight 132 is polymerized by a free-radical initiator in the presence of dodecyl mercaptan(CnH~sSH). The rate of polymerization is not depressed by the mercaptan. The purified polymerhas a sulfur content of 0.02%(w/w)and its DP,, is 450. If 80%of the kinetic chains are terminated by coupling and 20%by disproportionation, what should be the extent of terminal unsaturation of the chains ? A.swer : Molar mass of repeat unit = 132 g mo1-1, Polymerchains = [(450)(132 g mol-a)] Sulfur = (0.02 x 10-~)/(32 g g-atom

D--~,~= 450

In the presence of mercaptan,the following reactions take place [31]: M,~" RS-

+ RSH ~ M~-H + RS. +M ~ RSM" ~ etc. ~ RSM~"

Since the rate of polymerization is not reduced by the mercaptan, the transfer radical RS" evidently reacts readily with monomerto- start a new kinetic chain. Accordingly, it maybe assumedthat for one mol of mercaptan consumed one mol of polymerchains are formedand one mol of kinetic chains are initiated. Therefore 6.25 x 10-6 molesof polymerchains are producedby transfer to mercaptan. Remaining(1.68 x 10-5-6.25 x 10-~) or 1.055 x 10-5 mol of polymer chains result from termination by disproportionation and coupling. Out of every 100 kinetic chains, 80 terminate by coupling to produce 40 polymer molecules and 20 terminate by disproportionation to produce 20 polymer molecules of which10 contain terminal unsaturation [see Eq. (6.8)]. Therefore, terminal unsaturation = (1.055 x 10-5 mol) (10/60) = 1.76x 10-~ mol g-1 Hence (1.76×10-6)(100)/(1.68x10 -5) or 10.5% of polymer chains have a terminal unsaturation.

Degree of Polymerization Equations (6.129) and (6.130) for D---~n apply to free-radical polymerization following ideal kinetics in which termination of the growth of polymeric radicals is accounted for only by mutual reaction of two such radicals. Combining Eqs. (6.122) and (6.129) one may write DPn = 2 --

et:

~ --

znt

where z = (1 -- ere / 2). In Eq. (6.146), Re is the rate of termination chain radicals and zRt is the rate of production of dead polymer molecules by bimolecular termination mechanism, i.e., by combination [Eq. (6.9)]

Radical ChainPolymerization

497

and/or disproportionation [Eq. (6.10)]. (If combination is the sole mechanism, etc = 1 and z = 1/2, while for termination by disproporfionation etc = 0 and z = 1.) Since, as we have seen above, chain radicals can also be terminated by chain transfer reactions, Eq. (6.146) will nowbe amended to include transfer reactions. This can be easily done by redefining DPnas the ratio of rate of polymerization to the rate of formation of dead polymer molecules by all reactions, namely, by the normal bimolecular termination and various transfer reactions. Thus Eq. (6.146) takes the form

zRt + Rtr,M + Rtr,I

+ Rtr, S

zP~ + ktr,M [M’] [M] + ~r,I [M’] [I] + k, tr, S [M’] [S]

(6.147) since at steady state, Rt = R/. Noting from Eq. (6.15) that /~p k v[M] [M’], conveniently rearranged to the form

Eq. (6.147)

1

z_R/ [S] (6.148) ~DPn P~ CM -[- el -[[~] Cs [M---] where CM, CI, and Cs are the chain transfer constants for monomer, initiator, and solvent/chain transfer agent, respectively, defined as the ratio of ~tr for the respective material with a propagating radical to kv for that radical, that is, CM --

]Qr,M

CI - ktr,i

ktr,s

(6.149)

Equation (6.148), often referred to as the Mayoequation, shows the quantitative effect of various transfer reactions on the numberaverage degree of polymerization. Note that the chain transfer constants, being ratios of the respective rate constants for chain transfer (Rtr) to the rate constant for propagation (kp), are dimensionless quantities dependent on the types of both the monomerand the material causing chain transfer as well as on the temperature of reaction. Problem 6.22 Vinyl acetate has a relatively high monomerchain transfer constant (2x10-4 at 60°C). What is the upper limit of molecular weight of poly(vinyl acetate) madeby radical polymerization at 60°C? Answer: The upper limit of molecularweightcorrespondsto the lower limit of (l~--~n)

498

Chapter6

Eq. (6.148). The lower limit of the first, of Eq. (6.148) is 0. Hence, 1 -4 D-’-~---~lmin = CM= 2x10 DP,, = 5,000 M---, = (5,000)(86

g mol-’)

third and fourth terms on the right side

= 4,30,000

g mol-’

Problem6.23 For a solution (density 0.87 g/cma) of 1.0 M styrene and 0.01 Mtea-butyl peroxide in toluene, the initial rates of initiation and polymerization at 60°C are 4.0x10 -11 mol/L-s and 1.5×10-z tool/L-s, respectively. Calculate (a) how often on the average chain transfer occurs per each initiating radical from the peroxide and (b) the molecular weight (assuming coupling termination) of polystyrene produced. Use the following chain transfer constants: C’M = 6.0×10 -5, CI = 8-4x10 -4, and C’s -5 = .1.25x10 Answer: Given -1. [I] =0.01 moll -1, [M] = 1.0 moll -u -1 (Pg) = 1.5×10-z mol L-’ s-’. (/~i) = 4.0×10 mol L -~, s (a) Total rate of chain transfer (l~tr)tot = -~,-,M + i~’,S + P~t,-,I = ktr,M [M’] [M] + ktr,S [M’] [S] + /%,I [M’] [I] -1. Molar mass: styrene = 104 g mol-~; tert-butyl peroxide = 146 g moi [S] = (870 - 0.014x146 - 1.0x104)/92 = 8.31 mol -1 (/{t.)tot/~

= CM + Cs[SI/[M]

+ CI[I]/[M]

= (CM + CS[S] / [M]÷ 6’~ [I] / [M])(/%, (8.31 mol -1) = 6.0x10 -5 + 1.25x10 -5 (1.0 tool -1) + 8.4x

(0.01 mol L-x)] (1.5_x -7 ~ 10 -4 (1.0 tool -1) J \~× 1 -u]

= 0.646 (b) Substituting appropriate values in Eq. (6.148) and noting that z ---- 1/2, s. obtains D-~,~ = 3279 and so ~,~ = 3.41x 10

Problem6.24 Vinyl acetate is polymerized in benzene solution at 60°C using 2,2’-azobisisobutyronitrile(AIBN) as initiator and carbon tetrachloride as chain transfer agent. The initial monomerconcentration is 200 g/L and solution density is 0.83 g/cm3. Select concentrations of initiator and chain transfer agent that will give poly(vinyl acetate) with an initial molecular weight (assuming coupling) of 15,000 and 50%polymerization in 2 h. Neglect chain transfer to initiator in

499

Radical Chain Polymerization

calculations. [Data: ka = 8.45x10-6 s-l; k~ = 2.34x10a Idmol-s; ~ = 2.9x107 L/mol-s; f = 1.0; CM= 2.3x10-~; Cs(benzene)= 1-2x10-~; Cs(CCh)= 1.07 ] Answer: Molecular weights: vinyl acetate 86, AIBN164, benzene 78, CCI~154. [M]0 = (200 g L-1)/(86 g tool -1) -~ = 2.32 mol L Neglectinginitiator mass, [S]benzene= (830 g -- 200g)/(78 g mo1-1)= 8.08 moi -1. Assumingthat chain transfer has no effect on polymerizationrate, Eq. (6.33) can be used. Usingrelevant data and p = 0.5, this equation gives [I]0= 6.16x I0-3 tool-1 L -3x -I ----6.16x10 164 or 1.0 g L FromEqs.(6.125) and(6.126), (~-~,~)o

= k~ [M]o 1/2 (fkdkt

[[]o)

Substituting appropriate values, (~--~.)o = 4340. Desired DP,~ = 15,000/86 or 174. FromEq. (6.148), 1 1 IS]benzene + c~(ccm[S]cc~ q- CM q- Cs(benzene) D’~n -- (~-~)~ .... [Mlo -1 Substituting appropriate values, [S]ccl 4 = 0.0106 tool L = 0.0106 x 154 or 1.63 g -1. L

Determination of Chain ~ansfcr Constants Various methods can be employed, based on the Mayo equation, to determine the values of the chain transfer constants. The following sections review some of these methods. Deterrnlnation

of

CM

A special case of Eq. (6.148) is of interest. Consider polymerization in the absence of a solvent or added chain transfer agent, so that IS] = 0. For steady-state polymerization, Eq. (6.25) can be used to express Pq in terms of P~ as /~ -- kp2 ~ [M]

500

Chapter 6

and Eq. (6.26)

can be rearranged [I]

Substitution

to express [I] --

Rp2kt k i [M]2 f kd

of Eqs. (6.150) and (6.151),

1 + r Fnhi [M]2

CM "-~

(6.151)

and IS] = 0 into Eq. (6.148) CI

hi f

(6.152)

Since Eq. (6.152) is quadratic in ~, the plot of l~n vs. ~ is turned, the e~ent of w~ch varies with the initiator. However, the initial portion of the plot, co~esponding to small ~ values, is linear. By extrapolation of this linear part to ~ = 0 yields CMm the intercept. Moreover, the slope of the linear portion is given by 2zkt/~ [M]2 from which mode of termination or k~/kt may be dete~ined, provided the other is ~own. For AIBN initiator CI is negligibly small; as a result Eq. (6.152) is practically linear in ~ even at higher v~ues of ~.

Problem6.25 The following data of rate of polymerization

and degree of polymerization at low conversion were obtained in bulk polymerization of monomer M (initial concentration 8.3 tool/L) using different concentrations of thermal initiator [ at 60°C: [I]

Calculate

x 102, mol/L /~ x 10a, mol/L-s 0.018 0.005 0.072 0.010 0.280 0.020 1.74 0.050 4.48 0.086 7.80 0.115 13.20 0.15 (a) CM, and

DP~ 8267 5495 3296 1300 714 495 352

(b) k~,/k~t/2. Assumetermination by coupling.

Answer: The plot of 1/~--~,~ vs. P~ (Fig. 6.6) is seen to be linear at low P~ (i.e., first 3 points) according to Eq. (6.154). Therefore the values of the intercept slope are determined from a least-squares calculation using the first 3 data points, yielding CM = Intercept = 6.03×10 -5 and k~2[M]2 = slope

= 12.16

kp/kit 12 = 3.46 x 10-2 L112 mo1-1/2

-1/2 S

Radical Chain Polymerization

~01

32

24

0

/~

8 Rpx 5 10

12

~6

Figure6.6 Plot of Eq. (6.151) for the determination of M(Problem 6. 25). The monomer chain transfer constants are generally small for most monomers-being in the range 10 -5 to 10 -4 (Table 6.9). Chain transfer to monomer places the upper limit to the polymer molecular weight that can be obtained in the absence of all other transfer reactions (see Problem 6.22).

Table 6.9 Monomer Chain Transfer Monomer Acrylamide Acrylonitrile Ethylene Methyl acrylate Methyl methacrylate Styrene Vinyl acetate Vinyl chloride All CMvalues are for 60°C. Source: Data mostly from Ref. 3.

Constants (7 M X 104

0.6, 0.12 0.26 -0.3 0.4-4.2 0.036-0.325 0.07-0.25 0,30-0.60 1.75-2.8 10.8-16.

502

Chapter 6

Determination

of

CI

Several methods are available for the determination of CI. Equation (6.152) can be rearranged and divided through by P~p to yield 1 ~n

CM Rp -)

2zkt kp 2[M]2

q-

ktRp 3CIk~/kd[M]

(6.153)

A plot of experimental data as the left hand side of Eq. (6.153) versus P~p yields a straight line whose slope is ktt{p / (k2p f kd [M]3). The value of CI can be obtained from the slope provided other constants are known, or one can make use of the fact that

ted[M] 3 2--

(6.154)

[MIRp

and use experimentally measured P~p for a given [I] and [M]. When chain transfer to monomer is negligible (CM " 0), rearrange Eq. (6.153) and combine with Eq. (6.154) to yield

~2p2~]

= Ci

[~]

one can

(6.155 /

A plot of the left side of Eq. (6.155) versus [I]/[M] yields a straight line whose slope is CI. Some values of initiator transfer constants are listed in Table 6.10.

Table 6.10 Initiator

Chain Transfer

Initiator 2,2’-Azobisisobutyronitrile Benzoyl peroxide t-Butyl hydroperoxide t-Butyl peroxide Cumyl hydroperoxide Cumyl peroxide (50°C) Lauroyl peroxide (70°C) Persulfate (40°C)

Constants

CI for polymerization of Styrene Methyl metha- Acrylamide crylate 0.091-0.14 0.048-0.10 0.035 0.00076-0.00092 0.063 0.01 0.024 -

All CI values are for 60°C except where otherwise noted. Source: Data mostly from Ref. 3.

0.02 0.02 0.02 0.33 -

0.0026

Radica/Cha/n Polymerization

503

Problem 6.26 Fromthe polymerization data given in Problem 6.25, determine the chain transfer constant of the initiator I for the polymerizationof monomer Mat 60°C. Answer: The value of CM obtained in Problem 6.25 is 6xl0-~. Since it is very small comparedto 1/DP~, Eq. (6.155) can be used to determine Cl. For termination 2) by coupling, z = 1/9. and Fig. 6.7 thus showsa plot of (1/~-~n - ~P~/k~, [M] vs. [I]/[M]. Fromthe slope of the linear plot C’i = 0.066.

Determination

of Cs

A special case of Eq. (6.148) consists of the situation where transfer with the chain transfer agent is most important. In some instances, the solvent itself is the chain transfer agent, while in others it is an added compound.In such a case, the fourth term of the right side of Eq. (6.148) makes the

12 10

0

I

4

t

I

8

t

I

12

i

I

16

I

20

[l]x 3 ~o Figure 6.7 Plot of Eq. (6.155) for the determination of CI (Problem 6.26)

504

Chapter 6

biggest contribution to the determination of DP---n. For determination of Cs the polymerization conditions maybe adjusted so as to considerably @~plify Eq. (6.148). Thus, for a series of measurements one may keep [I]I/’/[M] constant, so the first term on the right side of Eq. (6.148) becomesconstant and also use an initiator for which OI is negligibly small (e.g., AIBN) that the third term becomesnegligible. Under these conditions, Eq. (6.148) takes the form

(6.156) + Cs D~ where (I/~.)0 isthevalue of(I/~) inthe absen~ ofsolvent thechaintransfer agent, andit represents thesumof thefirstthreeterms ontherightsideofEq.(6.148). Theslopeof thelinear plotof (I D/-D-~n) vs.[S]/[M]givesthe measureof ~’S-Figure6.8 showssuchplotsfor severai aromatic solvent as chaintransfer agentsin thepolymerization of styrene. Theplotsdearlyindicate dependence of chaintransfer constants on thechemical structure of chaintransfer agents.

0

5 10

1.5

20

25

Figure 6.8 Effect ofvafious chain ~ansfer agen~ on the degree ofpolyraerization of styrene at 100°C. (Af’terRef. 43)

Radical Chain Polymerization

505

Another method of determining Cs involves dividing the rate expression for transfer [Eq. (6.143)] by that for propagation [Eq. (6.15)] to yield dis]/dt ~,s IS] _ Cs IS] (6.~57) d[M]/dt - kp[M] [M] The value of Cs is obtained as the slope of the line obtained by plotting the ratio of the rates of disappearance of transfer agent and monomerversus

[Sl/[MI. Problem 6.27 In a free-radical polymerization of vinyl chloride the initial monomerconcentration was 1 mol/L and the concentration after 1 h was 0.85 mol/L. Chloroformwas present as a chain transfer agent and its concentration decreased from 0.01 mol/L at time zero to 0.007 mol/L after 1 h. Whatis the chain transfer constant CSfor vinyl acetate/chloroform ? (Neglect chain transfer to monomer, initiator, and solvent.) Answer: For chain transfer and propagationreactions, respectively, - d[S] / dt = knr,S [M’] [S] -d[Ul/dt = ~[M’][M] Dividing and rearranging, d IS] d[M]

[s-T= csIMp-

Integrating betweento and t, \ IS]o,] = CSIn ~ [M]o,] Thisgives, Cs = 1n(0.007/0.01)

ln(O.8~ / 1.o) = 2.19

Chaintransfer reactions, generally viewedas additional reactions disturbing the normal features of radical polymerization, are often advantageous in limiting the growth of polymer chains by design and hence in controlling their molecular weights to desirable ranges. Equation (6.148) can used to determine the concentration of the transfer agent needed to obtain a specifically desired molecular weight (see Problem 6.24). Transfer agents with large Cs are especially useful since they can be used in small ~ concentrations. The transfer constants for a number of solvents/additives for polymerization of styrene, methyl methacrylate, and vinyl acetate are listed in Table 6.11. The data indicate dependence of chain transfer constants on the chemical structure of both chain transfer agents and the monomer.The

506

Chapter 6

Table6.11 Transfer Constants for Solvents and Chain Transfer Agents

Transfer Cs × 104 for polymerization of. agent Styrene Vinylacetate Heptane 0.42 17.0 (50°C) Benzene 0.023 1.2 Toluene 0.125 21.6 Ethylbenzene 0.67 55.2 Isopropyltienzene 0.82 89.9 t-Butylbenzene 0.06 3.6 n-Butyl chloride 0.04 10 n-Butyl bromide 0.06 50 n-Butyl alcohol 1.6 20 Chloroform 3.4 150 n-Butyl iodide 1.85 800 Triethylamine 7.1 370 Di-n-butylsulfide 22 260 Carbontetrachloride 110 10,700 Carbon tetrabromide 22,000 390,000 n-Butyl mercaptan 210,000 480,000 All valuesare for 60°Cunless otherwisenoted. Source:DatamostlyfromRef. 3. CS value is influenced by the nature of the bonds which are broken and formed and the relative stabilities of both radicals Mn" and A" in reaction (6.135). In general, a given transfer agent (XA) is more reactive (Cs is greater) for a reactive radical (Mn") like those in ethylene or vinyl chloride polymerizations than for a resonance-stabilized radical like that of styrene. Similarly, when a given monomeris being polymerized, aliphatic compoundsthat yield tertiary radicals are more effective transfer agents than those that produce secondary radicals, and chain transfer activity is also enhanced by the possibility for resonance stabilization of radical A’. Chain

Transfer

to

Polymer

The previous discussion has ignored the possibility of chain transfer to polymer molecules. However, chain transfer to polymer can occur and it is very significant with very reactive propagating radicals like those in the polymerizations of vinyl chloride, vinyl acetate, ethylene, and other monomersin which there is no significant resonance stabilization. It is also likely to be significant in polymerizations carried to high conversions where the concentration of polymer in the system is relatively high.

507

Radical ChainPolymerization

Chain transfer to polymer results in the formation of a radical site on a polymer chain. Polymerization of monomerat this site produces a polymer with a long branch, for example, X

X Mn"

+ vvvwCH2-Cvww

~

MnH + vwwCH2-C,

vww

x wvwCH2Cwwv

At low conversions the polymer concentration is low and the extent of transfer to polymer is negligible. Thus ignoring chain transfer to polymer in Eq. (6.149) does not present difficulty in obtaining precise values CM,(7i, Cs, as these are determined from data at low conversions. It should be noted that the occurrence of chain transfer reaction (6.158) does not change the number of monomer molecules which have been potymerized nor the number of polymer molecules over which they are distributed. Chain transfer to polymer thus has no effect on DPnand so it is not included in Eq. (6.147). It, however, causes a change in the molecular weight distribution. The distribution becomes broader because the polymers which are already large are more likely to suffer transfer reactions and become yet bigger due to branching. Chain branching normally makes the polymer less crystalline, weaker in mechanical properties, and less resistant to heat, solvents, and chemicals. The effect of chain transfer to polymer thus plays a very significant role in determining the physical properties and the ultimate applications of a polymer. The transfer constant Cp for chain transfer to polymer is not easily obtained as it involves the difficult determination of the numberof branches produced in a polymerization relative to the number of monomermolecules polymerized. There are thus relatively few reliable Cp values available in the literature Cp values are about 10-4 for manypolymers. Flory [44] has derived the equation

to express the branching density Pb as a function of the polymer transfer constant Cp and the extent of reaction p. The branching density Pb is the -4 number of branches per monomermolecule polymerized. For a Cp of 10 and an 80%conversion, one thus calculates from Eq. (6.159) that there will be one branch for every 104 monomerunits polymerized. The extent of branching is greater in polymers, such-as poly(vinyl acetate), poly(vinyl chloride), and polyethylene, which have very reactive propagating radicals. The extent of branching in polyethylene varies considerably depending on the polymerization temperature and other reaction conditions, and may

508

Chapter 6

reach as high as 15-30 branches per 500 monomer units. The branches in polyethylene are of two types: short branches (ca. less than 6 monomer units) and long branches. The long branches are formed by the "normal" chain transfer to polymer reaction [Eqs. (6.158)]. The short branches, which outnumber the long branches by a factor of 20-50, are mostly nbutyl branches. A typical polyethylene, for example, contains five n-butyl branches and one or two each of ethyl, n-amyl, and n-hexyl branches per 1000 carbon atoms. The generally accepted mechanism for the formation of short branching in polyethylene involves a "backbiting" intramolecular transfer reaction in which a radical at the end of the polymer chain abstracts a hydrogen atom from a methylene unit in the same chain (Fig. 6.9). This is a very important process in the free-radical, high-pressure polymerization of this monomer. Branched polyethylene from this process has lower crystallinity than linear polyethylene produced by a low-pressure process and as a consequence it tends to be less rigid and tougher and form clearer films than the latter.

j H -,~CH "CH l"~.CH [ CH 22 /CH2

lntramotecutar chain transfer. (self-transfer)

I

CH2 ,CH2-CH 3 ~CH~" 2CH = CH

~

CH 3 A second I CH isetf-transfer I CH jCH-CH2- CH 3 ~CH 2

I

/.~CH2 H2 H

~ ~CH

~CH 2

/CH-CH2-CH 3

I

Propagation

Propagation

fC, H3 branchL~H 2 ..-..--~

!H 2

CH.CH ~CH~" ~CH2- CH3

CH2 I ~ C.l-I

Ethyt branch

.CH2- CH 3 2- CH ~ Butyt branch

Figure 6.9 Mechanism of ethyl and butyl branching

in polyethylene.

Radical Chain Polymerization Allylic

509

Transfer

Chain transfer to monomerwas described earlier in general terms for vinyl monomers[cf. Eq. (6.138)]. Such reactions are particularly favored with allylic monomerssuch as allylic acetate which have the structure CH~_=CHCH.~Xwith a C-H bond alpha to the double bond described as an allylic C-H. The propagating radical in the polymerization of such monomersis very reactive, while the altylic C-H bond in the monomeris quite weak, resulting in facile chain transfer to monomer: H H I I t,,~. M "~CH2-C"I + CH2=CH-C-H ----* CH~X H! ,~’*CH~-C~H2 + CH~=CH-C. e---~

cr 2x

H "CH2-CH=C

(6.160)

Since the allylic radical which is formed by the transfer reaction has high resonance stability, it is particularly unreactive and does not initiate new chains, with the result that the allylic monomerpolymerizes at abnormally low rates and the degree of polymerization, which is independent of the polymerization rate, is very low (for example, only 14 for allyl acetate). These effects are the consequenceof degradative chain transfer (see Table 6.8) to monomer,also knownas autoinhibition. In this polymerization, the propagation and termination reactions will have the same general kinetic expression resulting in the unexpected dependence of the rate on the first powerof the initiator concentration (see Problem6.28). Problem6.28 Suggest a kinetic schemeto account for the following characteristics of free-radical polymerizationof allylic monomers: (a) P,,p is very lowwith first order dependenc e in initiator concentration. (b) DP, is very low and independent of monomerand initiator concentrations. (c) Deuteratedallylic monomer possessing allylic C-Dbondhas significantly higher/~0 than the normalallylic monomer. Answer: (a) Since the allylic radical formed[Eq. (6.162)] has high resonance stability it does not initiate newchains. Chaintermination occurs predominantlyby chain transfer. At steady state, = /~ - ktr,M [M’I[M] = 0 dt Combinationwith /~p = kl. [M-] [M] leads to Rr -- ktr,M

_ 2fk d ktr,M

(P6.28.1)

0’6.28.2)

510

Chapter 6

For degradative chain transfer, kp << ktr,M. Therefore R~is very low. Equation (P6.28.2) showsthat r i s f irst o rder i n [I]. = v = ~ Rr -l k~[ M’I[M ~ 1~ ktr,M [M’] stant and very low. (b)

k~

D-~.

[M]

ktr,M"

Thus DP~is con-

(c) The C-D bond is stronger than the C-H bond due to its lower zero point energy. The degradative chain transfer would therefore be decreased in the deuterated monomerwith consequent increase in /~. The reluctance of a-olefins like propylene or of 1,1-dialkyl olefins such as isobutylene to form high polymers in free-radical reactions is a result of degradative chain transfer with the atlylic hydrogens. It should be pointed out, however, that other monomerssuch as methyl methacrylate and methacrylonitrile, which also contain allylic C-Hbonds, do not undergo extensive degradative chain transfer to monomer.This is probably because the propagating radicals are conjugated with and stabilized to some extent by the ester and nitrile substituents. The propagating radicals are thus less reactive than those in conventional allyl polymerizations and therefore have lower tendencies for transfer to monomer.Simultaneously, the reactivity of the monomertoward propagation is enhanced. These monomers, unlike the a-olefins and 1,1-dialkyl olefins, yield high polymers in free-radical reactions. DEVIATIONS

FROM IDEAL

KINETICS

A given polymerization conforming strictly to the reaction scheme (6.3) to (6.10) is commonlyconsidered as an ideal polymerization. Any system deviating from this pattern of reaction is to be considered as a case of nonideal polymerization. The ideal behavior prescribes constancy of the term (Rp~ / [I] [M]2), according to Eq. (6.26) expressed

[I]

[M] 2 --

fkd

(6.161)

But studies of the kinetics of polymerization of different monomers,under different conditions and chemical environments, indicate that ideal behavior is probably more an exception than the rule. Most practical free-radical polymerizations will deviate to a greater or lesser extent from the standard conditions outlined in the reaction scheme (5.3) to (5.10) either because the actual reaction conditions are not entirely as postulated in the ideal kinetic scheme or because some of the assumptions that underlie the ideal scheme are not valid. According to the ideal kinetic scheme, which yields Eq. (5.151), the initiation rate and initiator efficiency f are independent of monomercon-

511

Radical Chain Polymerization

centration in the reaction mixture. These may, however, depend on [M], with a consequent 3/2-order dependence of /~p on [M], if primary radicals escape from their solvent cage by reaction with the nearest monomer molecules (see Problems 6.8).

PrimaryRadical Termination Primary radicals under ideal conditions wouldcontribute to chain initiation only. But in certain systems and under special conditions in certain others, they mayalso contribute to chain termination (primary radical termination or primarytermination), partly or exclusively, giving rise to significant deviations from the ideal kinetics (see Problem 6.29). Problem 6.29 Showthat in the case of radical polymerization where both bimolecular and primary radical terminations occur simultaneously, the parameter (R~ / [I] [M]2) is not constant but is dependenton (P~ / [M]2). Hencesuggest methodof analysis of the primaryradical termination effect from the experimental P~ data. Answer: The reaction schememaybe written as:

(a/ ~ ~ (b/ R- + MAC,MI" (c)

M."

+ M ~ M.+~"

(d)

M.-

+ Mm" ~

(e)

R- + g~. ~

(n~X)

where P denotes a dead polymer. If bimolecular and prima~ radical te~inations, (d) and (e), occur simultaneously, then considering steady concentrations of [M’] and [R-] separately, namely, dt

= ki[R’l[M]

d(R’] dt one mayobtain

- 2kt[M’] ~ - ~,[M’][R’]

= 0

~y~[t]- ~, [~.] [M]- ~ [~.] [M.]= 0

~, [M-p _ f~[~]

~, [R’] [M] - ~,~ [R-] [Mq ~, [R’] [M] + ~,~ [R’] [M-]

Substituting (~ / k~ [M]) for [M’] and rearranging gives

(P6.29.1)

(P6.~9.~) (P6.~9.3)

512

Chapter 6

2 [I][M]

+ k~kv "~ ]

Unlikein ideal kinetics [Eq. (6.161)1whereR~/ [I] [M]2 is constant, Eq. (P6.29.4) showsthat P~ / [I] [M]2 2. dependson P~ / [M] Considering that for vinyl polymerization, (k,v / kik~) (~ [M] ~) ~ 1, Eq. (P6.29.4) can be simplified to the form: [I] [M]2

~k~

exp kik,

Ta~ng loga~thm 2[I] [M]

kk* fkd -- kkik,

j ~M]2

Equation (P5.29.5) permits a plot of the left hand side against (~/[M]Z), negative slope for the linear plot then being indicative of pfimau radical termination. The magnitude of the slope also ~ves a measure of the parameter

Initiator-Monomer ComplexFormation There are many reports on polymerization of different monomersshowing different degrees of kinetic complexity where the observed effects are well explained and understood on the basis of equilibrium complex formation between the initiator used and the monomer[45-47]. Problem 6.30 Formation of complex (C) between initiator (I) and monomer (M) leads to nonideal kinetic behavior in manycases. A simplified scheme chain initiation based on the concept of equilibrium complexformation between initiator and monomermaybe given as K (a) I + M ~ C (initiator-monomer complex)

c (c) R" + M --~ RM" where K is the equilibrium constant of the initiator-monomer complexationreaction (a) and the complexC is the true source of initiating radicals. Assuming that the usual bimoleculartermination is the only prevailing modeof chain termination derive an expression for the rate of polymerization Rv and show howK can be evaluated from the experimental Rv vs. [M] data.

513

Radical ChainPolymerization ~llSWer

:

For reaction (a), since usually [M] >> [C], K --

[c]

(P6.30.1)

= ([I]0 - [C])[M] where[I]o is the initial concentrationof the initiator. Rearranging Eq. (P6.30.1), K[M][I]0 [C] -- 1 + K[M]

(P6.30.2)

Therate of initiation is given by

(r6.30.3)

= [c] Combinationof Eqs. (P6.30.2) and (P6.30.3) with Eq. (6.25) t~

=

K

(r6.30.4)

\T,]

Equation (P6.30.4) desc6bes a change in reaction order with respect to monomer from 1.5 to 1 with increasing [M]. Equation (P6.30.4) maybe transfo~ed follows: [M] ~ ~ R~ - k~k.K [I1o + k~k~ [I]o (P6.30.5) 3/~ Equation (P6.30.5) pewits a plot of [M] vs.. [M] to give a straight line such that the quotient of the slope and the intercept is equal to ~.

Degradative Chain Transfer Initiators under ideal conditions wouldcontribute only to chain initiation by dissociation [Eq. (6.3)] into primary radicals (R’). But in certain systems they also contribute to chain termination, partly or exclusively, giving rise to significant deviations from the ideal kinetics (see Problem 6.31). The degradative chain transfer to initiator (I) maybe written M~" -[-I’-~

Mn q- I"

(6.162)

where I" is a radical product from I, inactive or less active than the primary radical R" from I.

514

Chapter 6

Problem 6.31 The degradative chain transfer to initiator (I) may be written k~r,I M~" + I ----* M, + I" where I" is a radical product from I, inactive or less active than the primary radical R- from I. Derive suitable expressions for the rate of polymerization considering (a) an extreme case where chain termination occurs exclusively by the degradative initiator transfer and (b) a more general case where chain termination takes place by simultaneous occurrence of the degradative initiator transfer and the usual bimolecular mechanism. In both cases, assume that the radical I- formed by the chain transfer to initiator I is too inactive to reinitiate polymerization. Answer: (a) For termination of the kinetic chains exclusively by the degradative initiator transfer process, the rate of termination is Rt = ktr,i [M’] [I] Using the steady-state concept for the chain radicals, d[M’] -- Ri - t~t dt

= 2fkd[I]

-- ktr,i[M’][I]

= 0

[M’]= --2fkd kt~,I The rate of polymerization is then given by Rr = k,[MI[M’]

(P6.31.1)

which shows that Rp is independent of initiator concentration but it still first order dependence on monomerconcentration.

retains

(b) Under steady state, the following relationship holds good when chain termination occurs both by the degradative process and the usual bimolecular mechanism: dt

-- 2fkd [I]-

kt,,i

[M’]

On elimination of [M’] by substituting rangement one obtains

[II - 2k, [M.I2 = 0 [M-] = /~p/k~[M] and on further

Equation(P6.~l.~) allows a plot of _R~/[I]

rear-

[~]~vs.P~/[M], anda negative

slope of the linear plot is indicative of termination by the degradativeinitiator transfer reaction in addition to bimolecular termination. Fromthe slope of the plot, the parameter (kt,,i I kp) can be obtained and from the intercept fkd can be calculated, if the kinetic parameter (k~ / ~) for the monomeris known. When ~ becomes independent of [I] for a given [M], Eq. (P6.31.3) is unsuitable for the analysis of the degradative effect and Eq. (P6.31.2) may then be employed.

Radical Chain Polymerization

515

Problem6.32 Nonideal behavior in polymerization

of a given monomer is reflected in the variability of R~/[I] [M]2 at a constant temperature. Derive an equation for R~/[I] [M]2 representing in the most general form the perturbations introduced by primary radical termination and various chain transfer reactions. Neglect, however, the recombination reactions of primary radicals and of radicals formed as a result of chain transfer, as also cross-combination of these types of radicals, all of which are likely to be insignificant except under unusual conditions. Discuss, on the basis of the derived equation, the factors which lead to decreased rate of polymerization. Answer: Omitting the recombination reactions of primary radicals and radicals which result from chain transfer, the following scheme of reactions may be considered: Reaction

Rate

(a) I k--Ka 2R-

2A/~ [I]

(b)

R" + M ~ MF

ki[R’]

[M]

(c)

MI" + M -~. M2" ~ k~ [M’t [MI

(d) Mm"+ M22Y_, Mm+l" (e) Mm"+ Mn* ~-~ 2Polymer 2kt [M’] (f) (g)

Mn" + R" ~ Polymer ktp [M’] [R’] M."

+X~M.+X"

(h) X" + M ~2K, MI" (i)

M,F + X" ~ Polymer

ki~ l [X’] [NI k~, [M’I[X’l

ProceN Production of primary radicals Initiation of polymer chain Propagation Termination of growing chains by mutual deactivation Chain termination by primary radical Chain transfer where X may be monomer,solvent, initiator, or any additive Re-initiation Chain termination by a radical formed due to transfer

In reaction (a), f~ represents the total efficiency of the initiator defined [48] as the fraction of primary radicals that come out of the solvent cage escaping recombination therein and take part in initiation and primary radical termination of growing chains. Evidently f~ should be greater than the conventional f used in ideal polymerization equation (6.26). Application of stationary-state conditions to the concentrations of R’, X" and M" (denoting all radicals of size MI" and larger) leads to the equations: 2y~,kd[I] = ki [R’] [M] + ~, [R’] [M’] k,,. x [M’] IX] = ki, [X’] [M] + k~, [M’] [X’] k, [R’} [MI + k,~ [x.] [M] = 2/~ [M’]2 + k,~ x [M’] IX]

(P6.32.1 (P6.32.2

(P6.32.3 +/% [M’I JR.] + kt~ [M.] [X-] Combining Eqs. (P3.32.1)-(P3.32.3) and simplifying leads to the following equation relating [M’], [I], [M], and IX]:

516

Chapfer 6

--

-

k,=

[M’])

’ (P6.32.4)

The summation in this equation extends for all the ~ansfer-active present in the system. Equation ~6.32.4) can easily be simplified to

kt[M.]

~ _ (1

k~,[M’]~k~[M]

-Z

J k,~[X][M’]~

components

(ra-z -s)

[Note that when both prima~ radical termination and degradative chain transfers are neglected, Eq. (P3.32.5) transforms to Eq. (6.24) for ideal polyme~ation with ~ Nven by Eq. (6.14).] Unless the polymeric chains are ve~ short, one may use the long chain approximation /~ = h~[M’I[M] (P6.32.6) so that we have from Eq. (P6.32.5)

[I][M]2

kt 1 + kikr ) [M]2.]

k~[X]C ~ k,~k~[I] [MI

(P6.32.7)

(

where CX is the transfer constant kt,,x/kp. Equation (P6.32.7) represents in the most general form the perturbations introduced by primary radical termination and chain transfers. The first term on the right side represents the effect of primary radical termination and the second term that of degradation. Evidently, the greater the value of ~p the greater is the decrease in the rate of polymerization. On the other hand, the greater the values of Cx and kt~/ki~, the greater is the perturbation due to the chain transfer. In physical terms, this predicts that the more stable is the radical formed due to chain transfer, the greater is the degradative effect of chain transfer on the rate of polymerization. The degradative chain transfer causes a lower specific rate of initiation, k~, of the newly formed radical compared to that of a primary radical derived from an initiator. Case L No degradative chain transfer The termination of chain radicals occurs by combination with primary radicals and by the normal processes of mutual deactivation. In this case, Eq. (P6.32.7) simplifies R] _ f~kek~ 1 kik~, [M]2,] a[I1 [M] k, 1 + kik~ [M]2,]

(P6.32.8)

517

Radical Chain Polymerization or

- 0.868

,og : ,og

(P6.32.9)

for hnpP~ //¢ik~ [M]2 KK1. Case II. No primary radical termination. The termination of chain radicals occurs by degradative chain transfer and by combination with the resulting radicals besides the normal processes of mutual deactivation. Equation (P6.32.7) can then be written as

_ fkkdk~

(P6.32.10)

[M]" (1

Problem6.33 Reconsider the case of degradative

initiator transfer as described in Problem 6.31 but assume now that the radical I" formed due to chain transfer is capable of reinitiating polymerization. In addition to the normal modes of initiation, propagation, and bimolecula,r termination of chain radicals as in ideal polymerization, other reactions that may follow as a consequence of this degradative initiator transfer are shownin the following reaction scheme: Reaction (a) Mn" + I -~ Mn + I" ktr,I

Rate [M’] [I]

(b) I" + M." ~ Polymer hnl [I’]

[M’]

Process Chain transfer Termination by I"

Reinitiation by I" following transfer Derive an expression for the rate of polymerization and indicate how it can be used to obtain a measure of the degradative initiator transfer from experimental kinetic data. (c)

I" + M ~-~ MI"

kil [I’] [M]

Answer: Degradative effect will be prominent due to the occurrence of reaction (b). In the absence of reaction (b), degradative effect would still be measurable and important due to reaction (c) with kit < k~,. In the absence of other chain transfer reactions (i.e., other than with initiator) and primary radical termination, Eq. (P6.32.7) simplifies to the form (replacing f~ by the conventional f since primary radical termination is neglected):

[i]

[M]--~k~~[I] [Cskt,~[/ [~ 1 + ~k~-~-]2] 2[,1+ (kilk~,) ]/ ktlP~p ~1J

--

fkdk~kt

(P6.33.1)

518

Chapter 6

For (P~/[M]2) <<< 1, Eq. (P6.S3.1)reduces R~ { , k,l (P6.33.2) = ~ [I] [M]~ 1 "3i- lkt, . kilkt~ . C -~M] } " fkd Onrearrangement of Eq. (P5.33.2) and ta~ng logarithms, a more useful expression, Eq. (P5.33.3) is obtained for ~netic analysis of the degradative effect: 2In [I] [M]

(P6.33.3) .-, " kt" kilk~ q [M] The second term on the right side of this equation represents the part played by degradative chain transfer to initiator. Whenthis is not considered, Eq. (P6.33.3) leads to the ideal Eq. (6.26). Equation(P6.33.3) pewits a plot of ln R~/[I] ~ against [I]/[M]. Anegative slope for the linear plot will indi~te that the degradarive initiator transfer under ~nsideration is signifi~nt. Witha ~owledgeof the initiator transfer ~nstant (CI) and k~ / k~, the plot gives a measureof the ~netic parameters (~1/k~;k~) and fka from the slope and intercept, respectively.

Autoacceleration In view of Eq. (6.26) for ideal polymerization kinetics one would normally expect the reaction rate to fall with time, since the monomerand initiator concentrations decrease with conversion. However, the exact opposite behavior is observed in manypolymerizations where the rate of polymerization increases with time. A typical example of this phenomenonis shownin Fig. 6.10 for the polymerization of methyl methacrylate in benzene solution at 50°C [49]. At monomerconcentrations less than about 40 wt%in this case, the rate (slope of conversion vs. time) is approximately as anticipated from the ideal kinetic scheme described in this chapter, that is, the rate decreases gradually as the reaction proceeds and the concentrations of monomerand initiator are depleted. An acceleration is observed, however, at higher monomer concentrations and the curve for the pure monomer shows a dramatic autoacceleration in the polymerization rate. Such behavior is referred to as the gel effect. (The term gel used here is different than the usage in Chapter 5 as it refers only to the sharp increase in viscosity and not to the formation of a cross-linked polymer.) The autoaccelerative gel effect is also knownas the Tromsdorffeffect or Norrish-Smith effect after pioneering workers in this field. It should be noted that the gel effect is observed under isothermal conditions. It should thus not be confused with the acceleration that would be observed if a polymerization reaction were carried out under nonisothermal conditions such that the reaction temperature increased with conversion due to exothermicity of the reaction. The gel effect observed for higher monomerconcentrations is generally attributed to a decrease Of the termination rate constant/¢t due to increased viscosity at higher conversions. Theoretical e0nsiderations indicate that the rate constant for reaction between two radicals in low viscosity media (such as bulk monomer)would be very large, about 8 x 109 L/mol-s. Experiment-

Radic~l Chain Polymerization

519

100

"~ 60 >

o /-.0 2O

0

I

I

I

I

300

I

I

I

600

I

I

I

900

I

1200

Time, min Figure 6.10 Conversion-time plots for the polymerization of methyl methacrylate in benzene at 50°C. The labeled curves are for the indicated monomer concentrations. (After Ref. 49.) ally determined kt values for radical polymerizations, however, are considerably lower; usually by two orders of magnitude or more (see Table 6.7). Thus diffusion is the rate-determining process for termination. Problem6.34 Chaintermination is a diffusion-controlled reaction best described [50-52] as occurring by a three step process: Step 1. Translational diffusion of two propagatingradicals (i.e., movement of the wholeradicals) until they are in close proximityto each other: M." + Mrs" ~=- M." ....

M~"

(i)

Step 2. Segmentaldiffusion (movementof parts or segmentsof a polymer, chain relative to its other parts) of the two chains so that the two radical ends are sufficiently close for chemicalreaction: M." .... M,." ~- M,~"/M,~" k4

(ii)

520

Chapter 6

Step 3. Chemical reaction of two radical ends: M,’/M,~"

~ dead

polymer

.

(iii)

Assumingthat diffusion is the rate-determining process for termination obtain an expression for the rate of termination. Simplify the expression for two limiting situations of slow translational diffusion and slow segmental diffusion. Answer: For diffusion control of termination, kc >> k4 and the reverse reaction in step 2 [Eq. (ii)] is neglected. Assuminga steady state for [M~".... M,,’], d[M." .... M,~’] dt Therefore, °[M,~

kl[M.’I[M.~’] - k2[M." .... M,~’] - k3[M." .... M~’] = 0

= k2 + k3 [M~’][M.~’]

and /~ = ka [Mn" .... M,~-] klk3 2[M’]

(P6.34.1) k2 + k3 where [M-] is the concentration of all chain radicals. For slow translational diffusion, kz >> k2 and Eq. (P6.34.1) then reduces /~ = kl [M’] 2 (P6.34.2) For slow segmental diffusion, k2 >> k3 and Eq. (P5.34.1) then reduces klk3 Rt -- k2 [M’]2 (P6.34.3) The experimentally observed termination rate constant kt thus corresponds to kl and k~k3/k2, respectively, for the two limiting situations.

To ascertain the. effect of conversion on polymerization rate, it is useful to determine the ratio P~ / [M] [I]1/2 instead of percent conversion as it takes into account the chafigeg in- monomerand initiator concentrations with time (or conversion). A plot of P~p / [M] [1] 1/2 versus conversion (Fig. 6.11) show that three stages can be distinguished in some polymerizations. Stage ! involves either a constant rate (IA) or declining rate (IB) with time. Stage H represents the autoaccelerative gel-effect region, while Stage 11I involves either a constant (IIIA)or declining rate (IIIB). Stage I behavior is caused by the opposite effects of the increase in segmental diffusion and decrease in translational diffusion at higher conversions. The increase in segmental diffusion is attributed to the fact that with increasing conversion the polymerization medium becomes a poorer solvent due to the increased polymer concentration. The size of the randomly coiled up propagating radical in solution (referred to as coil) thus becomes smaller, resulting in an effective higher concentration gradient

Radical ChainPolymerization

521

Ilia \

\ Ill II

Conversion(*/.) Figure 6.11 Effect of conversion on polymerization rate. across the coil. Segmentaldiffusion of the radical end out of the coil to encounter another radical therefore increases with conversion. Simultaneously, however, the increasing polymer concentration decreases translational diffusion, as the reaction mediumbecomesmore viscous. At sufficiently high conversion, the polymer radicals are more crowded and entangled with each other leading to an even faster decrease in translational diffusion. Stage IA behavior observed for manymonomersresults when the increase in segmental diffusion is counterbalanced by the decrease in translational diffusion (i.e., kt remains constant). If, however, the initial increase in segmental diffusion is greater than the decrease in translational diffusion, k¢ increases and the polymerization rate decreases producing Stage IB behavior of decrease in/~ /[M] [1] 1/2 with conversion. Moderate Stage IB behavior has been observed in polymerizations of several monomersincluding styrene and methyl methacrylate [53,54]. As polymerization proceeds and viscosity increases, at some point the translational diffusion decreases faster than the increase in segmental diffusion and rapid autoacceleration or the gel effect (Stage II) occurs. When the polymer concentration becomes high enough, the chain radicals become more crowded and entangled with each other. As a result, the rate

522

Chapter 6

of diffusion of the polymer radicals and the frequency of their mutual encounters decrease. The rate of termination thus becomes increasingly slower. Termination involves the reaction of two large polymer radicals whereas propagation involves the reaction of small monomermolecules and only one large radical. High viscosity thus affects the termination reaction much more than the propagation reaction, that is, kt decreases muchmore 1/2 in than kp. The net result in this case is an increase in the ratio kv //~ Eq. (6.26) and hence an increase in the rate of polymerization. Since vinyl polymerizations are exothermic (see later) the increased polymerization rate associated with the autoacceleration effect can cause a temperature rise and faster initiator decomposition, leading to runaway reactions or explosions, if the heat of reaction is not removedefficiently. In Stage II, the kp//c~/2 ratio and the /~p/[M][I] 1/:~ value increase rapidly at first and then taper off as /cv is also affected in the later stages of reaction. At ve~ high conversions, /~p becomessufficiently affected that the /~p/[M] [I] 1/2 begins to level off (Stage Ilia behavior) or decrease (Stage IIIB behavior). Stage IIIB behavior which is much more common than Stage IIIA, is sometimesreferred to as the glass effect. The decrease in polymerization rate can be extremely pronounced and polymerization can end appreciably short of full conversion when the reaction temperature is below the glass transition temperature of the polymerization reaction system [55].

INHIBITION

AND RETARDATION

Somesubstances suppress free-radical polymerization of monomersby reacting with primary radicals or polymerradicals to yield either nonradical products or radicals that are of too low reactivity to undergo propagation. Such polymerization suppressors are called inhibitors or retarders depending on their effectiveness. Inhibitors stop every radical and polymerization is completely halted until they are consumed.Retarders are less efficient and stop only a fraction of the radicals. In this case, polymerization is not halted but it occurs at a slower rate. Figure 6.12 compares these effects on the rate of free-radical polymerization. Polymeriztion is completely stopped by an inhibitor during an induction or inhibition period (curve 2). At the end of this period, whenthe inhibitor has been consumed, polymerization proceeds at the same rate as in the absence of inhibitor (curve 1). A retarder lowers the polymerization rate without an inhibition period (curve 3). Somesubstances exhibit complexbehavior (curve 4). This behavior is not at all uncommon and may be exhibited by a substance that initially acts as an inhibitor but is converted to a product which acts as a retarder after the inhibition period. Impurities present in a monomermay act as inhibitors or retarders and those are usually the cause of the irreproducible polymerization rates observed with insufficiently purified monomers.On the other hand, inhibitors are invariably added to

523

Radical Chain Polymerization

I

2

> 0

Time Figure 6.12 Comparison of conversion-time plots for normal, inhibited,

and retarded free-radical polymerization. Curve 1: normal polymerization; curve 2: inhibition; curve 3: retardation; curve 4: inhibition followed by retardation. commercial monomers to prevent premature thermal polymerization during storage and shipment. These inhibitors are removed prior to polymerization or, alternately, an appropriate excess of initiator may be used to compensate for their presence. The difference between inhibitors and retarders is simply one of degree and not kind. Both are either chain transfer agents or act by addition processes to provide an alternative reaction path to propagating polymer radicals: Monomeraddition M,~"

¯

Chain transfer to XA> Addition to Q

:~

M~+I" MnX + A"

(6.163a) (6.163b)

M,~Q"

(6.163c)

If the new radicals A" and M~Q"do not react readily with monomer, there will be a decrease in the concentration of reactive radicals and a consequent fall in the rate of polymerization. If the rate of reaction (6.163b) or (6.163c) is very much greater than that of reaction (6.163a) the new radicals A" and M,~Q" do not add monomer then high-molecularweight polymer will not be formed and the rate of polymerization will be effectively zero. This is a case of inhibition (see Table 6.8). In retardation

524

Chapter 6

the rate of polymerization is slowed down but not reduced to zero. This occurs (a) if the rate of reaction (6.163b) or (6.163c) is close to that normal propagation reaction (6.163a) and the new radicals A" and M,~Q" not reinitiate (Table 6.8), or (b) if reactions (6.163b) and (6.163c) compared to reaction (6.163a) but the new radicals A" and MnQ"that are formed reinitiate slowly. Quinones are probably the most important class of inhibitors. The following transfer reactions may take place in the presence of quinone:

(6.164) HC-R I -~--CH 2

R

The inhibitor radicals formed in the above reactions are stabilized by resonance to such an extent that they do not start chains and initiate polymerization. They disappear partly through disproportionation (forming quinone and hydroquinone):

o

HO

6 HO (6.165)

0

HO

HO

0

and partly by combining with each other (dimerization) or with new chain radicals. Reaction (6.165) results in the regeneration of one inhibitor molecule per each pair of inhibitor radicals. Therefore this wouldlead to a 2:1 stoichiometry between the number of kinetic chains terminated [such as by reaction (6.164)] and the number of quinone molecules consumed. Disappearance of inhibitor radicals by dimerization wouldlead to a 1:1 stoichiometric ratio. Hydroquinoneand Other dihydroxybenzenes such as t-butyl catechol also act as inhibitors, but only in the presence of oxygen. The inhibiting effect is due to their oxidation to quinone. A large number of other substances are also active inhibitors. These include oxygen, NO(one of the most effective inhibitors, so muchso that some highly reactive monomerscan be distilled

525

Radical Chain Polymerization

only under an atmosphere of NO), aromatic nitro compounds, numerous nitroso compounds, sulfur compounds, amines, phenols, aldehydes and carbamates. Aninteresting inhibitor is molecular oxygen. Being a diradical, oxygen reacts with chain radicals to form the relatively unreactive peroxy radical: M,~- + O2 --4 Mn--O--O" (6.166)

Kinetics

of Retarded

Polymerization

In treating the kinetics of inhibited or retarded polymerization the usual kinetic schemeof initiation, propagation and termination can be used with an added inclusion of the transfer reaction between the chain radical and the inhibitor molecule Z M" + Z ~2~

Z"

(6.167)

For the sake of simplicity it will be assumedthat the inhibitor radicals (Z’) do not initiate new polymer chains and also do not regenerate the original inhibitor molecule by reactions such as Eq. (6.165). In the presence of the inhibitor, the rate of radical termination will be given by Rt = 2~ [M’] 2 + kz [Z] [M’] Since at the steady state, P~ = Rt, one may write 2kt [M’]2 -t- kz [Z] [M’] -- /74 = 0 Substituting

(6.168)

(6.169)

for [M"l from P~p = kp [M] [M’], Eq. (6.169) yields ~ [M]-------~

P~ = 0 + [M]

(6.170)

where Cz is the inhibition constant defined as Cz

-

k~

(6.171)

Twolimiting cases may now be considered: (a) When~ << 1, the polymerization

rate is not retarded.

Equation

(6.170) then simplifies to the usual polymerization rate expression: /~ = kp[M]

\~tt]

(6.172)

526

Chapter 6

(b) When~p >~ 1, termination of chain radicals will be mostly caused reaction with the inhibitor. The normal bimolecular termination can thus be neglected and Eq. (6.170) reduces

~ [z] Cz

P~ = 0

(6.173)

or, diM] dt -

[M]P~ Rp - Cz[z]

(6.174)

The rate of strongly retarded polymerization is therefore dependent on the first powerof the initiation rate R4. This is in contrast to the polymerization in the absence of inhibitor or Cz << 1 where the rate is dependent on to a power one-half [Eq. (6.172)]. Mildly retarded polymerization falls between these two limits. Equation (6.174) also shows that P~p is inversely dependent on the inhibitor concentration. Since inhibitor molecules combine with primary and chain radicals as and when they are formed, the inhibitor concentration decreases with time. If y is the number of radicals terminated per inhibitor molecule, the concentration of inhibitor, [Z], after a time t can be given by

Nt

[z] = [z]0

Y Combination of Eqs. (6.174) and (6.175) yields d[M] dt

Ri [M]

(6.175)

(6.176)

Cz [Z]0 or, by rearrangement,

1 dln[M]/dt

-

[~Cz Czt °Y

(6.177)

In terms of monomerconversion, p -- ([M]0 -- [M]) /[M]0, Eq. (6.177) can be written as

_

1 d~n(1 - p) /

_ Cz [z]0 Czt P~

(6.17s)

A plot of the left side of Eq. (6.178) vs. time is linear and the values Cz and y can be obtained from the intercept and slope, respectively, if [Z]0 and P~/are known (see Problem 6.35). The method, however, involves

527

Radical ChainPolymerization

difficult experimentation, since the polymerization rates being measuredare quite small, especially if Cz is large. Problem6.35 Polymerization of vinyl acetate at 45°Cin the presence of 0.2 M benzoyl peroxide and 9.3 x 10-4 M2,3,5,6-tetramethylbenzoquinone(duroquinone), used as inhibitor, yielded the following data of monomer conversion versus time: Time (min) 0 100 200 300 400 500

% Conversion, pxl00 0 0.10 0.30 0.66 1.18 2.45

Determinethe inhibitor constant (Cz) at 45°C and the numberof radicals termi-7 nated per inhibitor molecule. [For benzoyl peroxide at 45°Ctake ke = 2.8 x 10 s-1, f = 0.LS] Answer: Froma polynomialfit of ln(1 - p) vs. time (t) data, dln(1- p)/dt is obtained by differentiation. The linear plot in Fig. 6.13, accordingto Eq. (6.178), yields slope --

Cz

-- -92.2 Y intercept -- CZ [Z]o :_ 5.1

x 10 4

min

-7×60 min-1)(0.2 Ri = 2(0.25)(2.8x -6 = 1.68x10 mot L-~ -~ min CZ = (5.1x104 y = 92.1/92.2

mol L - ~)

-6 mol L-lmin-1) min)(1.68x10 (9.3 x 10-4 -~) tool L ~_ 1

= 92.1

Table 6.12 shows Cz values for various systems. The value of Cz for a compound is seen to depend largely on the monomer being polymerized. The value of Cz is large for inhibitors and relatively small for retarders. A given compound can thus act as a strong inhibitor for a monomerwhile acting only as a mild retarder for another monomer. ENERGETIC

CHARACTERISTICS

Rate of Polymerization Increasing reaction temperature usually increases the polymerization rate. Quantitative relationship between polymerization rate and temperature,

528

Chapter 6

7.0 6.0 5.0

4.o 3.0 2.0 1.0 I

100

I

1

I

200 300 400 Time(min)

I

500

Figure 6.13 Plot of Eq. (6.178) from data of Problem6.35. is complex,since the rate dependson the combinationof three rate constants kd, kp, and kt. For a polymerizationinitiated by the thermal decomposition of an initiator for which Eq. (6.26) is applicable, the rate ~ depends on the ratio kpkd /kt ¯ Expressing each of the rate constants kp, kd and let by an Arrheniustype relationship k

= Aexp(-E/RT)

(6.179)

the temperature dependenceis given by

(6.180) A compositeor overall activation energy for the rate of polymerization,

529

Radical Chain Polymerization

Table 6.12 Inhibitor

Constants

Inhibitor

Monomer

Nitrobenzene

Methylacrylate Styrene Vinyl acetate

Cz = k~ / k~ 0.00464 0.326 11.2

1,3,5-Trinitrobenzene

Methyl acrylate Styrene Vinyl acetate

0.204 64.2 404

p-Benzoquinone

Acrylonitrile Methyl methacrylate Styrene

0.91 5.7 518

DPPH

Methyl methacrylate (44°C)

2,000

FeCla

Acrylonitrile (60°C) Styrene (60°C)

3.3 536

Oxygen

Methyl methacrylate Styrene

33,000 14,600

Phenol

Methyl acrylate Vinyl acetate

0.0002 0.012

p-Dihydroxybenzene

Vinyl acetate

0.7

1,2,3-Trihydroxybenzene

Vinyl acetate

5.0

2,4,6-Trim ethylphenol

Vinyl acetate

0.5

All values are for 50°Cunless otherwise noted. Source: Data from Ref. 3.

ERp, may be defined

as

2 Since Rpis given by Eq. (6.26), one maywrite 1/2] lnRp = In Ap (Ad~ k,~t ] j + ln[(f[I])l/2tM]] From the slope

and intercept

of the linear

plot of

ERe (6.182)

lnRp versus lIT

we

can determine -/~RP and Ap (Ad/At) 1/2. For most monomers, Ep and Et are in the range 5-10 kcal/mol and 2-5 kcal/mol, respectively, while for most initiators commonly used in thermal homolysis Ed is in the range

530

Chapter 6

30-35 kcal/mol. Therefore, -/~RP is about 20 kcal/mol. A temperature rise of 10°C will thus cause a two- or three-fold increase in the rate of polymerization. For redox initiation, however, Ed ~--10-15 kcal/mol and so /~Rt’ ~-- 10 kcal/mol which is one-half the value for nonredox initiators. In photo-initiation, on the other hand, the energy for the dissociation of the initiator is supplied by the radiation quanta; the initiation is therefore independent of temperature and hence Ed = 0. From Eq. (6.181), ERp "~ 5 kcal/mol. This low value of activation energy signifies that the polymerization rate in photo-initiated systems is relatively insensitive to temperature.

Problem6.36 Whenan organic peroxide is used as the thermal initiator

in the polymerization of styrene, the time it takes to convert 20% of the monomeris 40 min at 70°C and 23 min at 80°C in two experiments in which only the temperature is varied. On the other hand, when styrene~ is photopolymerized, the corresponding times for 20%conversion are 20 min at 60°C and 18 min at 70°C. If the half-life of the peroxide at 60°C is 153 h, what is the half-life at 90°C ? Answer: For polymerization initiated by thermal decomposition of initiators, Eq. (6.182) for two experiments 1 and ln(P~)2

(E RP)therm

we get from

(1 1)

Similarly for photopolymerization [see Eq. (5.81)], in (P~)------~ -- (ERP)ph°t° (~1

(P6.36.2) ~2) Since in photopolymerization, Ed = 0, we get from Eq. (5.181), (P6.36.3) (ERP)therm Ed/ 2 + (E Re)photo The rates of polymerization in two experiments are inversely proportional to the times required to attain the same conversion, i.e.,

R

In {’/~)~ (Rp)l

t~ - In

(e6.36.4)

For the peroxide initiated polymerization, we get from Eqs. (P6.36.1) and (P6.36.4), (ERP)therm

ln(tl

=

-

ln(40 min / 23 min) = 6,749°K (1/343OK - 1/353OK)

Similarly for photoinitiated (ERe)photo R

_ ln(20

polymerization from Eqs. (P6.35.2) and (P5.35.4), min/18

min)

= 1197OK

(1/333°K - 1/343OK)

FromEq. (P6.36.3), Ed/-R = 2(6749°K - 1197°K) = 11,104°K

Rad/cal Cha/n Polymerization

531

Expressing ka by ka = -44 exp(- Ea / RT temperatures T1 and T2,

we obtain for experimentsat two

In[(k~)2 R

(1/TI

- 1/T2)

Since to.5 = ln2 / ka,

h) -- (ll’104~K)(~K to.~ at 90~C = 9.7h

Degree of Polymerization It is obvious from Eq. (6.125) for kinetic chain length (u) that the temperature dependence of u is determined by the ratio kp / (kdkt) 1/2. Using the Arrhenius expression (6.179) for the individual rate constants one obtains

In

kp (kd~)l/z

= In

[Av / (AdAt) 1/2]

- EDp / RT (6.183)

where EDp, the activation energy for degree of polymerization, is given by

= Ep 2Ed

Et

2

(6.1S4)

For ter~nation by coupling or combination [cf. Eq. (6.125)],

lnDP.

= In (Ad-~t)U ~. + In (f[i])l/~

EDp

RT

With the usual values of Ep, Ed, and Et, Eq. (6.184) gives EDp"~ -- 15 kcal/mol. Accordingto Eq. (6.185), DPnwill therefore decrease rapidly with temperature increase. For a purely photochemical polymerization, Ed = 0 and hence EDpis positive, being approximately 5 kcal/mol. The molecular weight therefore increases with increase of temperature. Problem 6.37 Solution polymerization of monomerMinitiated by peroxide P at two temperatures, using the sameconcentrations of Mand P, gave the following

532

Ghapter 6

results : Run A (50°C) 20% 100,000

Conversion in 60 min Initial molecular weight

Run B (65°C) 58% 60,000

Calculate the energy of activation (Ea) for the decomposition of

Approximating (1 - e-kat/2)

in Eq. (6.33)

by kat/2,

(P6.37.1) Only K and ~ depend on temperature. by coupling, 1/2 M. = (DP.)Mo

--

From Eq. (6.125),

assuming termination

K [M] M0 v~u(Y[1])

(P6.37.2)

Taking ratio of (P6.37.1) and (P6.37.2), ln(1 - p) _ ku f[I]°

(P6.37.3)

Run A (50°C):

- ln(O.8O) 100,000

=

(kd)50°

I [~]o(6omin)

(P6.37.4/

Run B (65°C): - 1~(0.42)

f [I]o 60,000 - (ke)~~

(60 rain)

(P6.37.5)

TaNngratio of (P6.37.5) to (P6.37.4), 100,000 1n(0.42)

60, oooln(o.8o)

~ [(~)~ / (~)~1= ~.a~9 = (1.987 cal Ed

tool -1 °K-l)

39.5oK Z3g°K

-! 2.7 x 104 cal mol

Polymerization-Depolymerizafion Equilibrium A question may be asked, that under the prevailing

"why does polymerization occur ?" The answer is experimental conditions the free energy decreases

533

Radical ChainPol~rnerizafion

during polymerization, the extent of reaction being determined by the magnitude of this decrease. The free energy of polymerization is given by AG~, = AH~,TAS~ (6.186) where A(Tp, AHp, and AS:~ are the differences in free energy, enthalpy, and entropy, respectively, between 1 mol of monomerand 1 mol of repeating units in the polymeric product. (The thermodynamicproperties of a polymerization relate only to the propagation step, since polymerization consists of single acts of initiation and termination and a large number of propagation steps.) Chain polymerizations of alkenes are exothermic (A/-/ is negative) because the process involves the exothermic conversion of higher-energy rr-bonds in monomermolecules into lower energy g-bonds in the polymer. The entropy change in polymerization (AS) is also negative which arises from the decreased degrees of freedom (randomness) for the polymer relative to the monomer. It is thus seen from Eq. (6.186) that polymerization is favorable from the enthalpy viewpoint but unfavorable from the entropy viewpoint. According to Eq. (6.186), /k(7 v will become less negative as the contribution of the TASv term increases at higher temperatures. If side reactions due to chemical degradation do not intervene, a temperature Tc may be reached at which AGp= 0. It will therefore not be possible to produce high-molecular-weight polymer at temperatures > Tc (just as liquid does not aggregate into crystals at temperatures above the melting point). In a dynamic equilibrium at To, forward (propagation) and reverse (depropagation or depolymerization) reactions proceed at equal rates. Thus reaction (6.66) should be written more generally (6.187) M ~ Mn+ k@ where ]¢dp is the rate constant for depropagation or depolymerization. Equation (6.15) is then replaced Mn" 1" -I-

diM] Rp - dt - /~p [M] [M’]

-- k,

dp [M’]

(6.188)

The overall effect of temperature on polymerization is complex due to the presence of both propagation and depropagation reactions. Whenthe temperature is initially increased for the polymerization of a monomer, the polymerization rate increases as kp increases. However, at higher temperatures the depropagation rate constant k, dp, which was initially zero, increases and becomes significant with increasing temperature (see Problem 6.38). Finally, a temperature is reached at which the propagation and depropagation rates are equal. This temperature is called the ceiling, temperature, To. So at To, -d[M]/dt -= 0 and, from Eq. (6.188), ]~p [M]e = ~dp, or ]¢p/kdp = g = 1/[Mle (6.189)

534

Chapter 6

-4 6x10

_-

kdp kp[M] e -~/

_ ckp

[M]e’-~~I ,

0

200

400

600

Temperature

Figure 6.14 Variation of kp[M]~ and kdp with temperature for styrene (Problem 6.38).

whereK is the equilibrium constant and [M]e is the equilibrium monomer concentration. One maythus predict that a ceiling temperature will be reached whenkp [M]eequals kdp, regardless of the variations of [M’]. The variations of k~ and ka with temperature will be given by Arrhenius expressions" k~ -- Ap exp(-

E~ / RT)

kd~ = Adp exp(-Edp/

nT)

(6.190) (6.191)

where Ap and Adp are pre-exponential factors and Ep and Edp are the activation energies for propagation and depropagation, respectively; Ep Edp = AHpis the enthalpy change for the overall reaction. It is obviousfromthe abovedefinition Of the ceiling temperaturethat To will dependon the monomer concentration in the system. If polymerization is performed at a temperature T and [M] is greater than [M]e at that temperature, as calculated from Eq. (5:189), then the polymerization wouldproceeduntil [M]falls to the value of [M]e. Conversely,for a given [M]e, a temperature satisfying Eq. (5.189) is the ceiling temperature (To)

Radical Chain Polymerization

535

above which a high-molecular-weight polymer cannot be obtained. The predicted variation of Tc with [M]e can be found by inserting Arrhenius expressions in Eq. (6.189), whence

the

Tc =

zXHp

(6.~9~)

Equation (6.192) shows that there is a series of ceiling temperatures corresponding to different equilibrium monomerconcentrations. For any monomersolution of concentration [Mle there is a temperature at which polymerization does not occur. In fact, for each [M]e there is a corresponding plot analogous to Fig. 6.14 in which kdp = k v [M]e at its Tc. Stated another way, the polymerization of a monomersolution at a particular temperature proceeds until equilibrium is established, that is, until the monomerconcentration decreases to the [M]e value corresponding to that temperature which, in turn, is the Tc temperature corresponding to the particular [M]e. Thus higher monomerconcentrations are required with increasing temperature in order to observe a net production of polymer before equilibrium is reached. There is an upper temperature limit above which a polymer cannot be obtained even from pure monomerat equilibrium. The apparent designation of a singular Tc value, often referred to in the literature as "the ceiling temperature," usually refers to the Tc for the pure monomeror in some cases to that for the monomerat unit molarity. Problem6.38 For pure styrene (density 0.905 g/cm3 at 25°C) calculate ka0 [M]e and k@at different temperatures, and hence determine the ceiling temperature, using the following data [60]: Ap = l0 s L mo1-1 S-I; A@ = 1013 s-l; Ep = 6.5 kcalmol-1; E@= 6.5 + 16.1 = 22.6 kcalmo1-1. Answer: For bulk styrene at 25°C, zL-1) -3) [M]=(1000 cm (0.905 gcm

-1 =8.7mol

L

Assumethat ~]~ = 8.7 mol L-~. The values of k~[M], and kd at any given temperature ~n then be ~lculated from the following expressions: ~(6-5xl0zcalm°l-~) k~ [M]¢ = (10~L mol -~ s -~) [ e~ L(1.987~IV~~K). j (8.7mol

-~)

(P6.38.~) x-1 mo~ [ G~OK -~) (T °K)J (P6.~S.~) ~ = 0°’~ ~-~) ~P[(~.~ Values of G [M],, kd,, and (G [M]~- ka) ar~ ~lculated for different assumed

536

Chapter 6

values of T, starting from roomtemperature (T = 300°K), and these are plotted in Fig. 6.14. The extrapolated curve of (kp [M], ka) cuts the temperature axis at T = 580°K. Therefore, Tc = 580 - 273 = 307°C. Alternatively, since A//p = E~ - Ea = -16.1 kcal tool -1, one obtains from Eq. (6.192), -I) (-16.1 x 103 cal tool

-~) ln[(106 -~ s-1) (8.7mol’-~) /(101~ s(1.987cal°K-1 mol Lmol 1)] = 5S0OK(= 307°c)

An alternative approach to the problem of determining ceiling temperature is based on the recognition of Tc as the temperature at which AGp = 0, and hence from Eq. (6.186),

Tc = AHp/ ASp

(6.193)

where AHp and ASp are the heat and entropy changes under the prevailing experimental conditions. Since Eq. (6.193) contains no reference to the mode of polymerization, Tc is characteristic of the polymer and not of its method of synthesis. For example, the same Tc applies to all polystyrenes of given molecular weight and tacticity, regardless whether they were polymerizedby anionic, cationic, or free-radical initiation. Comparing Eqs. (6.192) and (6.193), one may write ASp = Rln(Ap/ddp)

Rl n[M]e =

AS~ +

Rl n[M]e (6

.194)

where AS’~ is the entropy change for [M]e = 1 mol/L. Therefore, Tc = AHp/(AS~

-t"

Rln[M]e)

(6.195)

This equation implies that at the ceiling temperature (To) the monomer concentration in equilibrium with long chain polymer is [M]e.

Problem6.39 To study the polymerization-depolymerization equilibrium of amethylstyrene, solutions of this monomerin tetrahydrofuran were polymerized [61] at several temperatures using a sodium-naphthalene complexas initiator. The polymerization reaction was allowed to proceed for as long as 2.5 to 65 hours, after whichthe reaction was terminated with water and the polymerprecipitated in methanol, dried, and weighed. The concentration of monomerremaining at equilibrium was obtained from the difference in the original monomer and the polymer formed. The equilibrium concentration values obtained at several temperatures are given below:

537

Radical Chain Polymerization

Time Temperature (°K) (h) 273.2 2.5 7 273.2 5.5 298.2 298.2 22.5 309.2 65 47 320.2 328.2 16 Source: Data from Ref. 61.

Initial monomer Equil. monomer (mol/L) (mol/L) 1.535 0.715 " 0.767 0.737 4.03 2.21 3.44 2.23 5.52 3.50 4.60 4.53 7.57 6.65

Determine the heat of polymerization and the corresponding entropy value. Answer: The data show that solutions with different initial monomerconcentrations give the same equilibrium monomer concentration for each temperature, showing that the equilibrium concentration depends only upon the reaction or ceiling temperature. Equation (6.195) can be rewritten ln[M]~

AHv (

1 )

AS’;

(P6.39.1)

As shownin Fig. 5.15, the given data afford a linear relationship of ln[M], against reciprocal temperature (To = T), in agreement with Eq. (P4.39.1), yielding Slope = AHp/R = - 3.53 × 103 °K AHp= (- 3.53 x 103 °K) (1.987 cal°K-1 mo1-1) ---AS from Eq.(P6.39.1) = 25.0 cal°K-lmo1-1

For polymerization dard states [60],

- 7.0 × 103 cal mo1-1

with the monomer and polymer in appropriate

= T S;

stan-

(6.196)

All conditions for a standard state are fixed except temperature, which is always the temperature of the system. Standard-state properties are therefore functions of temperature only [52]. For equilibrium situation, AGo = 0 and the ceiling temperature may thus be given by T~ : AH~ / AS~ (6.197) Thus the value of Tc really revolves around the actual values of A/-/° and AS° for polymerization. Table 6.13 lists A/-/~, AS~ and calculated values of AG~at 25°C for several olefin and aldehyde polymerization systems. The values of Tc calculated from Eq. (6.197) using AH~ and AS~ values ~5°C, with the assumption that standard enthalpy and entropy changes have no significant temperature dependence, are also shown in Table 6.13. It is

538

Chapter6

Table 6.13 Standard Enthalpies, peratures

Entropies, Free Energies, and Ceiling Temfor Polymerization of Various Monomer-PolymerSystems at 25°C -AS~ -AG~ (cal/deg-mol) (kcal/mol)

Ceiling Temperature (°C) (To = AH~

Monomer

-AH~ (kcal/mol)

Butadiene Ethylene

17.6 21.2

20.5 24

11.5

585

14.0

610

Formaldehyde

7.4

19

1.7

116

Isobutylene

12.9

28.8

4.3

175

Isoprene

17.9

24.2

10.7

466

Methyl methacrylate

13.2

28

4.9

198

a-Methylstyrene

8.4

24.8

1.0

66

Styrene

16.7

25.0

9.2

395

Tetrafluoroethylene

37

26.8

29

1100

Unless otherwise specified, data refer to standard states of pure liquid for the monomer and amorphousor slightly crystalline state for the polymer. ’ Source : C.T. Mortimer, Reaction Heats and Bond Strengths, PergamonPress, NewYork (1962), Chap5; R. M. Joshi and B. Z. Zwolinski, in Vinyl Polymerization, Part I (G. H. Ham,ed.) Dekker, NewYork (1967), Chap. 8; E SI Dainton and K. J. Ivin, Rev. Chem. Soc." (London), 22:61 (1958); T. Ohtsuka and C. Walling, J. Am. Chem.Soc. 88:4167 (1966).

539

Radical ChainPolymerization

seen from Table 6.13 that while A/-/_* has a wide range of values, AS°_ vanes httle between monomers. The range of AS~ values (Table 6.13) from -19 to -28 cal/deg-mol. The reason for the narrow range is that the dominant factor in A.S~ is the loss of translational entropy caused by the large reduction in the number of molecules present, and this factor is relatively constant from system to system. The ceiling temperature is therefore dominated by the magnitude of A_H~. Vinyl monomerswith 1,1° values and hence lower Te than disubstitution generally have lower AH_ the corresponding singly substituted anaffogs. Thus the calculated Tc values of polystyrene and poly(c~-methyl styrene) are 395°C and 66°C, respectively. Solving Eq. (6.195) for [M]e yields in

[M]e - AHp RT~

AS~ R

(6.198)

which shows the equilibrium monomerconcentration [M]e as a function of the reaction or ceiling temperature Te. Equation (6.198) can be used calculate how much monomerwill be in equilibrium with high-molecularweight polymer at any temperature. This information is useful because many monomersare toxic or have offensive odors and it is often necessary to limit their concentrations in their polymers. Table 6.14 shows the monomer concentrations at 25°C for a few monomers.The data do indicate that the polymer obtained in any polymerization will contain some concentration of residual monomeras determined by Eq. (6.198).

Table 6.14

Polymerization-Depolymerization Equilibria

Monomer [Mlc at 25°C -6 Styrene 1 x 10 c~-Methylstyrene 2.2 -9 Methylacrylate 1 x 10 .3 Methylmethacrylate 1 x 10 .9 Vinylacetate 1 x 10 Ethylene Isobutene Source:Datafromvarious sources.

T~ for pure monomer(°C) 310 61 220 400 50

Problem6.40 The standard enthalpy of polymerization (AH~)of vinyl chloride at 25°C is -72 kJ/mol and the standard entropy of polymerization (AS~)can taken to be approximately--100 J/°K-mol. Canthe polymerbe safely used at the

Chapter 6

540

ambient temperature (25°C), in view of the fact that vinyl chloride is carcinogenic Answer: For T~ = 298°K, Eq. (6.198) gives (-72

in [M], =

(8.314

× 103 J mo1-1) J °K-’ mol-’)

(100 J °K-1 mo1-1)

(a00°K) (8.314 J °K-1 mol-’)

= -16.84 or [M], = 4.9×10 -s -1 tool L So only a negligible concentration (< 0.0005 ppm) of monomerwill be in equilibrium with high-molecular-weight polymer at 25°C. If unreacted monomercan be purged from the polymer, no significant concentration will develop thereafter at room temperature because of the polymerization-depolymerization equilibrium.

Problem6.41 If a free-radical

polymerization of 1.0 M solution of methyl methacrylate was being carried out at 100°C, what would be the maximumpossible conversion of the monomerto polymer, that is, till the polymerizationdepolymerization equilibrium is reached ? (Take relevant data from Table 6.13.) Answer: To calculate an approximate value of [M]~, consider the monomerand the polymer in their standard states and assume that there is no significant temperature dependence of enthalpy or entropy of polymerization so that the AH~and AS~ data of Table 6.13 can be used. For Tc = 373 °K, Eq. (6.198) then yields, (-28 cal °K-’ mol-’) (-13.2 x 103 cal mol-’)

in

--

-

(1.987 cal °K-1 mo1-1) (373 °K) (1.987 cal °K-’ mol-’) = - 3.72 [M]¢ = 0.024 mol L-’ Maximum conversion = (1 - [M]~)x 100 97 .6%

Problem 6.42 Using the AHi~and AS~ data for styrene from Table 6.13 calculate kp at 60°C the depolymerization rate constant of polystyrene at 100°C. [Data: = 165 L/mol-s; Ep = 26 kJ/mol.] Answer: From the Arrhenius expression for k~, o (k/~)100 (26 x 103 J mo1-1) [ 1 In (165 L mol-’ s-’) = (~--..~1~ oK-X mol-’)~335°K (kp)100o

l

]

37~°K

= 452 L mo1-1 s-’

For Tc = 373 °K and assuming no significant temperature dependence of enthalpy and entropy of polymerization, so that AH~and AS~ values at 25°C from Table 6.13 can be used, Eq. (6.198) yields (25.0 -a cal °K-1 mol (- 16.7 x 10z cal mo1-1) In [M]~ = + -~) -1) (1.987 cal °K-1 mol (373 °K) (1.987 cal °K-’ tool = -9.95

Radical Chain Polymerization

541

[M]~ ---- 4.77x10 -2 -1 mol L From Eq. (6.189),

kd, = k,[Mh = (452L mo1-1 s-1)(4.77× -2 mo l L-1) = 2.15 x 10-2 -~ s

While for many alkene monomers the position of the propagationdepropagation equilibrium is far to the right under the usual reaction temperatures employed(that is, there is essentially complete conversion of monomerto polymer for all practical purposes), there are some monomers for which the equilibrium is not particularly favorable for polymerization. For example, o~-methylstyrene in a 2.2 Msolution will not polymerize at 25°C and pure o~-methylstyrene will not polymerize at 61°C (see Table 6.14). In the case of methyl methacrylate, though the monomercan be polymerized below 220°C, the conversion will be appreciably less than complete. For example, the value of [M]e at 110°C is found to be 0.139 M [3] which corresponds to about 86%conversion of 1 Mmethyl methacrylate. Since Eqs. (6.195) and (6.196) contain no reference to the of initiation, they apply equally well to ionic and ring-opening polymerizations. Thus the lower temperatures of ionic polymerizations often offer a useful route to the polymerization of many monomers that cannot be polymerized by radical initiation because of their low ceiling temperatures. Though a monomercannot be polymerized above its ceiling temperature, it should not be assumed that a polymer will depolymerize and thus be useless above the ceiling temperature. A dead polymer that has been removed from the reaction media or made at a temperature below Tc will, in fact, be stable and will not depolymerize unless an active end is produced by bond cleavage of an end group or at some point along the polymer chain. Whensuch an active site is produced by thermal, chemical, photolytic or any other means, depolymerization (unzipping) may follow until the monomerconcentration becomes equal to [M]e for the particular temperature. Poly(oz-methylstyrene) with Tc = 61°C and poly(methyl methacrylate) with Tc ~ 165°C exhibit such behavior. For a polymer with this behavior, depolymerization is a valuable method for recovering scrap quantities of the polymer as monomer. The thermal behavior of many polymers, however, is much more complex. Moreover, degradative reactions other than depolymerization often occur at temperatures below the ceiling temperature without yielding major quantities of monomer. It should be noted that Eq. (6.198) gives no information as to how quickly the equilibrium monomerconcentrations will be attained at the ceiling temperature. Polymers may in fact be quite useful above their ceiling temperatures if depolymerization processes are kinematically hindered. For example, poly(formaldehyde) h.as a Tc of 126°C, but the polymer can be made sufficiently stable for melt processing at temperatures above 200°C. This may be achieved by esterifying or etherifying the thermolabile hydroxyl ends of the macromolecule, copolymerizing with small concentrations of ethylene oxide, and using basic additives as stabilizers. The under-

542

Chapter6

lying principle of these expedients is to retard the initiation or propagation steps of chain reactions that could "unzip" the polymer to monomer.

MOLECULARWEIGHT DISTRIBUTION The molecular weight distributions in radical chain polymerization are more complex than those in step-growth polymerization (Chapter 5). This is because radical chain polymerization involves several possible modesof chain termination-disproportionation, coupling, and various chain transfer reactions. The situation is further complicated by the fact that the molecular weight of the polymer produced at any instant varies with conversion due to changes in the monomerand initiator concentrations and in the propagation and termination rate constants. However, at low conversions these kinetic parameters remain approximately constant and the polymer molecular weight does not change with conversion. Molecular weight distributions can thus be relatively easily calculated for polymerizations restricted to low conversions.

Low-ConversionPolymerization Weassume here that the concentrations of monomerand initiator remain essentially constant during polymerization and that any dependence of termination rate constants on polymer chain size and concentration or autoacceleration effects can be neglected. Molecular weight distributions derived under these conditions will not obviously apply to commercial polymers, whose polymerizations are often finished at high conversions. These highconversion polymers mayhave distributions that differ from those calculated here. A later section discusses the size distributions of such polymers. A given monomer-ended radical may add a monomer molecule or undergo chain transfer or termination. The probability that it will add monomer is

P =

/~ P~ + Rtr + Rt

(6.199)

where /~p, /:~tr, and -~tr are the rates of propagation, chain transfer, and termination, respectively. These are given by Rp

=

k v [M] [M’] ntr = ktr,M[M’][M] q- ]~r,I[M’][I] /~t = 2 (ktc q- ktd)[M’] 2

(6.200) q- k, tr,S[M’][S]

(6.201) (6.202)

Note that the effects of any chain transfer agent are included in the term for solvent and that transfer to polymer is not included in the expression for R.tr because such transfer can be significant only at high conversion where sufficient polymer can be present.

Radical ChainPolymerization Termination

543

by Disproportionation

and/or

~ransfer

Wefirst consider the polymerization where each kinetic chain yields one polymer molecule. This happens when the growth of microradical chains is terminated by disproportionation and/or chain transfer (i.e., ktc = 0). The situation here is completely analogous to that for linear, reversible stepgrowth polymerization described in Chapter 5. If we select an initiator fragment at the end of a macromolecule, the probability that the monomer molecule picked up by this initiator radical has added another monomer molecule is P. Continuing in this way the probability that x monomer molecules have been added one after another is px-1 (see p. 347). Since the probability that the radical end of a growing chain has terminated is (1 -- P ), the probability that the macromolecule under consideration consists of essentially x monomerunits is _p~-I (1 -- P). Since such propabilities are equal to the corresponding mole fraction of this size molecule, n~, we have the expression nx = (1

--

P)P~-I

(6.203)

for the number distribution function. The weight distribution function wx is also given by direct analogy to that for linear, equilibrium step-growth polymerization (p. 348)

w~ = x (1 - P)2PX-1

(6.204)

By analogy with l~qs. (5.77), (5.80), and (5.81) we can D--P,~ ----

1/(1 -- P)

DPw = (1

+ P)/(1

DP~/DPn

= 1 +P

(6.205) (6.206)

-

(6.207)

Substituting t~q. (5.199) into l~q. (5.205) and noting that for most addition polymerizations, P~p )) Rtr + /~t (or high-molecular-weight polymer would not be formed),

DPn =

P~ + Rtr + Rt Rv - Rt~ + R~ -/~tr + Rt

(6.208)

Insertion of I~qs. (5.200)-(6.202) into Eq. (6.208) and simplification (noting that ktc = 0) gives

1 DPn

-- CM -]-

CI,--,

-I-

[[MI_~]]

IS]

CS~-~ -1-

2/~d

k~[M]2Rp

(6.209)

which can be easily shownto be identical with lEq. (6.148). A consideration of Eqs. (6.203) and (6.204) indicates that highmolecular-weight polymer (i.e., large DPnand DPw)will only be produced

544

Chapter 6

if 2° is close to unity, i.e., if P~p >> }~tr -I- /~t. Equation (6.207) indicates that the size distribution D--~w/ DPn(also referred to as PDI, the polydispersity index) has a limiting value of 2 as 20 approaches unity. The situation is thus analogous to that for linear step-growth polymerization considered in Chapter 5 [cf. Eq. (5.81)]. There is, however, an important difference between the distributions calculated for equilibrium, bifunctional step-growth polymerization in Chapter 5 and for the free-radical polymerizations with termination by disproportionation and/or chain transfer that are being considered here. Thus p in Eq. (5.81) is the extent of conversion, while 20 in the above equations the probability that a propagating radical will continue to propagate instead of terminating. There is a very important second difference. While the distribution functions in the step-growth case apply to the whole reaction mixture, in the free-radical polymerization this distribution applies only to the polymer fraction of the reaction mixture. Termination

by Coupling

To consider another extreme, suppose that each polymer molecule consists of two kinetic chains, that is, termination occurs only by coupling. Each polymer molecule thus consists of two sequences of units which grew independently and were joined together at their mutual termination. They are analogous to the dichain (f = 2) condensation polymers discussed Chapter 5 (p. 367). Further advantage may be taken of this analogy replacing p by 20, defined as the probability of continuation of either chain from one of its units to the next. Then 1 -- _P is the probability that the i-th unit in the growth of the chain reacts by termination; it is therefore equal to the ratio of terminated to total units. Since two units are involved in each termination step, one can write 1

--

P -

Rt

_

2k¢c

[M’]

Rv + Rt kp[M] + 2ktc[M’]

(6.210)

where ktc is the rate constant for termination by coupling. Substitution of [M’] ----- P~p / kp [M] from Eq. (6.15) into Eq. (6.210) and simplification gives

1_ 1+

(6.211)

Then, since each molecule consists of two chains, the number-average degree of polymerization will be given by [cf. Eq. (6.205)],

_

2 - 2 + (1- 20)

[M]

(6.212)

RadicMChainPolymerization

545

For the weight-averagedegree of polymerization, the_following equation can be shownto be applicable: 2+P D---~, -- 1 -- P (6.213) Hence,

= (2 + P)/2

(6.214)

The ratio in Eq. (5.214) has a limiting value of 1.5 at high polymer molecular weights whenR,p >> Rt and P approaches 1. This is narrower than the distribution producedin the absenceof termination by coupling. Thenumberfraction and weight fraction distributions [63] are readily shownto be n~ = (x - 1)(1 - P)eP~-~ Wx-~ ~127(a~ -- 1) (1 _ p)Z

(6.215) (6.216)

whereeach initiator radical is countedas a unit. [Notethat Eq. (6.216) obtained simply by multiplication of Eq. (6.215) by x/~--~n.] These expressions correspond nearly to Eqs. (5.115) and (5.116) Chapter 5 for dichain polymers(f = 2). Minordifferences occur because the latter contain a single central unit, whereastwo mustbe countedin the present case. Termination by Coupling, and Chain Transfer

Disproportionation

For polymerization wheretermination occurs by all three modes, namely, coupling, disproportionation and chain transfer, one can obtain the size distribution by a weightedcombinationof the abovetwo sets of distribution functions. For example,the weightdistribution can be obtained as w~ = wx(1-P)2P~-I

+-~ l(l_w)x(x_l)(

1 p)apz-2 (6.217)

wherew is the fraction of polymermaterial formedby disproportionation and/or chain transfer reactions [64]. Anexpression for P can be obtained by inserting Eqs. (6.200)-(6.202) into Eq. (6.199) and using the relation [M’] = Rp / kv[M]. This gives 1 p --

i

[I] IS] 2(k, tc q- ~d) + CM~ + Cs~ + k~ 2[M]~ /~

(6.218)

546

Chapter6

The valueof w is givenby 2]¢¢r,M[M’][M]-t- ktr,I ktr,M[M’][M] -1- ktr, CM[M] -{- CI[I] CM[M]q- CI[I]

[M "] [I] -]- ~r,s[M’][S]-1- 2ktd[M’] I[M’][I l 2q- kcr, s[M’][S] q- 2(kt¢ q-ktd)[M’]

d- Cs[S l -{- 2~dP~p/kp2[Ml

q- Cs[S] -b 2(k,

tc q-

(6.219)

To calculate _P and w, and hence wz, from Eq. (6.217)~ from kinetic and theoretical considerations for any polymer system, it is thus necessary to have the values of transfer constants (CM, CI, Cs), 2 /k n, le t, an d

Problem6.43 The bulk polymerization

of methyl methacrylate (density 0.94 g/cma) was carried out at 60°C with 0.0398 Mbenzoyl peroxide initiator [64]. The reaction showed first order kinetics over the first 10-15%reaction and the initial rate of polymerization was determined to be 3.93 x10-4 mol/L-s. From the GPC molecular weight distribution curve reported for a 3% conversion sample, the weight fraction of polymer of D---~ = 3000 is seen to be 1.7x10-4. Calculate the weight fraction from Eq. (6.217) to compare with this value. [Use the following data: CI _~ 0.02; CM = 10-5, fka = 2.7 x 10-6 s-l; kt = 2.55 x 107 L/tool-s; fraction of termination bydisproportionation = 0.85] A~lswer : From Eq. (6.26), [M]2 (fka) [I]

kt

(3.93 x 10-4 tool L-1 2s-1) (9.4 tool L-l) ~ (2.7 × 10-6 S-1) (0.0398 mol

=

-~ = 0.0163 L mo1-1 s

In as much as polymerization rates are measured at less than 10% conversion, it can be assumed that [M] and [I] in Eq. (6.218) are equal to their initial concentrations. Therefore, -~) --1 = 1 + 10 -5 + (0.02)(0.0398 molL -1)/(9.4 molL P +

(2) (3.93 x -4 mol L - ~ s - 1) (0.0163 L tool -1 s -1) 2(9.4 mol L-l)

= 1.0006 P = 0.9994 Since ktd / (kte -}- ktd) -~- 0.85 and knc +ktd ---- kt = -1, 2.55 × 107 L mo1-1 s it follows that kid = 2.167x10 r -~ Lmo1-1 s

547

Radical Chain Polymerization

k:/~d = Ck:/k)C~/~,d) = (0.0163L mo1-1 s-1)(2.55 x 107/2.167 x 107) = 0.0t92L mo1-1 -1 Substituting the appropriate values in Eq. (6.219) yields ~ = 0.871. Therefore, from Eq. (6.217), w3000 = (0.871)(3000)(1 - 0.9994) 2 ~°°°-1 (0.9994) + ½(1 - 0.871)(3000)(3000 - 1)(1 - 3 (0.9994 3°°°-2 = 1.8 × 10-4 -4 (el.) reported value = 1.7 × 10

Problem6.44 Consider a case of free-radical

polymerization where termination involves both disproportionation and coupling of chain radicals but chain transfer reactions can be neglected. Derive an expression for the distribution function for the degree of polymerization of polymer in terms of the kinetic chain length and the ratio of termination by disproportionation to that by coupling [65]. Simplify the expression for two limiting cases where (a) termination is solely by coupling and (b) termination is solely by disproportionation. Answer: Applying the steady-state radicals may be written:

approximation, the rate equation for propagating :r-mer

at - ~ [M][M~-x’]- ~ [M][M.’]- 2~ [M~’][M’]= 0 (P6.44.1) After division by [Mz-l’], followed by rearrangement, Eq. (P6.44.1) yields [M,-] _ ¯ [M,_,’]

k~[M] = (1+ 2k’[[M--M~])-~ k. [M] 9- 2k, [M’] k.

(P6.44.2)

From Eq. (6.123), kinetic

chain length,

~, -- ks [M] 2/% [M-]

(P6.44.3)

Equation (P6.44.2) may thus be written conveniently

An expression for the ratio [M~-] / Nil may be obtained plication of the ratios in (P6.44.4). Thus,

by successive multi-

or

If a~ is defined as the fraction of propagating radicals that have a degree of polymerization z, then, in view of Eq. (P6.44.6), wehave the relationship:

[M.]

(P6.44.7)

Chapter ~

548

Applying the steady-state approximation to the concentration of the smallest propagating radical, [MI"], and to the total radical concentration, [M’], one obtains -- Ri - k~o [M] [M,-] -- 2~ [M~’] [M’] = 0

dt

-- Ri- - 2kt[M’l 2 = 0

dt

(P6.4A.8)

(P6.44.9)

Equation (P6.44.8) may be rearranged [Ml’] = k~ [M] + 2k~ [M-]

(P6.44.10)

and Eq. (P6.44.9) [M’] - 2k~ [M’] Taking the ratio (6.44.3) gives [M~’] [M’]

_

(P6.44.11)

of Eq. (P6.44.10)

2hn [M’] k~ [M] + 2/~ [M-]

to Eq. (P6.44.11)

and combining with

= (1 + u)-~

(P6.44.12)

Substitution of Eq. (P6.44.12) into Eq. (P6.44.7) then yields a~

= -

1 +

(P6.44.13)

This equation represents the distribution of the degree of polymerization of the propagating radicals. The distribution of the degree of polymerization of the polymer will, however, be determined by a, and by the mechanismof termination of the propagating radicals, as shownin the following. The mole fraction of polymer molecules consisting of z mer units will be given by the rate ratio d[P,] / dt d[Pl /dt

n~ -- dips] / dt Z~ d[P~] / dt

(P6.44.14)

where the denominator represents given by dIP]

/ dt = (k,~

the total rate of formation of polymer and is "~ + 2k, d) [M’] z (P6.44.15)

(The factor of 2 appears because two polymer molecules are formed in termination by disproportionation.) To evaluate the numerator in Eq. (P6.44.14) we need only

Radical Chain Polymerization sum the rates of all the reactions in which P~ is formed. These reactions are: Coupling

Disproportionation

M=-i" + M1° ~ P= M=-2" + M2" ~ P~

M=" + Ml" -~ P, + P1 M=" + M2" ~ P, + P2

:

M=_,~"

(P6.44.16) (P6.44.17)

;

+ M~" ~ P~

MI" + Mz-l"

M~" + M,~" ~ P~ + P,~

(P6.44.18)

~ P~

Weshould note that in summingthe rates of the coupling reactions in (P6.44.16) to (P6.44.18), every reaction will be counted twice, with one exception. The one exception is that reaction in which m = x - m = z/2, and such a reaction between like radicals is possible only if the integer z is even. If z is even, then

at = ~,o[M=/2.][M=/~.]+½ ~ /~:o[M._r~.][M,~.]+~/~[M,’][M’] (P6.44.19) where k~ is the coupling termination rate constant for like radicals and k’t~ that for unlike radicals; the factor 1/2 corrects for the addition of each reaction between unlike radicals twice in summingthe coupling reactions (P6.44.16) to (P6.44.18). maybe noted that the rate constant for termination of unlike radicals is twice that for like radicals-a conse~quenceof the relative collision frequencies of like and unlike species - that is, kte = 2ktc. Therefore Eq. (P6.44.19) may be written dt

-- kt~ ~ [M=_,~’] [M,~’] + 2k,d [M=’] [M’]

(P6.44.20)

There can be no combination of like radicals to produce a polymer having an odd numberfor x. Equation (P6.44.20) is therefore generally applicable, i.e., both for even and odd values of x. Substitution of Eqs. (P6.44.7) and (P6.44.12) leads iv 2 1 +

d~

~-~.(1) "~ + -- 1 +

[M’]* (P6.44.21)

m=l

The summation is equal to (n - 1), so we have

Dividing by Eq. (P6.44.15), one finally obtains the mole fraction of polymer of met units, that is, 1

(

~)-=

(z-l)/v

+

Equation (P5.44.23) showsthat the distribution function fo~ the degreeof polymerization of polymer formed by a free-radical chain mechanism,in which chain transfer is absent, depends only on the k~netic chain length andthe ratio of disproportionation to coupling. The plot of n~ vs. x accordingto Eq. (P6.44.23) gives the

550

Chapter 6

number or mole fraction distn’bution of the degree of polymerization. The molecular weight distribution is easily obtained from Eq. (P6.44.23) since M where -Mo is the, molecular weight of the monomer. (a) Case 1: Termination is solely by coupling, i.e., thus reduces to

/eta = 0. Equation (P6.44.23)

Since /2 = /~//~t and P~ >> Rt for most addition polymerizations, easy to see that Eq. (P6.44.24) is equivalent to Eq. (6.215). (b) Case 2: Termination is solely by disproportionation, (P6.44.23) reduces n~

i.e.,

it is

/~, = 0. Equation

=

+ (P6.44.25) 1(~ It is e,hsy to see that Eq. (P5.44.25) is equivalent to Eq. (5,203) when termination is solely by disproportionation (i.e., R~r = 0).

Problem6.45 Show graphically

the number distribution of degree of polymerization for (a) termination by coupling and kinetic chain length (v) values of 300 and 400, and (b) for/2 = 300 and ktd/ktc = O, 0.2, 0.4, 1.0, and Show that there is a value of ~c for which the mole fraction of polymer with DP = z is independent ofkt~/ktc and this value is ~: = /2+ 1. Answer: The functional dependence of the distribution function in Eq. (P6.44.24) for termination solely by coupling is shown in Fig. 6.16(a) for v = 200, 300, and 400. The distribution is seen to become broader at higher values of v. The number distribution of degree of polymerization for v = 300 and different values of ktd/~c is plotted according to Eq. (P6.44.23) in Fig. 6.16(b). It seen that all the distribution curves pass through the same point. This means that there is a value of x for which the mole fraction of polymer with DP = z is independent of ~/ktc. To determine this value of z analytically, Eq. (P6.44.23) may be differentiated with respect to k~d/ktc and equated to zero. This gives

Problem6.46 Derive weight-fraction

distribution of degree of polymerization, wz, from the number fraction distribution, n~; described by Eq. (P6.44.23). Answer: Let N = total number of moles of polymer molecules; M0= molecular weight of monomer; DP, = average degree of polymerization of polymer molecules. Then Number distribution, N~ = n~N Weight of x-mers, W~ = xN~Mo = xn~NMo Total weight, W = N(~-~,~)Mo

Radical ChainPolymerization

2.0 [

551

9 : 200 (a)

1.5

0.5 0

200

/-.00 x

600

800

:2.0

~

~.o ~.5~ o¯5I L/~.~ /

I/-I

0

I

~~~~o.2 ~~o,~ I

I

200

I

I

~00

I

I

600

/0.0

I

800

X

Figure 6.16 Numberdistribution of degree of polymerization (Problem 6.45) (a) termination by combinationfor several values of kinetic chain length (u); distribution for u = 300 and several values of kta/ktc shownon the figure.

Weight fraction

of x-mers, wx -- W= n=

Since DPnis equal at any instant to the ratio of the rate of monomer disappearance to the rate at whichcompletedpolymermolecules are produced,that is, D~,,-- - diM] / dt d[P] / dt Combinationof this equation with Eqs. (6.20) and (P6.44.15) yields DP’--~=

k~, [M]

(k,o + ~,~)[M.]

(P6.46.2)

(P6.46.3)

552

Chapter 6

Substituting for [M-] in this equation from Eq. (6.123) and for ~ from Eq. (6.12) one then obtains -(P6.46.4) DPn = 2v ~ -l(k,t~ + kta’~ Substitution of Eqs. (P6.44.23) and (P6.46.4) into Eq. (P5.45.1) yields weightfraction distribution --

~ - 2~2 1 + ~ (x 1) / ~ + 9.(~d/~o)

(r6.a6.5)

Note that Eq. (P6.46.5) is equivalent to Eq. (6.216) for kta = 0 and P~ >>

High-Conversion

Polymerization

Molecular weight distributions in high conversion polymerizations are not nearly as predictable as those in low-conversion reactions and they also vary with polymerization conditions and from monomer to monomer. The size distributions for high-conversion polymerizations becomemuchbroader than those described above for incremental or low-conversion polymerization. The polymer molecular weight depends on the ratio [M]/[I] 1/2 according to Eq. (6.125). As a polymerization proceeds, [I] usually decreases faster than [M] and the molecular weight of the polymer produced at any instant increases with the conversion. The overall molecular weight distributions for high-conversion polymerizations are quite broad with DPw/~-~n ratio being in the range 2 to 5. Whenautoacceleration or gel effect occurs in polymerization there is even larger broadening. Since autoacceleration results from a reduced termination rate resulting in large increases in the kp/le~/2 ratio, it is always accompaniedby large increases in the average molecular weight and the breadth of the distribution. Thus, DPw/~---~n values as high as 5 to 10 may also be observed when the gel effect is present. If chain transfer to polymer can occur, this will be most significant at higher conversions when the polymer concentration is high. This results in a further broadening of the molecular weight distribution and can lead to D---~o / ~-Pn ratios as high as 20 to 50. This very extensive broadeningoccurs because chain transfer to polymer increases as the polymer size increases and branching caused by chain transfer thus leads to greater branching as polymerization proceeds. Broad molecular weight distributions are usually not desirable from the practical viewpoint since optimum values of polymer properties are usually obtained at specific molecular weights. Addition of multiple charges of initiator and/or monomerduring the course of polymerization is often practiced in commercial polymerization processes to minimize the molecular weight broadening due to change in [M] and [I]. It is, however, more difficult to minimize molecular weight broadening due to the gel effect and chain

Radical Chain Polymerization

553

transfer. Thus, for example, low temperatures minimize chain transfer polymer but maximize the gel effect.

to

POLYMERIZATION

PROCESSES

Free-radical polymerizationsare carried out by a variety of processes .that require different design considerations with respect to recipe of polymerization and physical conditions for the process and process equipment. Generally free-radical polymerizations are carried out by four different processes: (a) bulk or mass polymerization, (b) solution polymerization, (c) suspension polymerization, and (d) emulsion polymerization. Of the four processes, the last two are essentially of the heterogeneous type containing a large proportion of nonsolvent (usually water) acting as dispersion mediumfor the immiscible liquid monomer.Bulk and solution polymerizations are homogeneous processes, but some of these homogeneous systems may become heterogeneous with progress of polymerization due to the polymer formed being insoluble in its monomer(for bulk polymerization) or in the solvent used to dilute the monomer(for solution polymerization). Though all monomerscan be polymerized by any of the various processes, it is usually found that commercial polymerizatio~ of any one monomeris best carried out by one or two of the processes. The laboratory techniques for carrying out polymerizations have been discussed [66,67]. The kinetic schemes described in this chapter will apply to free radical polymerizations in bulk monomer,solution, or in suspension, but the kinetics of emulsion polymerization, to be discussed later, are different. Bulk Polymerization Polymerization in bulk, that is, of undiluted monomer,offers the simplest process with a minimumcontamination of the product. Thus for radical chain polymerization in bulk, the only additive to the monomer is an initiator and that again decomposesand reaches almost vanishing concentration at the end of the polymerization; the process thus results in optically clear polymers such as polystyrene and poly(methyl methacrylate). Bulk polymerization difficult to control, however, due to the characteristics of radical chain polymerization, namely, their highly exothermic nature, the high activation energies involved, and the tendency toward the gel effect, which combine to makeheat dissipation difficult. For dissipation of the heat liberated with progress of polymerization, continuous stirring of the reaction system is essential, but the stirring process and heat dissipation becomeprogressively difficult due to rapid increase in viscosity of the reaction system. The viscosity and exotherm effects which often become more acute due to the phenomenon of autoacceleration or gel effect make temperature control difficult . Local hot spots may occur and this may lead to discoloration, thermal degradation, branching, development of chain unsaturation or even cross-linking, thus giving rise to irreproducible and often inferior product

Chapter 6

554

quality. In the extreme case, uncontrolled acceleration of polymerization rate can also lead to disastrous runawayreactions. Because of the aforesaid problems and disadvantages, bulk polymerization is not used for chain polymerization as muchas for step polymerization. It is, however, used in the polymerization of ethylene, styrene, and methyl methacrylate. The heat dissipation and viscosity problems in such cases are circumvented by two approaches: (i) by carrying out the polymerization to low conversions with separation and recycling of unreacted monomer,as in the high pressure polymerization of ethylene using tubular reactor; and (ii) by accomplishing the polymerization in stages-to low (2030%)conversion in a large stirred reactor and to final conversion in thin layers (either on supports or free-falling streams), as in the preparation acrylic castings from methyl methacrylate monomerand in the manufacture of polystyrene from styrene. Solution

Polymerization

By carrying out the polymerization of a monomerin a solvent manyof the disadvantages of the bulk process can be avoided. The solvent acting as a diluent reduces the viscosity gain with conversion, allows more efficient agitation or stirring of the medium, thus effecting better heat transfer and heat dissipation. Although thermal control is much easier in solution polymerization compared to bulk polymerization, the solution method has its own demerits. The method often requires handling of flammable or hazardous solvents and removal or recovery of the solvent to isolate the polymer after the polymerization is over. The purity of the polymer may suffer due to retention of last traces of solvent in the isolated product. Unless the solvent to be used is chosen with appropriate consideration, chain transfer to solvent may also pose a problem..Solution polymerization is, however, advantageous if the polymer formed is to be applied in solution as in the case of making of coating (lacquer) grade poly(methyl methacrylate) resins from methyl methacrylate and related monomers. Suspension

Polymerization

Suspension polymerization is designed to combine the advantages of both the bulk and solution polymerization techniques. It is one of the extensively employed techniques in the mass production of vinyl and related polymers. Suspension polymerization (also referred to as bead or pearl polymerization) is carried out by suspending the monomer as droplets by efficient agitation in a large mass (continuous phase) of nonsolvent, commonlyreferred to as the dispersion or _suspension medium. Water is invariably used as the suspension mediumfor all water insoluble monomers because of the manyadvantages that go with it. Styrene, methyl methacrylate, vinyl chloride, and vinyl acetate are polymerized by the suspension

555

Radical Chain Polymerization

process. The size of the monomer droplets usually range between 0.1-5 mm in diameter. Suspension is maintained by mechanical agitation and addition of stabilizers. Lowconcentrations of suitable water-soluble polymers such as carboxymethyl cellulose (CMC)or methyl cellulose, poly(vinyl cohol), gelatin, etc., are used as suspension stabilizers. They raise the mediumviscosity and effect stabilization by forming a thin layer on the monomer-polymerdroplets. Water insoluble inorganic compounds such as bentonite, kaolin, megnesiumsilicate, and aluminumhydroxide, in finely divided state, are sometimes used to prevent agglomeration of the monomer droplets. Initiators soluble in monomer,such as organic peroxides, hydroperoxides or azocompounds-oftenreferred to as oil-soluble initiatorsare used. Each monomerdroplet in a suspension polymerization thus behaves as a miniature bulk polymerization system and the kinetics of polymerization within each droplet are the same as those for the corresponding bulk polymerization. At the end of the polymerization process, the monomer droplets appear in the form of tiny polymer beads or pearls and hence, the process is also knownas bead or pearl polymerization. The polymers are filtered, washedprofusely with water to removethe stabilizer as far as practicable, and dried. They, however, usually retain traces of stabilizers besides the residual initiator as contaminants. Heat and viscosity control in suspension polymerization is relatively easy compared to bulk polymerization. Another important advantage of the method is that the polymer product can be obtained directly in spherical bead form (which may subsequently be functionalized to make ion-exchange resins). Problem 6.47 A monomeris polymerized at 80°C (a) in benzene solution and (b) in aqueous suspension in two separate runs, both containing 60 g the monomer (density 0.833 g/cm3) and 0.242 g of a peroxide initiator in a total volumeof 1 liter. If the initial rate of polymerizationfor 1 liter of solution is 0.068 mol/h, whatis the expectedinitial rate for 1 liter of suspension? (Assume that rate constants and the initiator efficiency are samein both cases.) Answer: Let MA= molar mass (g/tool) initiator I. (a) Solution (1 liter):

of monomerA; MI = molar mass (g/mol)

[M], mol/L = 60/MA, I[I],

mol/L = 0.242/M

(b) Suspension(1 liter): (1000 cm3) -z) (0.833 gcm 833 M. M. ~ 1) (0.242 g)(lO00 L mol/L = (MI g mol-1)(60 g/(0.833 -a )

[M], mol/L = [I],

3.36

MI

556

Chapter 6 (Rp)a = -dt d[M]~ ]a = kp(fkd/kt)l/2[Mla[II~a/2

(R~)b = kp (fkd / kt) 1/2 [Mlb[I]~/~ (R=o)b (8 33/MA)(3.36/MI) 1/2 = 51.7 1/~ (R~)a (60/MA) (0.242/MI) (/~)b = 51.7 (0.068 molh-1 L-1) = 3.5 molh-~ L-’ For 1 liter suspension (72 cm3 monomer),rate = 3.5×0.072 or 0.252 mol -1

Emulsion

Polymerization

Emulsion polymerization refers to a unique process employed for some radical chain polymerizations. It involves polymerization of monomersin the form of emulsions. Like suspension polymerization, the emulsion process uses water as a heat sink. Polymerization reactions are thus easier to control in both these processes than in bulk systems because stirring is easier and removal of the exothermic heat of polymerization is facilitated. Emulsion polymerization, however, differs from suspension polymerization in the type and smaller size of particles in which polymerization occurs and in the kind of initiator employed. The process is also quite different from suspension polymerization in its mechanismand reaction characteristics. The emulsion method of polymerization was developed in the United States during the 1940s for the production of synthetic styrene-butadiene rubber when the supplies of natural rubber were cut off during World War II. The.process is now extensively used for polymerizing conjugated dienes such as butadiene and isoprene and also for polymerizing vinyl monomers such as styrene, vinyl chloride, vinyl acetate, acrylates, and methacrylates to produce homopolymersand copolymers. Emulsion polymerization yields a stable water-dispersed product of the polymer in the form of colloidal size particles. Someof the advantages of the method are: (a) thermal and viscosity problems are much less severe than in bulk polymerization; (b) in several applications, such as in the formulation of latex paints, coatings, finishes, adhesives, and floor finishes, the products of emulsion polymerization can be directly used without further separation from the suspending medium; (c) whereas in bulk, solution, and suspension polymerizations, molecular weight of the polymer can be increased only by reducing the rate of polymerization [see Eq. (6.130)], the emulsion polymerization method allows both polymer molecular weight and polymerization rate to be increased simultaneously. Qualitative

Picture

Emulsion polymerizations vary greatly, and no single reaction mechanism accounts for the behavior of all the important systems. Useful insights

557

Radical Chain Polymerization

can be obtained, however, by considering an "ideal" case in which the monomeris nearly insoluble in water and the polymer is soluble in its own monomer.The general effects of experimental variables for such a case are discussed below. The physical picture of emulsion polymerization is based originally on the qualitative picture of Harkins [68] and the quantitative treatment of Smith and Ewart [69] with later contributions by Ugelstad and Hansen [70], Gilbert and Napper [71], Gardon [72], and others [73,74]. The essential ingredients in an emulsion polymerization are water, a monomerwhich is not miscible with water, an oil-in-water emulsifier, and a compoundor compounds (initiator) which produce free radicals in the aqueous phase. Typical proportions (by weight) are monomers100, water 150, emulsifier 2-5, and initiator 0.5, although these ratios mayvary over a wide range. Practical recipes mayalso include small amounts of various other ingredients. Thus an emulsion stabilizer such as carboxymethylcellulose, poly(vinyl alcohol), gelatin, dextrin, etc., may be used to prevent emulsion breakdown with progress of polymerization and a chain length regulator such as mercaptan to control the polymer molecular weight. To minimize fluctuations in surface tension of the emulsion as polymerization progresses, small proportion of a surface tension regulator (usually a long chain fatty alcohol such as cetyl alcohol) is used. Small volumeof a selected buffer solution is added to minimize or eliminate variations of pH of the system due to hydrolysis or other reactions. The emulsion system is usually kept in a well-agitated state during reaction. Before describing a qualitative picture of emulsion polymerization a note on monomersolubility and type of surface active agents is in order. Monomersfor emulsion polymerization should be near~y insoluble in the dispersing mediumbut not completely insoluble. The solubility must be less than about 0.004 mol/L, as otherwise the aqueous phase will become a major locus of polymerization and the system will then not be typical emulsion polymerization..At the same time the monomermust be slightly soluble as this will allow the transport of monomerfrom the emulsified monomerreservoirs to the reaction loci (see later). Surfactants or emulsifiers play an important role in the emulsion process. They are composed of ionic hydrophilic end and a long hydrophobic chain. Examplesare : Anionic detergent Sodium laurate

CHa(CH2)10COO- +

Sodiumalkyl aryl sulfonate

+ CnH2n+l-~O~--SO~1

Na

Cationic detergent Cetyl trimethyl Anionic and cationic

ammoniumchloride detergent

+ ClaHa3N(CH3)3 CI-

molecules may thus be represented

by

558

Chapter 6

-----~- and ----~+, respectively, indicating hydrocarbon chains with ionic end groups. The locations of the various components in an emulsion system will now be considered. Because of the polar end the detergent dissolves in waterthe nonpolar hydrophobic end finding itself in a hostile environment. Thus when a minimumor critical concentration of surfactant is exceeded, the hydrophobic ends collect into aggregates knownas micelles. This is known as critical micellar concentration (CMC).A micelle of anionic surfactant can thus be depicted as a cluster of surfactant molecules ( ----~- ) with the hydrocarbon chains directed toward the interior (see Fig. 6.17)..Since the surfactant concentrations in most emulsion polymerizations exceed CMCby several orders of magnitude, the bulk of the surfactant is in the micelles, the shape depending on the surfactant concentration. At lower surfactant concentrations (1-2%) the micelles are smaller and spherical (20-100 ,~), micelle containing about 50-150 surfactant molecules. At higher surfactant concentrations, micelles are longer and rodlike in shape. Such micelles are 1000-3000 ~ long with diameters approximately twice the length of an emulsifier molecule. Whena relatively water-insoluble vinyl monomer,such as styrene, is emulsified in water with the aid of anionic soap and adequate agitation, three phases result (see Fig. 6.17): (1) aqueous phase in which a small amount of both monomerand emulsifier are dissolved (i.e., they exist in molecular dispersed state); (2) emulsified monomerdroplets which are supercolloidal in size (> 10,000 ~), stability being imparted by the reduction of surface tension and the presence of repulsive forces since a negative charge overcoats each monomerdroplet; (3) submicroscopic (colloidal) micelles which saturated with monomer.This three-phase emulsion represents the initial state for emulsion polymerization. Stage [ Stage I begins (see Fig. 6.17) when a free-radical producing water-soluble initiator is added to the three-phase emulsion described above. The commonlyused initiator is potassium persulfate, which decomposes thermally to form water-soluble sulfate radical ions:

s2o- 0-60oc___+ 2so

(6.220)

The rate of radical generation by an initiator is greatly accelerated when it is coupled with a reducing agent. Thus an equimolar mixture of FeSO4 and K2S208 at 10°C produces radicals by the reaction: $2082-

q- Fe 2+ ~ Fe 3+ q- SO42- -k SO~-

(6.221)

about 100 times as fast as an equal concentration of the persulfate alone at 50°C. (Redox systems are generally used for polymerizations at lower temperatures. Manyof these redox initiator couples were developed for the emulsion polymerization of butadiene and styrene, since the 5-10°C "cold recipe" yields a better rubber than the "hot" 50°C emulsion polymerization.)

Radical Chain Polymerization

559

560

Chapter 6

The sulfate radical ions generated from persulfate react with the dissolved monomermolecules in the aqueous phase to form ionic free radicals SO~-

+ (n

+ 1)M ~ -SO4(M)nM"

(6.222)

Because of the presence of a long hydrocarbon chain carrying an ionic charge at one end, these ionic free radicals will have surface active properties. These soaplike anionic free radicals (represented in Fig. 6.17 as -A---’) behave like emulsifier molecules and because of the existence of a dynamic equilibrium between micellar emulsifier and dissolved emulsifier they can at some stage be implanted in some of the micelles. Once implanted in a micelle, a soaplike anionic free radical initiates polymerization of the solubilized monomerin the micelle. The micelle, thus "stung," grows in size as the solubilized monomeris used up and to replenish it more monomer enters the micelle from monomer droplets via the aqueous dispersion phase. The ’stung’ micelle is in this way transformed into a monomerpolymer (M/P) particle (see Fig. 6.17). Thus in stage I, the system consist of aqueous phase containing dissolved monomer,dissolved soaplike free radicals, micelles, ’stung’ micelles, M/P particles, and monomer droplets. The rate of overall polymerization increases continuously since nucleation of new particles and particle growth occur simultaneously. For the same reason a particle size distribution occurs during stage I. At 13-20% monomerconversion, however, nearly all the emulsifier will be adsorbed on the M/Pparticles and the micelles will disappear. Since new particles mostly originate in micelles, with the disappearance of micelles the nucleation of new M/Pparticles essentially ceases. This marks the end of stage I. Problem 6.48 In the model for emulsion polymerization it is assumed that most of the soaplike free radicals produced in the aqueous phase enters the micelles rather than the emulsified monomer droplets. Howwouldyou justify this assumption? Answer: The assumption that the soaplike free radicals produced in the aqueous phase enters the micelles rather than the emulsified monomer droplets can be justified because of two reasons: (a) Micelles have a muchhigher surface area to volumeratio than the monomer droplets since the former are muchsmaller in size than the latter. Micelles will therefore havea greater probability of receivingthe soap-typeanionic free radicals. (b) The numberof micelles per unit volumeof aqueousphase is muchmore than the numberof monomerdroplets per unit volume, the typical values of these al. numbersbeing 1018versus 10 The numberof free radicals that might be c~iptured by the monomer droplets can therefore be conveniently neglected. The monomerdroplets can thus be considered to serve primarily as reservoirs of monomer.

561

Radical Chain Polymerization

Stage H This stage is characterized by a continued growth of the existing M/P particles and no new particle nucleation. Free radicals enter only the M/P particles where polymerization takes place as the particles are supplied with monomer from the emulsified monomer droplets via the aqueous phase. Stage II is thus featured by a constant overall rate of polymerization as depicted in Fig. 6.18. As explained below, polymerization within a M/P particle can be characterized by alternating periods of activity and inactivity with the growth of a chain radical within the particle occurring in an isolated state unhindered over the active period. This results in a rapid rate of polymerization with a simultaneous generation of very high-molecularweight polymer. Stage III This stage begins when the overall rate of polymerization begins to deviate from linearity and is characterized by nonlinear growth rate with diffusion controlled regime. The nonlinearity may appear as a decrease in rate due to dwindling monomer concentration in M/P particles or as an increase if the Tromsdorff effect is important. The monomer reservoirs will have long since disappeared in stage II and the ratio of monomer to polymer within the particle will have dwindled to the point where the reaction becomes diffusion-controlled. The three stages in an emulsion polymerization system are depicted in Fig. 6.18.

/

y

J

diffusion controtted regime

-7L-~ 60 % StageII : Constantrate of growth--noparticle / /

~

/

.°° 7

~,~ !/I t" ~ zo*~ xo ou-/,

Emulsified monomer -.’"-’y .............. droptets disappear

15 % Stagel : SimuLtaneous particte nucteation and growth Time

.Figure 6.18 Schematic conversion-time curve for emulsion polymerization showmg three stages.

562

Chapter 6

Kinetics of Emulsion Poly~nerization The kinetic analysis here is based on quantitative considerations of the ideal emulsion polymerization systems which have been described qualitatively in the preceding sections. The treatment centers only around stage I and stage II (Fig. 6.18), as no general theory for stage III is available. The treatment applies to styrene-like monomers,meaning those monomerswith low water solubility and those in which monomerand polymer are completely miscible over all ranges of composition. Stage I This is the nucleation period. Ideally, all M/P particles are generated in stage I. The particles grow in volume and adsorb surfactant molecules, resulting in diminution of micelles. If the area occupied by each surfactant molecule is the same on particle and micellar surfaces and the particles are completely covered by surfactant, the combined areas of micelles and particles in unit volume of the reaction mediumwill remain constant.. The particle surface area will grow at the expense of micellar surface area and whenthe total surface area of particles equals the total surface area occupied by the surfactant initially added, all micellar soap will have disappeared and further nucleation will cease. For simplicity of calculations it is assumedthat the volumeof a particle grows linearly with time with rate u, once polymerization has been initiated there. It maybe noted that particles are nucleated at different times during stage I. Hence at time t in stage I, the volume of a particle which is nucleated at ~- will be u (/; -- ~-) if the volumeof the initial micelle negligible. The area of this spherical particle, at,~-, will be at,~-

= [(4~r) 1/2 3u (t - ~_)]2/3

(6.223)

If the rate at which radicals nucleate micelles is constant at v (effective radicals/era 3 aqueous phase/s), then vd~" particles are generated in the time interval dr. The area At of all particles present at time t, is given by the sums of the areas of particles generated from t = 0 onward: At = [(4~)l/Z3u]2/3

"L ~(t ~_)2/3vd~_ =

0.6 [ ( 47r ) l/2 3u]2/3vtS/3

(6.224) If as is the area occupied by unit weight of surfactant, then no mice~es will remain when At = asS, where S is the weight concentration of suffactant. ~us stage I is completed at a time tc such that a~S = 0.6 [(4~) U~ 3u] ~/3vt~/3

(6.225)

It nowremains to calculate v, the rate of nucleation. ~e rate of radical generation per unit volume, ~, by thermal decomposition ~ an initiator is (6.226) ~ = 2NA~kd[I]

563

Radical Chain Polymerization

where NAyis Avogadro’snumber. Nucleation is effected by radicals that enter micelles rather than particles. The fraction of radicals that enter particles rather than micelles maybe taken to be the ratio of the area of all particles to that of the total surfactant/water interface, that is, At / asS. Thus, (6.227) v --- P~ [1 - A, / a~S] Combiningthe foregoing equations one obtains [59]: tc = 0.37 (a~S //~)0.6

/ u0.4

(6.228)

The value of u for a particular system can be calculated ap~ori from knowledge of the monomerconcentration and other monomercharacteristics. In emulsion recipes, typical values [72] are R.r "~ 1012 - 1014 radicals cm-3 s -1, ass "~ 105 cm2/cm3 aqueous phase, u "~ -20 10 cm3 s -1. The number of particles per unit volume of aqueous phase, at the end of stage I is °6 (P~/u) °4 Np = vtc = 0.37(asS) (6.229) This equation indicates that the particle number depends on the 0.6 power of the suffactant concentration and on the 0.4 power of the initiator concentration (since /%r o< [I]). Usual values of Np are 1015 - 1016 per cm3 of aqueous phase.

Problem6.49 Consider the following emulsion polymerization recipe: z) Styrene (density 0.9 g/cm 3) Water(density 1.0 g/cm K2S2Os

Sodiumlauryl sulfate

100 g 180 g 1g 4g

Calculatethe numberof polymerizingparticles per liter of water using the following additional data: surface area per surfactant molecule = 5×10-15 cmZ; rate of volume increase of latex particle = 4×10-20 cmZ/s; ka of K2S2Osat 60°C = -~ s-~. 6xlO Answer: -I Molarmass of sodiumlauryl sulfate = 288 g tool (5 × 10-15 cm~ molecule-1) (6.02 × 10~ -1) molecules tool -~) g tool (288 = 1.05 x 10"t cm2 g-1 ~ L- 1) g)(1000 = 22.22 g L1 3) (180 cm (a~S) °’~ = 1.049 × 10~ em1’~ -°n L S = (4.0

564

Chapter 6 (1.0 g)(lO00 a [I]

L- 1)

L1

= (270 g mo1-1)(180 3) = 0. 0206 to ol

= 2(6x10 -6 s -1)(0.0206 -ls--- 11.488×1017 L

tool L-1)(6.02x102a

-1) tool

(@)o.4= Lr(1488 x 1°’7t’-I J =4.u8 x lO,’ t,-o."

= 1.~9 x 10 ~ -~ L -~ Here L imp~esper liter of aqueous phase.

Stage II Stage II begins when the conversion becomes linear with time (see Fig. 5.18). It is assumed that this coincides with the disappearance of micelles and the cessation of particle nucleation. The number of particles Np in stage II is fixed at the value formed in stage I and given by Eq. (5.229). Rate ofpol)~nerization. Under general conditions employed in emulsion polymerization, a typical value of the rate of generation of free radicals (P~) in the aqueous phase is 1014 per second per milliliter and a typical value of the numberof polymer particles is 10"~5 per milliliter. If all the radicals generated eventually enter M/Pparticles, since the micetles will have already disappeared, the rate of radical entry in a particle will average out to about one every 10 seconds, which means that the free radicals will generally enter the particles singly. Whena soaplike free radical enters a M/P particle, polymerization takes place. However, when another free radical enters the same particle, it terminates the growing chain radical by combining with it. (Calculation using known~ values predicts that two radicals cannot coexist in the same polymer particle and they would terminate mutually within a few thousands of a second.) So the particle, remains inactive till another free radical enters and initiates the polymerization. Thus, if a radical enters a polymerparticle every 10 seconds as calculated above, the particle will grow in alternating periods of actMty and inactivity, each of 10 seconds duration. In other words, each particle will remain active for half of the total time (and inactive for the other half). This situation will be unchanged even if the rate of radical entry into the particle is decreased or increased. This can

R~clicalChainPolymerization

565

3Rrp

Rpp~0

2Rrp

Rpp

~

0

Time Figure 6.19 Effect of rate of radical entry into particle (/~) on rate polymerization per particle (/~). be seen in the bar diagramof Fig. 6.19, wherethe rate of polymerization within a particle,/~o, for variousrates of ~ree radical entry into a particle, Ptcp (which is proportional to rate of radical generation P~r in aqueous phase) is shown. The important feature to observe here is that the total area of active period (shaded) for each P~pcase is the same; consequently the rate of polymerizationper particle is independentof the rate of radical entry. The concepts embodiedin the foregoing discussions are known collectively as the Srnith-EwartTheory,CaseH. The rate of polymerizationin a M/Pparticle, /~p, is given by P~ = k, [M] (~ / NAy)

(6.230)

where [M] denotes the monomerconcentration in the M/Pparticles and ~ the average numberof radicals per particle; -NAyis Avogadro’snumber. Since accordingto the Smith-Ewarttheory, the growingparticle contains a free radical only half the time, that is, ~ = 1/2, the rate of polymerization per particle is p~_

k,[M] 2NAy

(6.231)

If 1~ is constant and the numberof pai-ticles per unit volume, .Np, is constant, then the overall rate of emulsionpolymerizationper unit volume,

Chapter 6

566 Rp, is simply given by diM] dt -

Rp

Np kp [M] = 2NAv

(6.232)

Substituting for Np from Eq. (6.229) gives _k~o[M] (6.233) 2~Av In stage II, then, the rate of polymerization will depend on the 0.6 power of the suffactant concentration and the 0.4 power of the aqueous phase initiator concentration [since ~ is related to the aqueous phase initiator concentration through Eq. (6.226)]. It must be noted that although Eq. (6.233) gives the rate of polyme6zation in stage II, t~e value of [I] appropriate to the ~ term in this equation is that present in stage I when the panicles are formed. If this modelis co~ect, addition of extra initiator during stage II would have no effect on the rate of polymerization. This is evident from Eq. (6.232) in which the rate of polymerization is related only to quantities that are assumed to.be invadant du6ng this pa~ of the process. ~e value of ~ in Eq. (6.230) is of critical importance in dete~ining the rate of polymerization in stage II. Three cases -- designated 1, 2, 3 -corresponding, respectively, to ~ < 0.5, ~ ~ 0.5, and ~ > 0.5 can be distinguished based on the work of S~th and Ewart [69] and others [70-74]. The ~netic treatment given above confo~s to Case 2 (~ = 0.5), which is the predominant behavior for emulsion polymerizations. It occurs when desorption of radicals does not occur or is negligible comparedto the Rp = 0.37 (ass)0.6

Problem 6.50 The experimental value of dynamic concentration of styrene in polymer latex particles under the conditions of constant rate in emulsion polymerization has been found to be 5.2 tool/liter. Assumingthis value to be applicable, calculate the rate of polymerization per liter of aqueousphase in stage II of the reaction of the emulsion polymerization recipe given in Problem 6.49. (Data: kv for styrene af 60°C = 165 L mo1-1s-1.) Answer: FromEq. (6.232), diM] dt

- gpkp[M]/2YAv

mo1-1S -1) 2 (6.02x 1023-I) mol -I = 0.012toolL-~ s HereL-Iimplies perliter of aqueous phase. (1.649

× 1019

t -1)

(165

t

(5.2

-1) mol L

Radical ChainPolymerization

567

rate of radicals entering particles (absorption) and the particle size is too small to accommodatemore than one radical. If radical desorption from particles and termination in the aqueous phase are not negligible, ~ can drop below 0.5 (Case 1). The decrease in ff is larger for small particle sizes and low initiation rates. In large particles, on the other hand, several radicals can coexist, and ~ can be larger than 0.5 (Case 3). Degree of Polymerization. Once inside the M/P particle, agates at a rate rp given by rp

= kp [M]

a radical prop(6.234)

where [M] is the concentration inside the M/P particle. The rate re at which a radical enters a polymer particle in stage II is given by

= / Np

(6.235)

where /~ is the rate of generation of free radicals per unit volume in the aqueous phase [Eq. (6.226)] and Np is the number of M/P particles per unit volume. Equation (5.235) is based on the assumption that all radicals generated in the aqueous phase eventually enters the M/P particles, that is, radical capture efficiency is 1. Since termination reaction takes place as soon as a radical enters an active M/P particle (i.e. in which a polymer chain is propagating), re is also equal to the rate of termination of a polymer chain. Hence the degree of polymerization will be given by DPn --

rp

_ Npkp[M]

(6.236)

assuming, however, that chain transfer of any kind is negligible. Equation (6.236) shows that the degree of polymerization DP,~, like the rate polymerization Pqo [Eq. (6.232)], is directly dependent on the number particles. Thus, unlike polymerization by the bulk, solution, and suspension techniques, that by the emulsion technique permits simultaneous increase in rate and degree of polymerization by increasing the number of polymer particles (Np), that is, by increasing the surfactant concentration, at a fixed rate of initiation. Substituting Eqs. (6.226) into Eq. (6.236) one obtains Np kp [M] DPn -- 2NAvkd [I] The initiator stage II.

(6.237)

concentration here is that which exists at a given instant in

Problem 6.51 A particular emulsion polymerization yields polymer with ~ = 200,000. Howwouldyou adjust the operation of an emulsion process to produce polymer with M~= 100,000 in stage II without, however, changing the rate of

568

Chapter 6

polymerization,particle concentration, or reaction temperature? Answer: According to Eq. (6.237), DPno~ 1/[I], sin~e N is constant. So DP, can reduced to half by adding moreinitiator in stage II so that [I] is doubled. The rate of polymerization[Eq. (6.233)] and the particle concentration [Eq. (6.231)] will not be altered becausethe [I] value whichis operative in these relations is that for stage I. Analternative procedurewouldinvolve addition of chain transfer agents in appropriate concentrations.

The combination of high molecular weight with high polymerization rate is one reason for the popularity of the emulsion technique. Seeded polymerizations can be useful for makinglarge-particle-size latexes by the emulsion technique. Thus a completed "seed" latex may be diluted to give the desirable value of Np particles per liter of emulsion. No additional surfactant is added, so no new polymer particles are formed. Whenmonomeris fed and initiator is added, polymerization occurs in the previously formed particles, so that each one grows as monomerdiffuses into it and is converted.

Problem6.52 A polystyrene latex produced by emulsion polymerization contains 10%by weight of polymerparticles (average diameter 0.20 #m). It is decided grow these particles to a larger size by slowly adding to the latex 2 kg of monomerper kilogram of polymer as polymerization proceeds at 60°C without further addition of emulsifier. Thereaction is to be carried out until all monomer has been added to the reactor and the weight ratio of monomerto polymer has decreased to 0.2. The unreacted monomeris then to be removedby steam stripping. Estimate the’time required for the reaction and the final particle diameter. [Data: k~, at 60°C = 165 L mo1-1s -1. Density: monomer= 0.90 g/cm3; polymer = 1.05 g/cm3; dynamic solubility of monomerin polymer = 0.6 g monomerper gram polymer.] Answer: The time of reaction has to be calculated in two consecutive periods, with zero order in monomer(constant monomerconcentration) until the monomerreserve is exhausted, and then with first order until the desired conversionis reached. Basis: 1000 g latex (900 g water + 100 g polymer) Monomerto be added = 2×100 or 200 g Monomerto be converted = x g

C00_ = 0.2,

Reaction to be stopped when \1--~~

Monomerto be stripped = (200 - 150) or 50

x = 150

Radical Chain Polymerization Monomerconcentration

569

when reaction

is to be stopped is [(50 g monomer)/(104 g mol-1)](1000 s L- 1) (250 g polymer)/(1.05 g -3) + (5 0 g monomer)/(0.90 gc -a) -1 ---- 1.64 mol L Dynamic concentration of monomerin panicles is [(0.6 g monomer)/(104 g mol-~)](1000 -1) (1 g polymer)/(1.05 g -3) d- (0. 6 g m onomer)/(0.90 g c -a ) -~ = 3.56 tool L Let the amount of monomer remaining unconverted at the end of zero-order period be y gram. Therefore, [(y g) / (104 g mol-~)] (1000 a L- 1) -1 = 3.56 tool L [(300 - y) / 1.05] a polymer ÷ (y/ 0 .9 ) cm 3monomer y ---- 112.4 g Numberof particles

(Np’) in 1 kg of original latex:

Initial diameter of particle = 0.2 #m ---- 0.2x10-4 cm Volume/particle = (~r/6)(0.2x10 -4 cm)3 = 4.19×10 -~ 3cm 1 (100 g polymer) Nz~’ -~ = )(1.05 g cm (4.19 x 10-is 3) cm ~ = 22.7 x 10 Rate of conversion in zero-order period is: (22.7 x 101~ particles kg-~)(165L -~ s-~)(3.56 mol -~) N~,’k~,[M]/2NAv = -~) 2 (6.02 x 102a particles mol -Stools-I = 11.1x10 kg -1 (= 4.0x10 -~molh-~ -~) kg Monomerconverted in zero-order period per 1 kg original latex = 200 - 112.4 or 87.5 g (= 0.84 mol). Therefore, Time for conversion ---- (0.84 mol kg-~) /(4.0 x 10-2 tool h-1 -~) kg ~- 21h First-order period From Eq. (6.232),

d* = Integrating and solving for t

=

2 ln([M]0/[M]) NAy

Here Np is the numberof particles per liter of latex. It can be taken, without any significant error, to be equal to the numberof particles in 1 kg of latex, N~,’. Thus, 21n[(3.56 mol -~) / (1 .64 mol L- l)] (6 .02 x ~ part icles tool -~) t = (22.7 × 10is particles L-1(165 L mol-is-~) (3600 s -1) 69 h

570

Chapter 6

Total time= (21 + 69) or 90h Final particle diameter, d: (d/0.2 so,

Other

3 #m)

= ( oo 0.oo g)

=

d=2.7#m

Theories

A number of workers have suggested that emulsion polymerization may not occur homogeneously throughout a polymer particle but either at the particle surface [75] or within an outer monomer-richshell surrounding an inner polymer-rich core [76]. The latter has been referred tO as the shell or core-shell model. The latter model has been proposed to explain the apparent anomaly between the observed constant rate behavior up to about 60 percent conversion, which according to Eq. (6.232) requires [M] to constant, and the considerable experimental evidence which indicates that emulsified monomerdroplets (serving as monomerreservoirs) disappear at 25 to 30 percent conversion and the monomer concentration drops thereafter. According to the core-shell model, the growing particle is actually heterogeneous rather than homogeneous, and it consists of an expanding polymer-rich (monomer-starved) core surrounded by a monomer-rich (polymer-starved) outer spherical shell. It is the outer shell that serves the major locus of polymerization and Smith-Ewart (on-off) mechanism prevails while virtually no polymerization occurs in the core because of its monomer-starvedcondition. Reaction within an outer shell or at the particle surface would be most likely to be operative for those polymerizations in which the polymer is insoluble in its own monomeror under conditions where the polymerization is diffusion-controlled such that a propagating radical cannot diffuse into the center of the particle.

1. J. A. Kerr, "Rate Processes in Gas Phase," Ch. 1 in Free Radicals (J. K. Kochi, ed.), vol. I, Wiley, NewYork,Q973). 2. V. N. Kondratiev, "Chain Reactions,’ Ch. 2 in ComprehensiveChemical Kinet&s (C. H. Bamford and C. E H. Tipper, eds.), vol. 2, American Elsevier, NewYork(1969). 3. J. Brandupand E. H. Immergut(eds.) with W.McDowell,PolymerHandbook, Wiley-Interscience, NewYork(1975).

Radical Chain Polymerizafion

571

4. E. S. Huyser, Free Radical Chain Reactions, Chap. 10, Wiley, New York (1970). 5. E. A. Collins, J. Bares and E W. Billmeyer, Jr., Experiments in Polymer Science, Chap. 5, Wiley-Interscience, NewYork (1973). 6. T. Koenig, "The Decomposition of Peroxides and Azoalkanes," Ch. 3 in Free Radicals (J. K. Kochi, ed.), vol. 1, Wiley, NewYork (1973). 7. G. C. Eastmond, "The Kinetics of Free Radical Polymerization of Vinyl Monomers in Homogeneous Solutions," Ch. 1 in Comprehensive Chemical Kinetics (C. H. Bamford and C. E H. Tipper, eds.), vol. 14A, American Elsevier, NewYork (1976). 8. A. D. Jenkins, J. Polym. Sci., 29, 245 (1958). 9. L. M. Arnett and J. H. Peterson, J. Am. Chem. Soc., 74, 203 (1952). 10. G..Bonta, B. M. Gallo, S. Russo and C. Uliana, Polymer, 17, 217 (1976). 11. K. C. Berger, P. C. Deb and G. Meyerhoff, Macromolecules, 10, 1075 (1977). 12. P. D. Bartlett and H. Kwart~ J. Am. Chem. Soc., 72, 1051 (1950). 13. M. S. Matheson, E. E. Auer, E. B. Bevilacqua and E. J. Hart, J. Am_Chem. Soc., 73, 1700 (1951). 14. T. Sato, M. Abe and T. Otsu, Makromol. Chem., 178, 1951 (1977). 15. K. F. O’Driscoll, P. E Lyons and R. Patsiga, J. Polym. Sci., A3 1567 (1965). 16. S. E. Morsi, A. B. Zaki and M. A. E1-TGayami, Eur. PolyP. J., 13, 851 (1977). 17. P. Ghosh and S. N. Maity, Eur. Polym. J., 14, 855 (1978). 18. C. J. Sahani and N. Indictor, J. Polym. Sci., Polym. Chem. Ed., 16, 2683, 2997 (1978). 19. S.P. Manickam,U. C. Singh, K. Venkatarao and N. R. Subbaratnam, Polymer, 20, 917 (1979). 20. N. Mohanty, B. Pradhan and M. C. Mahanta, Eur. Polym. J., 16, 451 (1980). 21. R. K. Samal, M. C. Nayak and P. L. Nayak, J. Macromol. Sci.-Chem., A12(6), 815 (1978). 22. K. R. Ahmed, L. V. Natarajan and Q. Anivaruddin, Makromol. Chem., i79, 1193 (1978)., 23. G. S. Misra and S. L. Dubey, Z Polym. Sci. Polym. Chem. Ed., 17, 1393 (1979). 24. D. Pramanick and A. K. Chatterjee, J. Polym- Sci. Potym. Chem. Ed., 18, 311 (1980). 25. K. Kaliyamurthy, P. Elayaperumal, T. Balakrishna and M. Santappa, Makrotool. Chem., 180, 1575 (1979). 26. R. G. Norrish and J. P. Simons, Proc. Roy. Soc. (London), A251, 4 (1959). 27. E. A. Lissi, M. V. Encina and M. T. Abarca, J. Polym. Sci. Polym- Ed., 17, 19 (1979). 28. W. D. Graham, J. G. Green and W. A Pryor, J. Org. Chem., 44, 907 (1979). 29. J. Lingnau, M. Stickler and G. Meyerhoff, Eur. Polym. J., 16, 785 (1980). 30. N. J. Barr, W. I. Bengough, G. Beveridge and G.. B. Park, Eur. Polym. J., 14, 245 (1978). 31. P. J. Flory, Principles of Polymer Chemistry, Comell Univ. Press, Ithaca, New York (1953). 32. J. Lingnau and G. Meyerhoff, MacrrmOlecules, 17, 941 (1984). 33. R. H. Gobran, M. B. Berenbaumand A. V. Tobolsky, J. Polym. Sci., 46, 431 (1960). 34. K. E O’Driscoll and P. J. White, J. Polym. Sci., BI: 597 (1963); A3:283 (1965). 35. K. F. O’Driscoll and S. A. McA_rdle,J. Polym. Sci., 40, 557 (1959). 36. C. Walling, Free Radicals in Solution, Chaps. 3-5, Wiley-Interscience, New York (1957).

572

Chapter6

37. E Brier, D. L. Chapmanand E. Waiters, J. Chem. Soc., 562 (1926). 38. W. A. Noyes and P. A. Leighton, The Photochemistry of Gases, p. 202, Reinhold, NewYork (1941). 39. M. S. Matheson, E. E. Auer, E. B. Bevilacqua and E. H. Hart, J. Am. Chem. Soc., 71,497(1949). 40. P. Penchev, Mal~omol. Chem., 177, 413 (1976). 41. K. E O’Driscoll and H. K. Mahabadi, J. Polym. Sci. Polym. Chem. Ed., 14, 869~(1976). 42. C. H. Bamford, G. C. Eastmond and D. Whittle, Polymer, i0, 771 (1969). 43. R. A. Gregg and E R. Mayo, Disc. Faraday, Soc., 2, 328 (1947). 44. E J. Flory, J. Am. Chem. Sot., 69, 2893 (1947). 45. C. Walling and L. Heaton, J. Am. Chem. Soc., 87, 48 (1965). 46. E Ghosh and E W. Billmeyer, Advances in Chemistry Series, 91, 75 (1969). 47. P. Ghosh and A. N. Banerjee, J. Polym. S¢i. Polym. Chem Ed., 12, 375 (1974). 48. E C, Deb and G. Meyerhoff, Eur. Polym. J., 10, 709 (1974). 49. G. V. Schulz and G. Haborth, Makromol. Chem., 1, 106 (1948). 50. A. M. North and G. A. Reed, Trans. Faraday Soc., 57, 859 (1961); J. Polym. Sci., A1, 1311 (1963). 51. H. K. Mahabadi and K. E O’Driscoll, J. Polym. Sci. Polym. Chem. Ed., 15, 283 (1977); Macromolecules, i0, 55 (1977). 52. J. M. Dionisio, H. K. Mahabadi, K. E O’Driscoll, E. Abuin and E. A. Lissi, J. Polym. Sci. Polym. Chem. Ed., 17, 1891 (1979). 53. J. M. Dionisio and K. E Driscoll, J. Polym. Sci. Pokym. Chem. Ed., 18, 241

198o). 54. H. K. Mahabadi and A. Rudin, J. Polym. Sci., 17, 180 ,(1979). 55. N. Friis and A. E. Hamielec, ACS Symp. Set, 24, ’Polymerization, ,, 82 " 56. t1976~j . L. ce, J. Am. Chem. Soc., 76, 627 (1954). 57. E R. Mayo and C. Walling, Chem. Revs. 46, 191 (1950). 58. H. W. Melville and L. Valentine, Proc. Roy. Soc. (London), A200, 337, 358 (1950). 59. E D. Bartlett and H. Kwart, J. Am. Chem. Soc., 72, 1051 (1950). 60. E S. Dainton and K. J. Ivin, Qtly. Rev., 12, 61 (1958). 61. H. W. McCormick,J. Polym. Sci., 25, 0. 488 (195" 62. J. M. Smith and H. C. Van Ness, Introduction to Chemical Engineering Thermodynamics, McGraw-Hill, New York (1987). 63. G. V. Schulz, Z. Physik. Chem., B43, 25 (1939). 64. W. B. Smith, J. A. Mayand C. W. Kim, J. Pofym. Sci., A2, 365 (1966). 65. A. M. North, The Kinetics of Free Radical Polymerization, Pergamon Press, New York (1966). 66. J. A. Moore (ed.), Macromolecular Syntheses, Collective vol. 1, WileyInterscience, NewYork (1977). 67. S. R. Sandier and W. Karo, Polymer Syntheses, Academic Press, NewYork, vols. 1 and 2 (1974), vol. 3 (1980). 68. W. D. Harkins, J. Am. Chem. Soc., 69, 1428 (1947). 69. W. V. Smith and R. W. Ewart, J. Chem. Phys., 16, 592 (1948). 70. J. Ugelstad and E K. Hansen, Rubber Chem. Technol., 49, 536 (1970). 71. R. G. Gilbert and D. H. Napper, J. Chem. Soc., Faraday I, 70, 391 (1974). 72. J. L. Gardon, "Emulsion Polymerization," Ch. 6 in Polymerization Processes (C. E. Schildknecht, ed., with I. Skeist), Wiley-Interscience, NewYork (1977). 73. I. Piirma (ed.), Emulsion Polymerization, AcademicPress, NewYork (1982). 74. B. S. Casey, I. A. Maxwell, B. R. Morrison and R. G. Gilbert, Makromol. Chem. Macromol. Symp., 31, 1 (1990). 75. A. Sheinker and S. S. Medvedev, Dokl. Akad. Nauk SSR, 97, 111 (1954). 76. M. R. Grancio and D. J. Williams, J. Polym. Sci., A-I, 8, 2617 (1970).

Radical ChainPolymerization

573

EXERCISES 6.1. Whena.peroxide P is heated to 60°C in an inert solvent it decomposes by a first order process and 20%of the peroxide decomposes in 60 min. A bulk monomeris polymerized using this initiator at 60°C, the initial concentration of the letter being 4.0x10 -4 mol/L. What fractions of the monomer and the initiator should remain unconverted after 10 min ? At 60°C, the system parameters are k~/~ = 22.34 L mo1-1, f = 0.8. [Ans. Monomer0.67;. Initiator 0.963.] 6.2. A solution of 100 g/L acrylamide in methanol is polymerized at 25°C with 0.1 mol/L isobutyryl peroxide whose half life is 9.0 h at this temperature and efficiency in methanol is 0.3. For acrylamide, k~/~ -1 = 22 L/mol-1 s at 25°C and termination is by coupling alone. (a) What is the initial steady state rate of polymerization ? (b) Howmuch polymer has been made in the first 10 min of reaction in 1 L of solution ? [Ans. (a) 0.37 g -1 s-l; ( b) 8 9.4 g L-1.] 6.3. A dilatometer which has a 50 cm long capillary (diameter 0.2 cm) has total volume of 50 cmz (including the volume of capillary). The dilatometer was filled with a freshly distilled sample of methyl methacrylate (MMA) containing 0.25 wt% benzoyl peroxide and then immersed in a water bath (at 50°C) so that the capillary tube protruded from the water. Whenthe volume of the solution began to decrease after coming to thermal equilibrium and overflowing the capillary, the fall in liquid level in the capillary was determined periodically from the scale on the dilatometer. This yielded the following data : Time (s) 480 1200 1920 4080 8280 9600

Ah (cm) 1.48 4.09 6.46 13.56 27.24 31.91

Determine the kinetic parameter kp/k~/2 at 50°C, given that f = 0.80 and ka = 1.11×10 -6 s -1 at 50°C. [Density (g/cm a at 50°C): MMA0.893; PMMA

1.160]

[Ans. 0.0948 tool 1/2 L-1/2 -1/2] s 6.4. The polymerization of methyl methacrylate (1.0 Min benzene) is carried out using a photosensitizer and 3130 ,~klight from a mercury lamp. If the quantum yield for radical production in this system is 0.50 and light is absorbed by the system at the rate of 105 ergs/L-s, calculate the rate .of initiation. [Ans. 2.6×10-8 tool L-1 -1] s 6.5. Consider irradiation of pure acrylonitrile (density = 0.81 g/cm~) at 20°C with "/-rays, with a dose rate of 106 rads/h. Calculate (a) the rate of energy absorption and (b) the rate of initiating radical formation per unit volume of the monomer.(100-eV yield of initiating radicals = 5.0) [Arts. (a) 1.4×1016 eV cm-3 s-l; (b) 7,0×1014 radicals -3 s - 1]

574

Chapter 6

6.6. The peroxide (thermal homolysis) initiated polymerization of a monomer follows the simplest kinetic scheme represented by Eq. (6.26). For a polymerization system with [M]0 = 4 mol/L and [I]0 = 0.01 tool/L, the limiting conversion p~ = 0.10. To increase P~o to 0.20, (a) would you increase or decrease IM]0 and by what factor? (b) would you increase or decrease [I]0" and by what factor ~c) would you increase or decrease tlae reaction temperature [Ans. (a) no effect; (b) increase by a factor of 4.5; (c) decrease temperature.] 6.7. One hundred liters of methyl methacrylate containing 10.2 moles of an initiator (ti/2 = 50 h) in solution is polymerized at 60°C. Calculate (a) kinetic chain length in this polymerization and (b) the amount of polymer formed in the first 1 h of reaction. [Data: monomerdensity 0.94 g/cm3; _kp =515Lmo1-1 s-l; kt =2.55x107Lmo1-1 s-l; f =0.3] [~tns. (a) 1397; (b) 70 6.8. Calculate the time needed to convert half of charge of methyl methacrylate (10 g per 100 mLsolution) to polymer using benzoyl peroxide (0.1 g per mL solution) as initiator in benzene at 60°C. What number-average degree of polymerization will be expected initially ? Whatfaction of the initiator will remain unused after 50% conversion of the monomer? [At 60°C, ku = 4.47×10 -6 s-l; k~/kt = 10-~ L/moPs; f = 0.4; termination occurs by both disproportionation (58%) and coupling (42%).] [Ans. 24.7 h; 739; 0.67] 6.9. Initiator 11 has half the half-life that initiator I2 has at 80°C wheninitiator I1 is used for both and all the concentrations are the same. What is the ratio of degree of polymerization for M1 and M2if they are polymerized with initiator I1 and I2, respectively, the ratio of monomerconcentrations being 1:2 and that of initiator concentrations being 1:5 ? It can be assumed that both polymers terminate exclusively by coupling and that the initiator efficiencies are equal. [Arts. 3.16] 6.10. A vinyl polymer with a number-average degree of polymerization of 10,000 was produced by polymerization at 70°C using a peroxide initiator concentration of 4×10-4 mol/L. If 4.0% of the initial monomerpresent was converted to polymer in 60 min, what was the initial monomer concentration? The modeof termination is coupling and the initiator is knownto have a half-life of 1.0 h at 90°C and an activation energy of 30.0 kcal/mol. Assume f = 1.0. [Ans. -1] 6.0 tool L 6.11. Using carbon-14 labeled AIBNas an initiator, a sample of styrene is polymerized to an average degree of polymerization of 1.28×104. The AIBN has an activity of 8.97x107 counts per minute per mol in a scintillation counter. If 5.0 grams of the polystyrene shows an activity of 315 counts per minute, determine the modeof termination of polystyryl radicals. [Arts. Coupling 93%, disproportionation 7%] 6.12. In the bulk polymerization of methyl methacrylate at 60°C with azo-bisisobutyronitrile as the initiator the initial rates of initiation and polymerization are 1.7x10-6 mol/L-s and 8.8x10-4 mol/L-s, respectively. Predict the initial molecular weight of the polymer formed in this system, if the extent of disproportionation is 70%at 60°C. Neglect chain transfer reactions for the calculation. [Ans. 60,800]

Radical Chain Polymerization

575

6.13. Determinethe concentrations (g/L) of initiator (AIBN)and chain transfer agent (n-butyl mercaptan)that will give poly(vinyl acetate) with an initial molecular weight (assuming coupling) of 15,000 and 50%conversion monomer (initial concentration 250 g/L) at 60°Cin 30 rain. [Systemparameters (all at 60°C): k~/k~ = 0.1824 L/mol-s; tl/2 of AIBN= 22 h; f = 1; CS = 481 -1] [Arm.15.24 g L-l; 0.027 g L 6.14. The molecular weight of polymerwhenstyrene is polymerizedin benzeneis 400,000.Withall other conditions the same,addition of 4.23 mg/Lof n-butyl mercaptan decreases the molecular weight to 85,000. Whatconcentration (mg/L)of n-butyl mercaptanwill give a molecular weight of 50,000 (other conditions remaining the same)? -l] [Arm. 8 mgL 6.15. Accountfor the fact that propylene and isobutylene have low reactivity toward radical polymerization, and do not yield high polymersin contrast to monomers such as methyl methacrylate and methacrylonitrile which yield high polymersin radical polymerization. 6.16. Discuss howthe followingfactors wouldinfluence the gel effect: (a) polymer molecular weight; (b) solvent goodness; (c) chain transfer; temperature. 6.17. Modifythe Mayoequation (6.148) to take into accountthe effect of degradative chain transfer on the number-averagedegree of polymerization. For simplicity, assumethat onlythe chain transfer to solvent is degradative(i.e., the newradical formeddoes not initiate polymerization). 6.18. Considera free radical polymerization initiated by 10-3 MAIBN.At 70°C, kd is 4.0×10-5 s -1 and f is close to 0.6. If an inhibitor is to be used to suppress polymerization for an hour, what should be its concentration, if every inhibitor molecule accounts for one primary or monomer-ended radical ? [~ns. 1.73 x 10-4 M] 6.19. Assumethat the various reactions which may be brought about by an inhibiting or retarding substance, represented by Z, maybe reduced to the following simple schemeof three reactions: ktr,z (a) M~-÷ Z ~ Polymer q- Z"

Co)z.+ (c) Z’÷ Z-k-~Z~ Nonradica! products Derive an expression for the rate of consumptionof inhibitor and explain its dependenceon reaction parameters. 6.20. One ldlogramof a 25%Coy weight) solution of acrylamide in water at 27°C is polymerizedadiabatically with a redox initiator. The peak temperature reached is 90°C. Calculate the molar heat of polymerization using the following heat capacity data: monomerand polymer= 0.5 cal/g-°C, water 1.0 eal/g-°C, reactor 0.1 kcal/°C. -1] [Arm.17.4 kcal tool

576

Chapter 6

6.21. Methyl acrylate (1 mol) is polymerized using 0.001 mol succinic peroxide in 1 liter solution in benzene at 60°C. If the polymerization is carried out adiabatically, how much would the temperature rise in 30 min ? [Data (all at 60°C): tl/2 of initiator = 19 h; f = 1; k~/kt = 0.460/_]mol-s; AH~, = - 18.6 kcal/mol] [Ans. 6.0°C] 6.22. A vinyl monomeris photopolymerized in two experiments in which only the temperature is varied. In these experiments, the time to convert 20%of the original charge of monomerto polymer is found to be 30 min at 60°C and 27 min at 70°C. However, when an organic peroxide is used as the initiator, the corresponding times for 20% conversion are 62 min at 60°C and 29 min at 70°C. What is the activation energy for the dissociation of the organic peroxide ? [Ans. 29.7 kcal mo1-1] 6.23. The half-lives of azobisisobutyronitrile at 50°C and 70°C are 74 h and 4.8 h, respectively. What will be the half-life at 60°C? [Ans. 18.1 h] 6.24. In the bulk polymerization of styrene by ultraviolet radiation, the initial polymerization rate and degree of polymerization are 1.3×10 -3 mol/Ls and 260, respectively, at 30°C. What will be the corresponding values for polymerization at 80°C? The activation energies for propagation and termination of polystyryl radicals are 26 and 8.0 kJ/mol. What assumption, if any, is madein this calculation ? [Ans. 4.48×10-z tool L-1 s-l; 896] 6.25. A radical chain polymerization conforming to ideal behavior shows the indicated conversions for specified initial monomerand initiator concentrations and reaction times: z[I]0 x 10 Temperature Reaction Conversion [M]0 (mol/L) (mol/L) time (min) (%) (°C) 0.80 1.0 60 60 40 0.50 1.0 75 70 80 Calculate the overall activation energy for the rate of polymerization. [Ans. 63.8 kJ mo1-1 (= 15 kcal mol-1)] 6.26. The enthalpy and entropy of polymerization of a-methylstyrene at 25°C are -35 kJ/mol and -110 J/°K-mol, respectively. Calculate approximately the equilibrium constant for polymerization at 25°C and 50°C. Comment on the results. [Ans. 2.45, 1.22] 6.27. Calculate/~[M], and kap at different temperatures and hence determine the ceiling temperature for pure methyl methacrylate (density 0.940 g/cm3 at 25°C) using the following data: Ap = 106 L mo1-1 -l, Aa~ = 10az s - I -a. = 82 IO mol (assumed), /~ = 26 kJ tool -~, Edp

[Ans.T~= 488°K] 6.28. Calculate the equilibrium monomerconcentration [M]e for radical polymerization of styrene at 50°C, assuming that A/-/° and AS° are given by values in Table 6.13. Repeat the calculations for 75°C and 125°C. [Ans. 4.23x10-7 mol L-l; 2.98×10-6 tool L-l; 1.0×10-4 -~’] tool L

Radical Cha/_n Polymerizafion

577

6.29. The enthalpy and entropy of polymerization of methyl methacrylate at 25°C are -56 kJ/mol and -117 ld/°K-mol, respectively. For a solution of the monomer(1.0 mol/L), calculate the maximumattainable conversion at (a) 25°C, (b) 120°C and (c) 200°C. [Ans. (a) 99.98%; (b) 95.0%; (c) 6.30. For polymerization of tetrafluoroethylene, AH° and AS° values at 25°C are given as -37 kcal/mol and -26.8 cal/°K-mol. Calculate the ceiling temperature (To) from these two values. Account for the fact that in practice poly(tetrafluoroethylene) is found to undergo fragmentation well below the calculated [Ans. 1380 °K] 6.31. Using the enthalpy and entropy of polymerization data for methyl methacrylate from Table 6.13, calculate the depolymerization rate constant of poly(methyl methacrylate) at 100°C. [Data: kp at 60°C = 515 L/mol-s; Ep = 26.4 k J/moll [Ans. -1] 34.7 s 6.32. If a free-radical polymerization of 1.0 M solution of styrene were being carried out at 100°C, what would be the maximumpossible conversion of the monomerto polymer, that is, till the polymerization-depolymerization equilibrium is reached ? (Take data from Table 6.13.) [Ans. 99.995%] 6.33. What is the maximumbreadth of the size distribution to be expected for a low-conversion polymerization where termination is entirely by coupling ? Howwill each of the following situations alter the size distn’bution (a) chain transfer to mercaptan, (b) chain transfer to polymer, (c) high conversion, and (d) autoacceleration? [Arts. PDI = 1.5] 6.34. Styrene (density 0.90 g/cm3) was polymerized at 60°C with 0.01 Mbenzoyl peroxide as the initiator [W. B. Smith, J. A. May, and C. W. Kim, J. Polym. Sci., PartA2, 4, 395 (1966)]. The initial rate of reaction was obtained as 3.95x10 -s mol/L-s. From the GPCmolecular weight distribution curve reported for a 0.79% conversion sample, the weight fraction of polymer of DP~---- 3000 is seen to be 2.3x10-4. Calculate the weight fraction from theoretical distribution function to compare with this value. [Data: k~/kt at 60°C = 0.00119 L mol [Ans. -4] 2.2 x 10 6.35. MonomerA is polymerized in solution at 60°C using a peroxide initiator I which has half-life of 5.0 h at the same temperature. A 30%conversion of the monomeris obtained in 25 rain when the initial concentration of A is 0.40 Mand the initial concentration of I is 0.04 M. Polymerization of A in an emulsion of 8x 1017 particlesper liter at 60°C yielded a conversion rate of 18.1 mol/h/L when the concentration of A in the particles is constant at 4.0 M. Determine the termination rate constant of A at 60°C. [Ans. 9.73x10 6.36. What happens to (a) rate of emulsion polymerization, (b) number average degree of polymerization, and (c) polymer particle size, if more monomer is added to the reaction mixture during stage II polymerization ? Explain. [Ans. (a)’No change; (b) no change; (c) increases.]

578

Chapter 6

6.37. The rate of emulsion polymerization of styrene at 60°C during the constant rate period (stage II) is 5.6×10 -~ mol/cm3-min and the number of M/P particles is 1.40×10~5 per cm3. Taking kv from Table 6.7, calculate the dynamic concentration of monomerin particles under these conditions. -1] [Ans. 4.8 tool L 6.38. In an emulsion polymerization of isoprene with 0.10 M potassium laurate at 50°C the estimated time required for 100%conversion at steady rate is 30 h. The final latex has 40 g of polymer per 100 mLwith particles of 450 ~, diameter. During stage II, the growing swollen polymer particles contain 20 g of monomerper 100 mL of swollen polymer. Assuming that there is no change in total volume on polymerization, estimate the polymerization rate constant from these data. Assume that the polymer has a density of 0.90 3.g/cm [Ans. 2.4 L tool -1 -1] s 6.39. Consider a typical reactor charge for the production of polymer latex: monomer(s) 100, water 180, sodium lauryi sulfate (surfactant) 4, potassium persulfate (initiator) 1 (all quantities are in parts by weight). What effects do the following changes have on the polymerization rate in stage II ? (a) Using 8 parts surfactant; (b) using 2 parts initiator; (c) using 8 parts surfactant and 2 parts initiator; (d) adding 0.1 part butyl mercaptan (chain transfer agent). [Anz. (a) P~ increases by a factor of 1.52; (b) /~ increases by a factor of 1.32; (c) /~ increases by a factor of 2; (d) /~ unchanged.] 6.40. A 10%(by weight) latex of poly(methyl methacrylate) produced by emulsion polymerization contains particles that average 0.2 mumin diameter. In order to grow the particles to a larger size it is decided to feed 4 kg of monomerinto the latex per kilogram of polymer as polymerization proceeds at 60°C without further addition of emulsifier. The reaction is to be carried on until all monomeris added to the latex and the weight ratio of monomer to polymer has decreased to 0.2. The unreacted monomer is then to be recovered by steam stripping. Calculate the total time that will be required for reaction and the final particle diameter. [Data: kp at 60°C = 515 L mo1-1 s -~. Density: monomer 0.9 g/cm3; polymer 1.2 g/crn 3. Dynamic solubility of monomer in polymer = 0.5 g monomer per gram polymer.] [Ans. 42 h; 0.64/zm]

Chapter Chain Copolymerization INTRODUCTION In the preceding chapter we have considered free-radical polymerizations where only one monomer is used to produce a homopolymer. However, chain polymerizations can be carried out with mixtures of two or more monomersto form polymeric products that contain two or more different structures in the polymer chain. This type of chain polymerization process in which two or more monomers are simultaneously polymerized is termed a copolymerization and the product is a copolymer. It is important to note that the copolymer is not an alloy of two or more homopolymers but contains units of all the different monomersincorporated into each copolymer molecule. The process can be depicted, for copolymerization of two monomers, as MI + M2-----o

MIM2MIM2M2M2MIMIMIM2MIMI

(7.1)

The two monomersenter into the copolymer in overall amounts determinedby their relative concentrations and reactivities. The chain copolymerization may, however, be initiated by any of the chain initiation mechanisms, namely, free-radical chain initiations considered in the preceding chapter, or ionic chain initiations, which will be described in a later chapter. Chain copolymerizations involving more than two monomersare generally referred to as multicomponent copolyrnerizations. For systems of three monomers,the specific term terpolymerization is commonlyused. Chain copolymerization is important both from academic and technological viewpoints. Thus much’of our knowledgeof the reactivities of monomers, free radicals, carbocations, and carboanions in chain polymerization comes from copolymerization studies. The behavior of monomersin copolymerization reactions is especially useful for studying the relation betweenchemical structure and reactivity of monomers. From the technological viewpoint, 579

580

Chapter 7

copolymerization of two or more monomersis an effective way of altering the balance of properties of commercial polymers. While polymerization of a single monomeris relatively limited as to the numberof different products that are possible, copolymerization, enables the polymerscientist to tailormake polymers with specifically desired properties. Thus by variations in the nature and relative amounts of the two monomersin a copolymerization, an almost unlimited number of products with different properties can be synthesized. A notable example of the versatility of the copolymerization process is the case of polystyrene: Polystyrene is a brittle plastic with low impact strength and low solvent resistance. Copolymerization greatly increases the usefulness of polystyrene. Styrene copolymers are useful not only as plastics but also as elastomers. Thus free-radical copolymerization of the hydrocarbon monomerstyrene with 20-35% of the relatively polar monomeracrylonitrile produces a transparent copolymer with increased impact and solvent resistance, while copolymerization with 1,3-butadiene leads to elastomeric properties. On the other hand, free-radical copolymerization of styrene with acrylonitrile and 1,3-butadiene improves all three properties simultaneously. Similarly, although polyisobutylene is elastomeric, the polymer product consisting of saturated hydrocarbon chains cannot be crosslinked by sulfur vulcanization. Cationic copolymerization of isobutene with 1-3 mol%isoprene at very low temperatures yields a polymer with sufficient unsaturation to permit vulcanization by modified sulfur systems. The copolymerdescribed by Eq. (7.1), referred to as st atistical co polymer, has a distribution of the two monomerunits along the copolymer chain that follows somestatistical taw, for example, Bernoullian (zero-order Markov) or first- or second-order Markov. Copolymers formed via Bernoullian processes have completely random distribution of the two monomer units along the copolymer chain and, according to IUPACterminology, are referred to as randomcopolymers. Statistical copolymers are those in which the distribution of the two monomersin the chain is essentially random but influenced by the individual monomerreactivities. The reader is cautioned that the distinction between the terms statistical and random, recommended by IUPAC[1], has generally not been followed in the literature and most references ~ase the term random copotymer independent of the type of the statistical process involved in synthesizing the copolymer. There are three other types of copolymerstructures-alternating, block, and graft. The classification of copolymers according to these structural types and the nomenclature for copolymers have been described previously in Chapter 1. The present chapter is primarily concerned with the simultaneous polymerization of two monomersto produce random, statistical, and alternating copolymers. Graft copolymers and block copolymers are not synthesized by the simultaneous polymerization of two monomers. These are generally

581

Chain Copolymeriza~ion obtained by other types of reactions (see p. 641). BINARY

COPOLYMER

EQUATION

The composition of a copolymer produced by simultaneous polymerization of two monomersis usually different from the composition of the comonomer feed from which it is produced. This shows that different monomershave different tendencies to undergo copolymerization. These tendencies often have little or no resemblance to their behavior in homopolymerization. Somemonomersare more reactive in copolymerization than indicated by their rates of homopolymerization, and some monomers are less reactive. Thus, vinyl acetate polymerizes about twenty times as fast as styrene in a free-radical reaction, but the product in free-radical polymerization of a mixture of vinyl acetate and styrene is found to be almost pure polystyrene with practically no content of vinyl acetate. By contrast, maleic anhydride, which has very little or no tendency to undergo homopolymerization with radical initiation, undergoes facile copolymerization with styrene forming one-to-one copolymers. The composition of a copolymer thus cannot be determined simply from a knowledge of the homopolymerization rates of the two monomers. The simple copolymer model described here, however, accounts for the behavior of manyimportant systems and the entire process is amenable to statistical calculations which provide a good deal of useful information from few data. Thus, it is possible to calculate the distribution of sequences of each monomerin the macromolecule and the drift of copolymer composition with the extent of conversion of monomersto polymer. To predict the course of a copolymerization we need to be able to express the composition of a copolymer in terms of the concentrations of the monomersin the reaction mixture and the relative reactivities of these monomers. In order to develop a simple model, it is necessary to assume that the chemical reactivity of a propagating chain (which may be freeradical in a radical chain copolymerization and carbocation or carboanion in an ionic chain copolymerization) is dependent only on the identity of the monomerunit at the growing end and independent of the chain composition preceding the last monomerunit [2-5]. This is referred to as the first-order Markovor terminal model of copolymerization. Let us consider the case for the copolymerization of two monomers M1and M2. Although copolymerization has been more extensively studied using radical initiation, and radical copolymerization is also more important than ionic’copolymerization, we will consider here the general case without specification as to whether polymerization occurs by a free-radical or ionic mechanism. To generalize, an asterisk(*) will be used-instead of the con-

582

Chapter 7

ventional dot(" ) used in radical polymerization or plus (+) and minus signs used in ionic polymerizations-to indicate the active center of chain growth. Copolymerization of the two monomers M1 and M2 would thus lead to two types of propagating species-one with M~at the propagating end and the other with M~,where the asterisk represents either a radical, a earbocation ion, or a carboanion, depending on the modeof initiation. If it is assumed that the reactivity of the propagating species is dependent only on the monomerunit at the end of the chain (referred to as end or ultimate unit), four propagating species are then possible. Representing the propagating chains simply by their asterisked end units, these propagation reactions can be written as: M~

+

M1

~

M~

(7.2)

M~

+

M2

~

M~

(7.3)

M~

+

M1

~

M~

(7.4)

M~ + M2 k-~ 22 M~ (7.5) where the first subscript on the rate constant refers to the active center of the propagating chain and the second to the monomer.Reactions (7.2) and (7.5) where the reactive chain end adds the same monomerare often referred to as homopropagationor self-propagation. Propagation reactions involving addition of another monomer(Reactions 7.3 and 7.4) are referred to as cross-propagation or cross-over reaction. It is assumed in the above scheme that the reaction is carried out below the ceiling temperature of both the monomersand the various propagation reactions are irreversible. In order to simplify the kinetic formulation of copolymerization it is assumed that a steady state mechanismapplies, in which the concentration of each propagating chain type, that is, the concentration of each of M~and M~, remains constant. This assumption requires that the rate of conversion of M~to M~must equal that of M~to M~, or in mathematical terms k12 [M~] [M2] -- k21 [M~] [MI]

(7.6)

The rates of disappearance of the two types of monomersare given by [see Eqs. (7.2) to (7.5)1

dt

dt

-- kll. [MI] [M1] -}- k21 [M~] [M1]

-

[M2]+ [M;]

(7.7)

(7.S)

It should be noted that the rate of disappearance of each type of monomer is synonymouswith its entry into the copolymer chain.

583

Chain Copolymerization Dividing Eq. (7.7) by Eq. (7.8) and combining the resultant with Eq. (7.6) we obtain

diM1]

_

2]gXlk21 [M~][M1] k12[M2] q- k21[M~][M1]

d [M2]

k21[M~21[M1] q- k22 IMP] [M21

By defining the parameters rl and r2, representing ratios, as rlkl 2 , and r2 --

equation

(7.9)

monomerreactivity (7.10)

and substituting them into Eq. (7.9) after dividing the top and bottom the right side of this equation by k21 IMP] [M2] one finally obtains d[Ml_~!] d[M2]

= [M1] rl [M1] + [M~] [M2] [M~] + r2[M2]

(7.11)

This is the so-called copolymer equation or the copolymer composition equation. The ratio d [M1]/d [M2] in Eq. (7.11), representing the ratio the rates at which the two monomers M1 and M2 enter the copolymer, gives the molar ratio of the two monomerunits in the copolymer (being formed at a given instant), and hence is referred to as the copolymer composition. According to Eq. (7.11), the copolymer composition depends on the concentrations of the two types of monomersin the feed, namely, IMp] and [M2] and the monomerreactivity ratios rl and r2. As defined by Eq. (7.10), the monomerreactivity ratio is the ratio the rate constant for a reactive propagating species adding its own type of monomerto the rate constant for its addition of the other monomer.The monomerreactivity ratio can thus be looked upon as the relative tendency for homopropagation and cross-propagation. The tendency of two monomers to copolymerize is noted by r values between zero and unity. An rl value of zero would mean that M~is incapable of undergoing homopolymerization in the presence of M2. While an r~ value greater than unity means that ~ preferentially adds M1instead of M2, an rl value less than unity means that M~preferentially adds M2. For example, an rl value of 0.5 would mean that M~ adds M2twice as fast as M1. It is evident from Eq. (7.10) that the values of rl and r2 refer to pair of monomers undergoing copolymerization. Thus the same monomer can have different values of ri in combination with different monomers, e.g., acrylonitrile has (rl, r2) values of (0.35, 1.15), (0.02, 1.8), (1.5, and (4.2, 0.05) at 50°C in flee-radical copolymerization with acrylic acid,

584

Chapter 7

isobutylene, methyl acrylate, and vinyl acetate, respectively, each one being designated as M2 and the other monomer, acrylonitrile, as M1.

Problem7.1 The above derivation

of the copolymer composition equation [Eq. (7.11)] involves the steady-state assumption for each type of propagating species. Show that the same equation can also be derived from elementary probability theory without invoking steady-state conditions [6-8].

Answer: Let = Pll

probability that M~will add MI rather than M2, where the first subscript designates the active center and the second the monomer.

P12 = probability

that

M~will add M2 rather

than M1

Since termination occurs rarely in the formation of high polymer, we can neglect it for the purpose of this analysis. This means that Plx + PJ.2 In the same way,

= 1

(PT.l.1)

(P7.1.2) P22 ÷ P21 = 1 Theprobability thatpropagating species M~ addsan M~ unitis equalto therate of thisreaction divided by thesumof theratesof allreactions available to this radical. Thisis theprobability P~Ithatan MI unitfollows an M~ unitin the copolymer.Hence Pll

=

kll [M~][M1]

= rl

[M~]

kl, [M~][Ma]+ k,~ [M~][M2]rl [M~]÷ [Mz]

(P7.1.3)

Similarly, the other probabilities are obtained as P12 = p~l

=

[M2]

n

+ [M~]

[Mx]+ [M2]

p22 = ~’~ [M2] [M~] ÷ r2 [M2]

(P7.1.4) (P7.1.5) (P7.1.6)

Let "~(M1) = number average sequence length of monomer M~, that average number of M~monomerunits that follow each other consecutively in a sequence uninterrupted by M2 units but bounded on each end of the sequence by M2 units. Similarly, 5(M2) = number average sequence length of monomerM2. order to evaluate 5(M~) and 5(M2), it is necessary to determine the distribution of sequence lengths of each monomerin the copolymer.

585

Chain Copolymerization

To determine the distribution of sequence lengths of M1, an M1unit in the copolymer is selected at random. If this unit is part of a sequence of x number of M1units, reaction (7.2) would have been repeated (x - 1) times. Since probability of one such an event is Ply, as shown above, the probability that it occurs (z - 1) times is (Pll) z-1. If the M1sequence is exactly x units long the (x - 1) reactions of M~with M~must be followed by reaction (7.3). placement has the probability P~2 = 1- Pu [see Eq. (P7.1.1)]. It can thus concluded that the probability that the original M~unit was part of a sequence of x such units is PI{-~ (1 - P~I). But the probability that the sequence contains x number of M1 units is also the fraction of all M~sequences which contain x units. That is to say, it is the numberdistn’bution function n~(M1)for M~-sequence lengths : (P7.1.7) nz(M1) = Vii z-l(1 _Pll) = -~ O12 _p11~-x Similarly, units is

the fraction of all M2sequences that contain exactly x number of M~_

n~(M2) = P22~-1 z-1 (1 - P22) = P21 P22 The number average sequence length of M1 is then given by

(P7.1.S)

oo

5(M~) = ~-~n~(M,)

(e7.1.9)

[This is completely analogous to the definition of number average molecular weight M,~ in Eq. (4.3). A number average quantity is always the sum of products values of that quantity times the corresponding fraction of the whole sample which is characterized by the particular value.] Substituting Eq. (P7.1.7) into Eq. (P7.1.9), -~ 5(M1)

= £XPlzPI{ = P~2 (1 + 2P1, + 3Pll ~ + 4P~1a + ......

)

(PT.l.10)

For P~ < 1 which holds in a copolymerization, the expansion series (P7.1.10) is 1/(1 -- Pll) 2 and l~q. (PT.l.10) becomes 1 P12 ~(M1) = (1 - Pll) 2 - P12 = 1 -1-

r1~-~] IMp]

in Eq.

(P7.1.11)

In a similar manner one obtains P21 2 ~(M2) = (1 - P22)

--

P21

1

--

[M~]

1 q- r2 IMp]

(P7.1.12)

The mole ratio of monomers M~and M2 contained in the copolymer is given by the ratio of the two number-average sequence lengths, 5(M~) _ diM1] [M~] (r l [M 1] + [M~]) ¯ (M~) d [M2] [M2] ([M1] -]- r2 [M2])

(P7.1.13)

586

Ghap~er7

whichis exactly the sameresult as Eq. (7.11). Thusthe copolymerequation holds for copolymerizations carried out under both steady-state and non-steady-state conditions providedthat the reactivity of a propagatingspecies is dependentonly on the end unit, depropagationdoes not occur, and high polymeris formed. Equation (P7.1.13) [and also Eq. (7.11)] describes firs t-order Mark ov or t erminal model of copolymerization.

Range of Applicability

of Copolymer Equation

The simple copolymerequation [Eq. (7.11)] has been experimentally verified in innumerable comonomersystems. The equation is equally applicable to radical, cationic, and anionic chain copolymerizations, although the rl and r2 values for any particular monomerpair can be drastically different in the three types of chain copolymerization. For example, for the monomerpair of styrene (M1) and methyl methacrylate (M2) the I and r 2 values a re 0 .52 and 0.46 in radical copolymerization, 10 and 0.1 in cationic polymerization, and 0.1 and 6 in anionic copolymerization. Methyl methacrylate as expected has higher reactivity in anionic copolymerization and lower reactivity in cationic copolymerization, while the opposite is the case for styrene. Thus the copolymer obtained from an equimolar styrene-methyl methacrylate feed is approximatelya 1:1 copolymerin the radical case but is essentially a homopolymerof styrene in cationic copolymerization and a homopolymer of methyl methacrylate in anionic copolymerization. This high selectivity of ionic copolymerizationlimits its practical use. Since, moreover, only a small number of monomersundergo ionic copolymerization (see Chapter 8), the range of copolymer products that can be obtained is limited. On the other hand, almost all monomersundergo radical copolymerization and thus a wide range of copolymers can be synthesized. For any specific type of initiation (i.e., radical, cationic, or anionic) the copolymer composition equation is independent of many reaction parameters. Since no rate constants appear as such in the copolymer equation, the copolymer composition is independent of differences in the rates of initiation and termination or of the presence or absence of inhibitors or chain transfer agents. Thus the same copolymer composition is obtained irrespective of whether initiation occurs by the thermal homolysisof initiators (such as AIBNor peroxides), photolysis, radiolysis, or redox systems. Under a wide range of conditions the copolymer composition is also independent of the degree of polymerization. The limitation on the above generalization is that the copolymer be of high molecular weight. It may be recalled that the derivation of Eq. (7.11) involved an assumption that the kinetic chains

587

Chain Copolymerization

were long so that initiation and termination reactions could be ignored compared to propagation events.

TYPES OF COPOLYMERIZATION Before measurementsof reactivity ratios are reviewed, it is useful to consider what the absolute magnitudes of these parameters imply. Depending on the values of rl and r2, and of the rlr2 product four types of copolymerizations can be recognized.

Alternating

Copolymerization:

rl = r2 = 0

A zero (or a nearly zero) value for the reactivity ratio means that the monomeris incapable of undergoing homopolymerization and its radical prefers to add exclusively to the other monomer.This leads to alteration of the two monomerunits along the copolymer chain. For rl = r2 = 0, Eq. (7.11) reduces aIM1]

_ 1

(7.19,)

Copolymerization of the two monomerstherefore produces an alternating copolymer (in which the two monomerunits alternate in a regular fashion along the chain) irrespective of the composition of the monomerfeed.

Ideal Copolymerization:

rl = r2 = 1 and rlr2 = 1

A value of unity (or nearly unity) for the monomerreactivity ratio signifies that the rate of reaction of the growing "chain radicals towards each of the monomersis the same, i.e. kll ~ k12 and k22 ~-- k2~ and the copolymerization is entirely random. In other words, both propagating species M~and M~have little or no preference for adding either monomer.The copolymer composition is the same as the comonomerfeed with a completely random placement of the two monomersalong the copolymer chain. Such behavior is referred to as Bernoullian. Free-radical copolymerization of ethylene and vinyl acetate and that of isoprene and butadiene are examples of such a system, but this is not a commoncase. Randommonomerdistributions are obtained more generally in a situation where both types ofradicals have exactly the same preference for the same type of monomeras represented by the relationship

kll k~l -kl~ k~

or r~ --

1 r~.

(7.13)

Chapter 7

588

Equation (7.13) means that k11/k12 and k21/k22 will be simultaneously either greater or less than unity or in other words, that both radicals prefer to react with the same monomer.All copolymers whose ?’1r2 product equals 1 are therefore called Meal copolymers or random copotymers. Most ionic copolymerizations are characterized by the ideal type of behavior.

Problem7.2 Use simple probability concepts to justify the following statement: a value of unity for the product rlr2 signifies that the likelihood that an M1unit in the copolymerchain follows an Maunit is the same as the likelihood that it follows an M2unit.

According to the reaction scheme of the simple copolymer model, M1-Mabonds are formedonly by reaction (7.2). The probability that a propagating species (that is, endingin an M1 unit) adds an Maunit is equal to the rate of this reaction divided by the sumof the rates of all reactions available to this propagating species. This is the probability Pn given by [see Eq. (P7.1.3)]: Pn =

r~ IMp] rl [M~]+ [M21

(P7.2.1)

Similarly, the probability P2a that an M1unit follows is given by [see Eq. (P7.1.5)]: P2~ =

[M1] r~_ [M2]+ [Ma]

an M2 unit

in the polymer

(P7.2.2)

If r~r2 = 1, then Pn and P~ defined above are equal. That is to say, the likelihood that an M1unit follows an M~unit equals the likelihood that it follows an M2unit in the copolymerchain. The absolute values of the probabilities Pn and P2a will dependon the relative concentrations of monomers in the feed, but the equivalence of the probabilities is independent of the feed and copolymer compositions.

For rl = r2 = 1, Eq. (7.11) reduces diM1] _ [M~] diM2] [M2]

(7.14)

which means that the copolymer composition will always be the same as the feed composition. The relative amounts of the two monomerunits in

589

ChMnCopolymeriz~ion the copolymer chain are determined by the relative monomerunits in the feed. For rlr2 = 1, Eq. (7.11) reduces d [M1] diM2]

_ rl

concentrations of the

(7.15)

[M2]

The relative amounts of the two monomerunits along the copolymer chain are thus determined by the relative concentrations of the monomerunits in the feed and the relative reactivities of the two monomers.Thus a very important practical consequenceof ideal copolymerizations is that it becomes progressively more difficult to produce copolymers containing appreciable amounts of both monomersas the difference in reactivities of the two monomersincreases. It should be noted that the term ideal copo~ymerization does not in any sense connote a desirable type of copolymerization. Somecommercially important examples of random free-radical copolymerizations include styrene (rl ---- 0.8)-butadiene (r2 ---- 1.4) for which rlr2 = 1.1 and vinyl chloride (rl ---- 1.4)-vinyl acetate (rg. = 0.65) which rlr2 = 0.9. In these products the proportion of a given monomer in the copolymer depends on the feed concentrations and reactivity ratios [Eq. (7.11)]. Random-Alternating

Copolymerization:

0 < fir2

<

1

Copolymerizations in which 0 < r~r2 < 1 are intermediate between alternating and random. Most copolymer systems fall in this category. One can speak of such systems as belonging to random-alternating type. As ~’1/’2 product decreases, cross-propagation reactions are favored and the monomerunits in the copolymer chain show an increasing tendency toward alternation. On the other hand, as fir2 approaches 1, the copolymer chain composition becomes increasingly random. Whenone reactivity ratio is greater than unity and the other is less than unity, either propagating species will prefer to add monomersof the first type. Relatively long sequences of this monomerwill thus be formed if the reactivity ratios differ sufficiently. A special situation arises whenrl >> 1 and r2 << 1 or vice versa. In this case, the product composition will tend toward that of the homopolymerof the more reactive monomer.Such reactivity ratios reflect the existence of an impractical copolymerization. An example of this type of behavior is the radical chain polymerization of styrene-vinyl acetate system, where monomerreactivity ratios of 55 and 0.01 are observed. The large differences between the monomerreactivity ratios imparts a tendency toward consecutive homopolymerization of the two monomers. For example, when r~ >> 1 and r:~ << 1, both ~ and

590

Chapter 7

M~ preferentially add monomerM1till the monomeris consumed, which is then followed by homopolymerization of M2. This is therefore a case of consecutive homopolymerizations. Block

Copolymerization:

rl

> 1,

r2 > 1

An rl-value greater than 1 means that an M{ propagating species would add monomer M1 in preference to M2, and so many units of M1 would successively add to the growing chain until an M2unit happened to add, converting the growing chain from ME type to M~type. Since r2 is also more than 1, the ~ propagating species would then preferentially add manyM2units in succession until an M1 unit happened to add, converting the chain again to the ME type. This process would give rise to a block copo(ymer consisting of long sequences of each monomerin the copolymer chain. With an increase in the value of r, the tendency of each radical to add its own type of monomer would increase and in the limit when both rl and r2 became sufficiently large, the two types of monomerswould simultaneously homopolymerize in each other’s presence. However, such reactivity combinations are not knownin free-radical copolymerizations, but they can be found in other systems. Since there is no established instance of free-radical copolymerization where both rl and r2 are greater than unity, block copolymersby free-radical initiation are thus to be madeby homopolymerization using special techniques. These are discussed toward the end of this chapter. INSTANTANEOUS COMPOSITIONS FEED AND COPOLYMER

OF

The copolymer equation (7.11) can be converted to a more useful form expressing concentrations in terms of mole fractions. Let fl and f2 be the mole fractions of monomersM1and M2in the feed, that is, fl

= 1 -- f2 --

[M,] [M1] d-[M2]

(7.16)

and F1 and F2 be the mole fractions of monomers M1 and M2 in the polymer being formed at any instant, so that (7.17) d[M1] d([M~] + [M2]) CombiningEqs. (7.16) and (7.17) with the copolymer equation (7.11), obtains rifl 2 q- fir2 (7.18) -~ : r~fl ~ d- 2fir2 -t- ~ r2f2 F1

= 1-

F~

=

Chain Copolymerizafion

591

Equation (7.18), which is also called the copotymer equation, gives the mole fraction of monomerM1in the copolymer whose feed contained fl mole fraction of monomerM1. It is more convenient to use than its previous form [Eq. (7.11)]. It should be noted that/71 gives the instantaneous copolymercomposition and both fl and/71 change as the polymerization proceeds. The composition of the copolymeras a function of conversion can be derived by integration of Eq. (7.18). Equation (7.18) may be used to calculate the instantaneous composition of copolymer as a function of feed composition for various monomer reactivity ratios. A series of such curves are shownin Fig. 7.1 for ideal copolymerization, i.e., rlr2 = 1. The term ideal copolymerization is used to showthe analogy between the curves in Fig. 7.1 and those for vapor-liquid equilibria in ideal liquid mixtures. The vapor-liquid composition curves of ideal binary mixtures have no inflection points and neither do the polymercomposition curves for random copolymerization in which rlr~ = 1. Such monomersystems are therefore called ideal. It does not in any sense imply an ideal type of copolymerization. It is evident from Fig. 7.1 that only a small range of feed compositions give copolymers containing appreciable amounts of both components unless monomershave very similar reactivities. Suppose, for example, that it is desired to synthesize an ideal copolymer with 60 mol%M1in a system where z’l --= 0.5. FromFig. 7.1 it is seen that a feed composition of about 75% M1is required. However, since/71 ~ fl, there occurs a drift in monomer composition as copolymer is formed. Therefore copolymer composition changes with conversion. To obtain a constant copolymer composition of 60% M1, it would be necessary to maintain a constant feed composition of 75%, such as by adding fresh M1to the feed. For cases in which rl = r2 = 1, the composition of the copolymer (/71) will always be the same as the feed composition (fl)- For this copolymer system obviously there occurs no drift in composition with conversion.

Problem 7.3 It is desired to form a copolymer from CH~=CHX (M1) and CH2=CHY (M2), containing twice as manyX groups as Y groups. The monomers copolymerizeideally, with M~adding M1twice as fast as M2. Describe the procedure as well as the feed compositionyou might use to makethis polymer. Answer: Since (M~+ M1)reaction is twice as fast as (M~+ Ms)reaction, --- -- kn/ kl2 = 2. For ideal polymerization rlr2 = 1. Therefore, r2 = 0.5 It is desired to haveF1 = 2/3. Todeterminethe correspondingfl it is necessary to rearrange the copolymerequation (7.18) into a form which enables fl to calculated from /;’1 for a given pair of ~1 and-r2 values. Remembering f~ = 1 fl, then from Eq. (7.18), 2 q- flf2 -/7’I(’rl/? -1- 2:1f2q-~’2f~) = ’rlfl

- 1) + :i(I

- - + - /i)

592,

Chapter 7 f12rl (_~1 - 1) + f~2 (1 -.2ivy) + f~ (2F~ - 1) + F~r2 2 - 2 fl + 1)=

f,2 IF, (rl + r2 - ~.) + (1 - ~1)]+ f, [2F,(1 - ~) - 11+ y~r~

(],7.3.1)

This equation is a qua~dratic in fl and can be solved in the usual way. For the calculations required it is necessary to evaluate fl for F1 = 2/3, rl = 2, r2 = 0.5. Equation(P7.3.1) simplifies

2f12+ fl-- 1 -= 0 Only one of the two solutions will be meaningful. Solving, fl = 0.5. (To check the result, substitution of fl = 0.5 into Eq. (7.18) gives F~= 0.667.) Since the copolymerformedis richer in M1as comparedto feed, calculated amountsmust be added to the monomermixture, continuously or periodically, to maintain the compositionat fl = 0.5 as the reaction progresses. Figure 7.2 shows curves for several nonideal cases, that is, where fir2 ~ 1. It is seen that whenboth r I and r2 are less than 1 there exists somepoint on the Fl-versus-fl curve where the copolymer composition equals the feed composition and at this point the curve crosses the line F1 = fl (that is, the diagonal line). At this point of intersection, polymerization proceeds without change in either feed or copolymer composition. Distillation terminology is again borrowed for this instance. Azeotropic copolymerization is said to occur at such points and the resulting copolymers are called azeotropic copolymers. 1.0

0.4 0.2 0

~ 0.2

0./,

0.6

0.8

1.0

Figure 7.1 Copolymerizationdiagrams (without inflection points) showinginstantaneous compositionof copolymer(mole fraction F1) as a function of monomer composition (mole fraction fl) for copolymerswith the values of rl = 1/r~ for ideal copolymerization.

Chain Copolymerization

593

1.0

0.6

.

0.2

0

0.2

0./-.

0.6

0.8

1.0

f~ Figure 7.2 Copolymerizationdiagrams with inflection points showingcomposition of copolymerF~ as a function of monomer composition f~ for the values of the reactivity ratios rl/r~ indicated. Since all azeotropic copolymers must have a point of constant composition, the critical composition (fl)c for the azeotrope can be evaluated by solving Eq. (7.11) with d [M1] / d [M2] = [M~] / [M2] or Eq. (7.18) F1 = fl. Whenthis is done,

_ ([M,]~

(7.~)

1 - r2 (f,)c =(2 - rl -

(7.20)

1-

and

Note that f~ in the above equation is physically meaningful (0 _< f~ <_ 1) only if z’] and rg. are both either greater or less than unity. (If ’r 1 = ~’2 = 1, all values of fl are azeotropic compositions.) Since the case of ~’] > 1, r2 > 1 is unknownin free-radical systems, the necessary conditions for azeotropy in such copolymerizations is that rl < 1, r2 < 1 (see Fig. 7.2).

594

Chapter 7

Equation (7.20) predicts the feed composition that would yield invariant copolymer composition as the conversion proceeds in a batch reactor. It should be noted that comonomerratios that are near but not equal to the estimated azeotropic value may also produce copolymers whose compositions are constant for all practical purposes. It is seen from Fig. 7.2 that the permissible range of feed compositions for which this "approximate" azeotropy occurs is greater the closer the two reactivity ratios are to each other.

Problem 7.4 WhenmonomersMI and M2are copolymerized, an azeotrope is formed at the feed ratio of 1 mol of M1to 2 tool of M2. MonomerM1is known not to homopolymerize.Will a polymer formedat 50%conversion from an initial mixture of 4 mol of M1and 6 mol of M2contain more of M1or less of M1than a polymer formed at 1%conversion? Answer: Since M1does not homopolymerize,ra = 0. Azeotropic feed composition, fl = 1/3 = (1 - r~)/(2 - 0 - r2). This gives r2 = 0.5. For a feed composition 0.4 and f~ = 0.6, Eq. (7.18) with rl = 0 and r~ = 0.5 gives F1 = 0.36. Therefore the feed compositionwill drift toward higher fl at higher conversion and hence F1 at 50%conversionwill be greater than _b"l at 1%conversion.

INTEGRATED

BINARY

COPOLYMER

EQUATION

The copolymer equations, Eqs. (7.11) and (7.18), give inst antaneous copolymer composition, i.e. the composition of the copolymer formed from a given feed composition at very low degrees of conversion (approximately < 5%) such that the composition of the monomerfeed may be considered to be essentially unchangedfrom its initial value. For all copolymerizations except when the feed composition is an azeotropic mixture or where rl = re = 1, the comonomer feed and copolymer product compositions are different. The comonomer feed changes in composition as one of the monomerspreferentially enters the copolymer. Thus there is a drift in the comonomercomposition, and consequently a drift in the copolymer composition, as the degree of conversion increases. It is important to be able to calculate the course of such changes.

Problem 7.5 A monomerpair with rl -- 0.2 and r2 = 5.0 is copolymerized beginning with a molar monomerratio [M1]/[M2] = 60/40. Assumingthat the copolymercompositionwithin a 10 mol%conve.rsion interval is constant, calculate instantaneous monomerand copolymercompositions and cumulative average copolymercompositions at 10 mol%conversion intervals up to 100%total conversion. Showthe results graphically as change in composition of the copolymer and the monomermixture during copolymerization.

595

Chain Copolymeriza~ion Answer: In intervalI: fl = 60/(60+40) = 0.60.From Eq. (7.11),El = 0.2308. At the end of interval 1 (i.e.,

after 10 mol%conversion),

M~converted = 2.308 mol M1 remaining ---- 60 - 2.308 or 57.692 mol fl = 57.692/90 = 0.641 _b-’~= 2.308/10----- 0.2308 The residual mixture with fl = 0.641 will be the starting mixture for the interval 2. From Eq. (7.11) then, F1 = 0.2631. Hence at the end of interval 2 (20 tool% conversion), M1 converted = 2.308 + 2.631 = 4.939 tool M1 remaining = 60 - 4.939 = 55.061 mol fl = 55.061/80 = 0.6883 F1 = 4.939/20 = 0.247 The results obtained by proceeding in this way are tabulated below: mol% Conversion Interval (cumulative) FIa ~ f, a 0.23 1 10 0.60 0.23 0.25 2 20 0.64 0.26 0.27 3 30 0.69 0.31 4 0.74 0.37 0.29 40 5 50 0.80 0.45 0.32 0.58 0.37 6 60 0.88 7 70 0.95 0.79 0.43 8 80 1.0 1.0 0.50 9 90 1.0 1.0 0.55 10 0.60 100 1.0 1.0 aInstantaneous value at the beginning of each interval (assumedto be constant withinthe interval). bThis copolymeris really a mixture of different copolymersand, at higher conversions, a mixture of these copolymersas well as someM~homopolymer. The results are shown graphically in Fig. 7.3. Comment:If one makes the conversion interval smaller and smaller, this corresponds to an integration of the copolymer equation (see below).

To follow the composition drift of both the comonomer feed and the copolymer formed requires integration of the copolymer equation. This problem is rather complex. The most convenient approach utilizes a numerical or graphical method developed b} Skeist [9] for which Eq. (7.18) forms the basis. Consider a system initially containing a total of N moles of the two monomers; choose M1 as the monomer in which F1 > f~ (i.e.,

596

Chapter 7

20

40

60

80

100

Conversion(moL°/o) Figure 7.3 Change in the composition mixture during copolymerization.

of the copolymer and the monomer (fx)0 = 0.60, rl = 0.20, r2 = 5.0. (Problem

7.5). the polymerbeingformed contains moreM1than the feed). WhendNmoles of monomers have polymerized,the polymerwill contain FIdNmolesof M1and the feed content of M~will be reducedto (N - dN)(f~ - df~) moles.A material balancefor monomer M~requires that the molesof M~ copolymerized equal the difference in the molesof M~in the feed before andafter the reaction, or fin - (N - aN )(fl - dr1) = FIdN (7.21) which becomes (neglecting the small term dfldN) dN df~ (7.22) ~ = -~l

--

-fl

597

ChainCopolymerization Integration

gives lrI-N-- f(fl)fl° (-~’1-/~’0

dfl

-

(7.23)

where No and (fl)0 are the initial values of M and f~. For given values of rl and r2, the quantities F~ and 1/(F1 -- fl) are computed from Eq. (7.18) at suitable intervals for 0 < f~ < 1. The indicated integration may then be performed graphically or numerically to give the degree of conversion /9 (= 1 -- N/No) required for a change in feed composition from (fl)0 to f~. Through a repetition of this process for suitably chosen values of fl, it is possible to construct the relationship between fl and the degree of conversion p. The average overall copolymer composition for any conversion p can be determined by graphical integration of a plot of F1 versus f~ or from the amounts of residual monomers and those present initially (see Problem 7.6).

Problem7.6 Derive an equation that gives the cumulative or average composition of the copolymer formed at a given overall conversion of the monomers. Answer: Consider a batch polymerization mixture containing initially (N1)o tool of monomer M1 and (N2)o tool of monomerM2. After a fraction p of the initial monomers have been polymerized, the unreacted monomersare, respectively, N1 and N2. The mole fractions of monomersM1 and M2 in the feed after a degree of conversion p are A and f2, the corresponding initial values being (fl), and (f2)0- Then, (N1)o + (N2)o (N1)o = (fl)0No, (N2)0 (f ~)oNo NI = f~(1 - p)No,. Nu = fz(1 - p)No

(P7.6.1)

Since the average mole fraction of M~in copolymer, F~, is the ratio of the number of moles of M~converted divided by the total number of moles of M1 and M2 polymerized in the same interval,

F1 =

(N1)o NI [(N1)0 - N,] + [(N2)o -

Substituting from the above definitions ~11 = (fl)o

(P7.7.1), one obtains

- fl(1 (P7.6.2) P The cumulative average copolymer composition can be calculated in a straightforward manner by entering Eq. (P7.6.2) with the cumulative value of p and the initial value of (f~),0. [As a check, the value of ~ at p = 1 must equal

598

Chapter 7

Equation(7.23) has been integrated to the useful closed form [10]:

(7.9.~) whichrelates the degreeof (overall) conversion(1 N/No) tochanges in the monomer feed composition. Thezero subscripts indicate initial quantities and the other symbolsare given by r2 (1 -"/

= (1 --

rl 9-2)

’

(1 --

(1 - 9-z9-~) 9"1)(1 - r2)

(5 = (1 (2 -- 9"1 -- r2)

(7.25)

It shouldbe noted that there are certain difficulties with the use of Eq. (7.24) for conversion-composition calculations. Thus, singularities occur r: = l, r2 = 1, r: -F r2 = 2, f: = di, and (fl)0 = 0 or (f2)0 all of whichresult in division by zero. Thedifficulties with the reactivity ratios equat to unity can be circumventedby using the special solutions [10] for these two cases as follows: For ?’1 = 1 and r2 ~ 1, Eq. (7.24) becomes

N

11

i:{t(f:)oJ 9"2 1 (fl)o-

1 .] (7.26) I fl-

For rl ~ 1 and r2 = 1, Eq. (7.24) becomes 9" 1

In --

No

B

fl

rl

- 1

[{ }1

J 1

/Sx)0

1 ]

/

(7.27)

Equation (7.24) was formulated so as to make6 equal to the azeotropic composition(f~)c Ice. Eq. (7.20)] for those systemsexhibiting azeotropic

599

Cha/n Copolymerizafion

composition behavior. However, this results in division by zero when rl + r2 = 2. To overcome, this difficulty, the third right-hand term of Eq. (7.24) may be rewritten as [11]: [(fl)0

-- ~]7 (fl)0(2-

-~- -rl r--2} -’+--- --r2- ---- 1

]7

J = £(2 - rl - r2) + - 1

(7.28)

It should be noted, however, that the difficulties with the reactivity ratios can also be circumvented simply by making a small, but for practical purposes insignificant, change in the values of the reactivity ratios. The copolymerization behavior for the cases (fl)0 -- 0, (f2)0 = 0 and (fl)0 = ¢5 which give rise to singularities are not systems of actual interest and so may be ignored. Equation (7.24) or its equivalent has been used to correlate the drift in the feed and copolymer compositions with conversion for a number of different copolymerization systems [10,12,13].

Problem 7.7 A mixture of styrene (M1) and methyl methacrylate 0VI2) polymerized at 60°C with initial composition (fl)0 = 0.80, (f2)0 = 0.20 and polymer obtained by precipitation at appropriate intervals was analyzed. Some of the conversion-composition data so obtained are given below: Oxygen in polymer (Wox), Conversion (we) wt.% wt.% 11.74 8.32 29.32 7.92 46.18 7.61 65.88 7.43 86.37 7.00 Source: Data from Ref. 14.

0.8091 0.8229 0.8391 0.8769 0.9648

With rl = 0.52 and r 2 ~. 0.46, calculate from Eq. (7.24) the changes in instantaneous monomerand copolymer compositions as a function of conversion and compare the results graphically with the above experimental data. Also calculate the cumulative average copolymer composition at different conversions. Answer: The essential procedure for calculating the composition drift with conversion "is that fl is decreased or increased in suitable increments from (fl), to 0 or 1.0. For each value of fl, the corresponding degree of conversion is obtained from Eq. (7.24) and the corresponding instantaneous copolymer composition from Eq. (7.18). With the monomer mixture composition, fl, and the degree of conversion p = 1 - N/No thus known, it is then easy to also calculate cumulative average copolymer composition F1 from Eq. (P7.6.2). For the given monomersystem and feed composition, Eq. (7.18) shows that F1 < fl, i.e., the

600

Chapter7

polymeris richer in M2as compared to the monomer feed; fl is therefore to be increasedin step incrrmentsfrom(fl)0 in the computation. Therelation betweenF1 and the oxygencontent (Woxwt.%)of the copolymer is easily obtained frommassbalance, taking note that each MMA unit in the copolymeraccounts for 2 oxygenatoms. Takingmolar massesof styrene and MMA as 104and 100 g/mol, the followingrelation is obtained: 32 - wo~ 32 + 0.04wo~ To convert weight%conversion(we) of monomers into molefraction degree of conversion(1 - N/No),the followingrelation is readily derived: N [4(fl)0 + 100] 1 No - 1- [~-~-~-]~ j (1-we/100) [Since for the given systemthe differences between(fl)0 and fl are small, 1- N/No ~ we/100and mole fraction conversion nearly equals the weight fractionconversion.] Thecalculated values of fl, F1, and F1 are plotted against molefraction degree of conversionalong with the given experimentaldata in Fig. 7.4. The results for f2 followfromthe relation fl + f~ = 1, andsimilarly for F2 andF2.

EVALUATION OF MONOMER REACTIVITY RATIOS Most procedures for evaluating rl and r2 involve the experimental determination of the compositions of copolymersformed from several different comonomerfeed compositions. These data of corresponding feed and copolymercompositionsare used in conjunction with the differential copolymerequation [Eq. (7.11) or (7.18)]. Copolymerizationsare carried out to as low degrees of conversion as possible (ca. < 5%) in order insure that the feed compositionis essentially unchanged.This is necesSary in order to minimizethe drift of copolymermakeupand consequent errors in the use of the differential equation. The copolymercomposition is determined either directly by analysis of the copolymeror indirectly by analysis of comonomerfeed. While comonomer feed compositions are typically analyzed by high-pressure liquid chromatography(HPLC)or gas chromatography(GC), the copolymercomposition is measured, depending on the type of the copolymer,by elemental analysis, chemicalanalysis, or physical analysis (such as refractive index, IR, UV,NMR,etc.). Though the techniques for analyzing comonomer compositionsare inherently more sensitive than those for copolymer composition,this is offset by the fact that the determination of copolymercomposition by comonomer feed analysis requires the measurementof small differences between large numbers. Various methods have been used to obtain monomerreactivity ratios from the copolymercomposition data. Several procedures for extracting reactivity ratios from the differential copolymerequation [Eq. (7.11)

Chain Copolymerizafion

601

0.6

0,4

0.2

O.Z, 0.6 0.8 0 Degreeof conversion (1-N/N O)

1.0

Figure 7.4 Copolymerization behavior of styrene (M1) and methylmethacrylate (M2) for (f~)o = 0.80, (f2)~ = 0.20, and rl = 0.53, r2 = 0.56. Experimental points oI A are from Problem 7.7.

(7.18)] are mentionedin the following paragraphs. Twoof the simpler methodsinvolveplotting of rl versus ~’~ or fi’l versus Plot of rl Versus 7"2 Also knownas the "methodof intersections," the methodfirst described by Mayoand Lewis[3] has been widely used for computingreactivity ratios from data fitted to the differential copolymerequation. In this procedure, Eq. (7.11) is recast to the form

1-~ ] "’) =[~1L~--]~,] -~

(~~)

~h~pte~ 7

1.0

0.8

0.6

0.4

0.2

0

0.2

0.4

0.6

0.8

1.0

Figure 7.5 Graphical determination of rl and ’r2 according to Eq. (7.30) for the system styrene/methylmethacrylate (M1 = styrene; M2= methyl methacrylate). (After Ref. 3.) or, equivalently, Eq, (7.18) to the form

If one knows experimentally the copolymer composition (/~1) corresponding to a given feed composition (fl) then one can calculate ~’2 values corresponding to various assumed values of rl and thus obtain a straight line plot of rl versus r2. Each experiment thus yields one straight line in the rl -- r2 plane (see Fig.7.5). If this procedure is repeated for different values of F~ and f~, a series of straight lines with different slopes should result. Theoretically, the lines should intersect in a single point corresponding to the actual values of rl and r2. However, because of errors in experimental results, the lines may not pass through a commonpoint and the area of the region in which the intersections occur is assumed to give the best values of rl and

603

Chain Copolymerization

Plot of F1 Versus fl Fineman and Ross [15] rearranged

Eq. (7.18)

(1 - fl)F1 = ----

to the form

(7.31)

fl)2F~j r~ r2

or

G = rlH - r2

(7.32)

where G = X(Y-1)/Y, H = X2/y, X = [M1]/[M2] = fl/(1 :-- f~), and Y d[ M1]/d[M2] = /7 1/(1 /7 1). [s ee Eq (7.16) and (7.17)]. If the term on the left side of Eq. (7.31) or (7.32) can be plotted against the coefficient of rl, a straight line should result, the slope of which is rl and intercept is r2.

Problem7.8 The initial

concentrations of styrene (M1) and acrylonitrile (Mg_) employed in a series of low conversion free-radical copolymerizations are given in the table below together with the nitrogen contents (% N by wt.) of the corresponding copolymer samples produced: [M1] mol/L [M2] mol/L % N in copolymer

3.45 1.55 5.69

2.60 2.40 7.12

2.10 2.90 7.77

1.55 3.45 8.45

Determine r~ and r.~ for the monomerpair by the Fineman-Ross method. Answer: -~ Molar mass of styrene (M~) repeat unit = 104 g mol Molar mass of acrylonitrile %N =

(M2) repeat unit = 53 g mo1-1 (1 -- F1)(14 g -I) tool × 100 F~(104 g tool -1) -1 + )(1 - F1)(53 g tool

which upon rearrangement for F1 gives 1400 - (%N × 53) 1400 + (~oN x 51) Table below gives fl = [M1]/([M~ -t- [M2]) and F~ calculated from %Ntogether with the composite quantities required to make a plot according to Eq. (7.31).

_~ =

0.69 0.52 0.42 0.31

0.6499 0.5800 0.5501 0,5200

f1(2~1 - 1) f~(1 - F1) (I - fl)FI (1 -- fl)2F1 1.0268 2.6688 0.2989 0.8499 0.1319 0.4289 0.0346 0.1863

Chapter 7

604

The data are plotted in Fig. 7.6 from which slope = rl and intercept = -r2. This gives rl = 0.40 and r2 = 0.04. Analysis of the data may also be done by regression analysis.

The best values of r" are obtained from slopes rather than intercepts. While Eq. (7.23) gives rl as the slope, it can be rewritten in form which makes r2 the slope: G/H = -re/n -{- ?’1 (7.33) The experimental composition data are unequally weighted by the MayoLewis and Fineman-Rossplots with the data for the high or low compositions (depending on the equation used) having the greatest effect on the calculated values of rl and r2 [16]. For example, the experimental data obtained at low [M2] in Eq. (7.32) or low [M1] in Eq. (7.33) have the greatest influence on the slope of a line corresponding to these equations. The same set of experimental data can thus yield different (rl, r2) sets depending on which monomeris indexed as MI and which is M2.

1.0 0.8 0.6 0.4 0.2 I

I

1.0

2.0

3.0

Figure 7.6 Plot according to the Fineman-Rossmethod (Problem 7.8).

Chain Copolymerization

60~

Linear least-squares regression can also be applied to a series of experimental values of/’1 and fl to obtain the best values of ~’1 and ~’2 from Eqs. (7.32) and (7.33). This procedure is statistically unsound, however, since Eqs. (7.32) and (7.33) do not meet the statistical requirements linear least-squares computations [16]. Kelen and Tudos [17] refined the Fineman-Ross linearization method by introducing an arbitrary positive constant ot into Eq. (7.34) to spread the data more evenly so as to give equal weighing to all data points. Their results are expressed in the form

where = G/(a

q- H)

(7.34a)

# = H/(c~

-b H)

(7.34b)

rl

Plotting r/ against ~z gives a straight line that yields --r2/a and ~’1 as intercepts on extrapolation to /~ = 0 and # = 1, respectively. The values of a, chosen as a = (HminHmax)1/2, where Hmin and Hmax are the lowest and highest H values, respectively, distributes the experimental data symmetrically on the plot. Even with the Kelen-Tudosrefinement there are statistical limitations inherent in the linearization method. It has been shown[18] that the independent variable in any form of the linear equation is not really independent while the dependent variable does not have a constant variance. The most statistically sound method of analyzing the experimental composition data is the nonlinear methodwhich involves direct curve fitting to the copolymer composition equation. Direct

Curve Fitting

Fromthe slope of the experimental _~1 versus fl plot an estimate of r 1 and ~’2 can be made by comparison with curves based on Eq. (7.18). The best values of r’~ and r’2 can be selected by determining which theoretical curve best fits the data by trial and error. A limitation of the methodis the relative insensitivity of the curves to small changes in ~’1 and r2. The pros and cons. of both the linearization and nonlinear methods have been discussed in detail, along with approaches for the best choice of feed compositions to maximizethe accuracy of the I" 1 and v2 values [17,19,20]. A serious drawback in the use of a differential form of the copolymerization equation [Eq. (7.11) or (7.18)] is the assumption that the composition does not change during the experiment, which is obviously not true. One carries out the polymerization to as low a conversion as possible, but there are limitations since one must be able to isolate a sufficient sample of the copolymer for direct analysis, or, if copolymer analysis is done

606

Chapter 7

indirectly via the change in feed composition, there must be a significantly measurable change in the feed composition. These limitations can be overcome by the use of an integrated form of the copolymer composition equation [18,21]. The change in copolymer composition or feed composition with conversion is measured and the data are curve-fitted to an integrated form of the copolymer composition equation. For example, one method uses Eq. (7.24) relating the degree of conversion (1 N/N o) to changes in the comonomerfeed composition fl or fg.. Experimental data on the variation of feed composition with conversion are plotted as fl or f2 versus (1 -- N/No) to yield curves like those in Fig. 7.4. Using computational techniques one then determines the best values of/’1 and r2 that fit Eq. (7.24) to this experimental curve.

Table 7.1 MonomerReactivity Ratios in Radical Copolymerization Monomer (M1)

Monomer (M2)

rl

r2

T (°C)

Acrylic acid

Styrene Vinyl acetate

0.25 8.7

0.15 0.21

60 70

Acrylonitrile

Acrylamide Ethyl vinyl ether Methylacrylate Methyl methacrylate Styrene Vinyl acetate Vinyl chloride Vinylidenechloride

0.86 0.69 1.5 0.14 0.020 5.5 3.6 0.92

0.81 0.060 0.84 1.3 0.29 0.060 0.044 0.32

40 80 50 70 60 70 50 60

1,3-Butadiene

Methyl methacrylate Styrene Vinyl chloride

0.75 1.4 8.8

0.25 0.58 0.04

90 50 50

Ethylene

Acrylonitrile Tetrafluoroethylene Vinyl acetate

0 0.38 0.79

7.0 0.10 1.4

20 25 130

Maleic anhydride

Acrylonitrile Methyl methacrylate Styrene Vinylacetate Vinylchloride

0 0.01 0.005 0 0

6.0 3.4 0.050 0.019 0.098

60 75 50 75 75 .. (continued)

6O7

Chain Copolymeriza~ion Table 7.1 (Continued) r2

T (°C)

Monomer (Mr)

Monomer (M2)

rl

Methacrylic acid

Acrylonitrile Styrene Vinylchloride

2.4 0.60 24

0.092 0.12 0.064

70 60 50

Methacrylonitdle

Styrene Vinyl acetate

0.25 12

0.25 0.01

80 70

Methylmethacrylate

Styrene Vinyl acetate Vinyl chloride Vinylidene chloride

0.46 20.0 9.0 2.4

0.52 0.015 0.07 0.36

60 60 68 60

Styrene

Ethyl vinyl ether Vinylacetate Vinylchloride Vinylidenechloride

90 42 15 1.8

0 0 0.01 0.087

80 60 60 60

Vinyl acetate

Ethyl vinyl ether Vinyl chloride Vinylidenechloride

3.4 0.24 0.03

0.26 1.8 4.7

60 60 68

DatamostlyfromR. Z. Greenley,"FreeRadicalCopolymerization ReactivityRatios," pp 153-266in Chap.II in PolymerHandbook,3rd ed., (J. Brandrupand E. H. Immergut, eds.), Wileylnterscience, NewYork(1989).

Somerepresentative values of rl and r2 in radical copolymerization for a number of monomerpairs are shown in Table 7.1. These are seen to differ widely. The reactivity ratios obtained in anionic and cationic copolymerizations are given and discussed in Chapter 8. MONOMER AND RADICAL Resonance

REACTIVITIE,

S

Effects

The reactivity of a monomertoward a radical depends on the reactivities of both the monomerand the radical. The reactivities of various monomers can be considered by considering the inverse of the monomerreactivity ratio (l/r). The inverse of the monomerreactivity ratio gives the ratio of the rate of reaction of a radical with another monomerto its rate of reaction with its own monomer: 1/~1 = kl~/kll

(7.35)

608

Chapter 7

Table 7.2 Relative Reactivities (l/r) of Monomers with Various PolymerRadicals Polymerradical (reference) Methyl Vinyl Vinyl Monomer Acrylonitrile methacrylate Styrene Acetate Chloride Butadiene 50 4 1.7 29 Styrene 25 2.2 100 50 Methyl methacrylate 6.7 1.9 67 10 Methylvinyl ketone 1.7 3.4 20 10 Acrylonitrile 0.82 2.5 20 25 Methylacrylate 0.67 0.52 1.3 10 17 Vinylidenechloride 1.1 0.39 0.54 10 Vinylchloride 0.37 0.10 0.059 4.4 Vinylacetate 0.24 0.050 0.019 0.59 Valuesof 1/r calculated fromdata of Table7.1

Values of lit for different monomerpairs having the same monomeras M1 but different monomers as M2 would thus provide a comparison of reactivities of the M2monomers. Table 7.2 shows 1/r values calculated from the data in Table %1. The data in each vertical column show the monomerreactivities of a series of different monomerstoward the same reference polymer radical. (Note that the data in each horizontal row in Table 7.2 cannot be compared; the data can only be compared in each vertical column.) The monomershave been arranged in Table 7.2 in their general order of reactivity. The order of reactivity is approximatelythe same in each vertical columnirrespective of the reference radical. (The exceptions that occur are due to strong alternating tendency of certain monomerpairs.) Table 7.2 and other similar data of relative monomer reactivities indicate that substituents tend to enhance monomerreactivity according to the following order: -C6H5, -CH=CH2 > -CN, -COR > -COOR > --CI > -OCOR, R > --OR The order of monomerreactivities in the above series corresponds to the order of increased resonance stabilization (by the particular substituent) of the radical formed from the monomer.Substituents containing unsaturation are more effective in stabilizing the radicals because of the loosely held 7r-electrons, which are available for resonance stabilization. Resonance stabilization becomesmore significant in radical polymerization when the monomers contain conjugated C-C double bonds as in styrene, 1,3-butadiene, and similar molecules: H H H H H R*+ CH2=C--~ R-CH2-C’~’~’R-CH2-C~ R-CH2-.C~ R-CH2-C

06-6

(7.36)

Chain Copolymerization

609

Substituents such as halogen, acetoxy, and ether are increasingly ineffective in stabilizing the radicals because only the nonbondingelectrons on halogen or oxygen are available for interaction with a radical. Thus styrene has a radical whose resonance stabilization is high (84 kJ/mol) whereas vinyl acetate forms a very unstable radical. The spread in the effectiveness of the various substituents in enhancing the monomerreactivity is about 50-200 fold depending on the reactivity of the reference radical. The less reactive the radical, the greater is the spread in reactivities of the different monomers.The placement of a second substituent in the a-position, as in vinylidene chloride, lends an additive effect, provided steric hindrance does not dominate. As a reactive monomerforms a stable free radical, the radical reactivity will be the reverse order of the series given above. This means that monomerscontaining conjugated systems (styrene, butadiene, acrylates, acrylonitriles, etc.) will be highly reactive monomersbut will form stable and so relatively unreactive radicals. Conversely, nonconjugated monomers (ethylene, vinyl halides, vinyl acetate, etc.) are relatively unreactive toward free radicals but will form unstable and highly reactive adducts. A quantitative comparison of the order of radical reactivities can be obtained by multiplying the 1/r values in Table 7.2 by the appropriate propagation rate constants for homopolymerization (kll). This yields the values of k12 for the reactions of various radical-monomer combinations as shownin Table 7.3. The k12 values in any horizontal row in Table 7.3 give the order of radical reactivities toward a reference monomer.(The data in any vertical columnstill give the order of monomer reactivities, as was the case for the data in Table 7.2). It is seen fromTable7.3 that the order of radical reactivities is essentially the same irrespective of the monomerused as reference. The order of substituents in enhancing the radical reactivity, however, is seen to be the opposite of their order in enhancing monomerreactivity. A substituent that

Table 7.3 Rate Constants (k12) for Radical-MonomerReactions Polymerradical Monomer (M2) Methyl (Reference) Styrene methacrylate Acrylonitrile Butadiene 280 2,060 98,000 Styrene 165 1,130 49,000 Methyl methacrylate 314 515 13,100 Acrylonitrile 413 422 1,960 Methylacrylate 215 268 1,310 Vinyl chloride 9.7 52 720 Vinyl acetate 3.4 26 230 Valuesof k12 calculatedfromdata in Tables6.7 and7.2

Vinyl Vinyl acetate chloride 319,000 230,000 550,000 154,000 46,000 23,000 10,100 2,300

110,000 225,000 187,000 11,000 6,490

610

~]~ap~e~ 7

increases monomerreactivity does so because it stabilizes (and hence makes less reactive) the radical formed from the monomer. A consideration of data in Table 7.3 showsthat the effect of a substituent on radical reactivity is considerably larger than its effect on monomerreactivity. It thus turns out that resonance stabilization is more successful in suppressing radical reactivity than in enhancing monomerreactivity. Thus vinyl acetate radical is about 100-1000 times more reactive than styrene radical toward a given monomer, while styrene monomeris only 50-100 times more reactive than vinyl acetate monomertoward a given radical. These two effects tend to compensate each other as shownlSy the fact that the self-propagation rate constant (kp) for vinyl acetate is only about 16 times that of styrene (Table 6.7). Steric

Effects

The rates of radical-monomer reactions are also dependent on considerations of steric effects. It is observed that most common1,1-disubstituted monomers--for example, isobutylene, methyl methacrylate and methacrylonitrile-react quite readily in both homo- and copolymerizations. On the other hand, 1,2-disubstituted vinyl monomersexhibit a reluctance to homopolymerize, but they do, however, add quite readily to monosubstituted, and perhaps 1,1-disubstituted monomers.A well-known example is styrene (M1) and maleic anhydride (M2), which copolymerize with rl = 0.01 ~’2 = 0 at 60°C, forming a 50/50 alternating copolymer over a wide range of monomerfeed compositions. This behavior seems to be a consequence of steric hindrance. Calculation of k12 values for the reactions of various chloroethylenes with radicals of monosubstituted monomerssuch as styrene, acrylonitrile, and vinyl acetate showsthat the effect of a second substituent on monomerreactivity is approximately additive when both substituents are in the 1- or o~-position, but a second substituent when in the 2- or H-position of the monomerresults in a decrease in reactivity due to steric hindrance between it and the polymer radical to which it is adding. Polar Effects Monomerreactivity cannot be considered independent of radical reactivity and vice versa. One observes enhanced reactivities in certain comonomer pairs and this corresponds to an alternating tendency. Whicheverof the two radicals one might find at the growing chain end, the opposite monomeris always preferred in forming an alternating sequence. This behavior leads to the idea of a positive and a negative charge. Thus, if the radical containing end unit is positively charged, a negatively charged monomeris attracted and vice versa. Whether the addition of a monomerto a growing chain gives rise to a positive or a negative chain end depends on the substituents of the monomer. Thus, electron withdrawing (accepting) substituents, -C6H5,

611

Chain Copolymeriza~ion

-C1, -COOR, -CN, -COCH3, all decrease the electron density of the double bond in a vinyl monomer,whereas electron donating groups, e.g., -C~Hs, -CHa, -OR, -OCOCH3increase the electron density. (The phenyl group -C6H5 can be regarded as an electron acceptor as well as an electron donor. It becomesone or the other by inductive effects.) Thus monomerssuch as vinylidene chloride, acrylic esters, and acrylonitrile are polarized in such a way that the vinyl groups represent the positive part, and the substituents the negative part of a dipole:

CHz~C

tiCe

\ct’-

CH~CH--C-O-CHa

CH~CH-C~N

On the other hand, monomerssuch as styrene or vinyl ethers are polarized in the opposite sense such that the vinyl groups represent the negative part, and the substituents the positive part of a dipole:

~

2~CH

CH2~- CH

C 3 The polarity of any species is independent of whether it is in monomer or radical form for the free radical is a neutral entity. Thus assumingthat the above types of polarization are also possible with radicals, one can understand that a chain end at which there happens to be a structural unit with an electron-withdrawing substituent, prefers a monomerwith an electron-donating substituent and vice versa. Such consideration of polarity explains that, for example, acrylonitrile forms statistical copolymers with methyl acrylate (rlr2 = 1.26), while copolymerization of acrylonitrile with ethyl vinyl ethers leads to alternating structures (fir2 = 0.04). However, the opposite polarization caused by the electron-donating or electron-withdrawing substituents is not the only factor governing the reaction of monomersand radicals in copolymerization. Thus the magnitude of the resonance stabilization and in some cases steric hindrance are important too, and the actual behavior of the monomertherefore results from the overlapping of the three factors.

Problem7.9 Explain the following copolymerization results or observed behaviors considering the influence of resonance, steric, and polar effects on monomer reactivity: (a) Vinyl acetate and vinyl chloride are the most favored copolymerization componentsfor vinyl ethers, but both of themare reluctant to copolymerizewith styrene. (b) Neither maleic anhydride nor diethyl fumarate forms homopolymers but both react with styrene and vinyl ethers to form alternating copolymers. (c) In copotymerization, vinylidene chloride monomeris 2-10 fold more reactive whereas1,2-dichloroethyleneis 2-20 fold less reactive, as comparedto vinyl

612

Chapter 7

chloride. !d) Althoughthe reactivity of 1,2-dichloroethylenein copolymerizationis low, it Js still muchgreater than their reactivity in homopolymerization. 1,2-Disubstituted ethylenes do not homopolymerize. Ans wet: (a) Vinyl ethers have low resonancestabilization. In copolymerizationwith vinyl ethers, one should probably choose those monomerswhich have equally low resonancestabilization, but, if possible, no polarization, or only one of opposit~ sign. Thusvinyl chloride and vinyl acetate, whichhavelow resonancestabilization and also low polarity, should be chosenas copolymerizationcomponentsfor vinyl ethers. Styrene, on the other hand, has high resonance stabilization. A styryl radical therefore does not add a vinyl chloride or vinyl acetate monomer as this wouldlead to the formation of a higher energy radical. (b) Both maleic anhydride and diethyl fumarate do not form homopolymers due steric hindrance.But both react with styrene and vinyl ethers of oppositepolarities which help to overcomesteric hindrance and lead to the’formation of alternating copolymers. (c) The difference maybe attributed to steric hindrance. The effect of a second substituent on monomer reactivity is approximatelyadditive whenboth substituents are in the 1- or s-position. However,a second substituent whenin the 2- or flposition of the monomer results in a decrease in reactivity due to steric hindrance betweenit and the radical to whichit is adding. (d) The steric hindrance betweena fl-substituent on the attacking radical and substituent on the monomer is responsible for the inability of 1,2-disubstituted ethylenes to homopolymerize. The reactivity of 1,2-disubstituted ethylenes toward copolymerizationis due to the lack of fl-substituents on the attacking radicals (e.g., styrene, acrylonitrile, andvinyl acetate radicals).

The Q --

e Scheme

All the above factors controlling monomerand radical reactivities contribute to the rate of polymerization, but in a manner which makes it difficult to distinguish the magnitude of each effect. Attempts to correlate copolymerization tendencies based on these factors are thus mainly of a semiempirical nature and can, at best, be treated as useful approximations rather than rigorous relations. However,a generally useful schemewas proposed by Alfrey and Price [23] to provide a quantitative description of the behavior of different monomersin radical polymerization, with the aid of two parameters, for each monomerrather than for a monomerpair. These parameters are denoted by (~ and e and the method has been called the Q -- e scheme. It allows calculation of monomerreactivity ratios rl and r2 from properties of monomersirrespective of which pair is used. The scheme assumes that each radical or monomercan be classified according to its reactivity or resonance effect and its polarity so that the rate constant

Chain Copolymerization

613

for a radical-monomer reaction, e.g., the reaction of MI" radical with M2 monomer, can be written as kt2 = P1Q2exp(-ele~) (7.37) where P1 is considered to be a measure for the reactivity of radical MI" and Q a measure for the reactivity of monomerM2; e~ and e2, on the other hand, are considered to represent polar characteristics of the radical and monomer,respectively. By assuming that the same e value applies to both a monomerand its radical (that is, el defines the polarities of M1and MI", while eg~ defines the polarities of M~and M2", one can write expressions for kll, k22 and k21 analogous to Eq. (7.37). These can be appropriately combinedto yield the monomerreactivity ratios. Thus for kl~ one can write by analogy to Eq. (7.37), kll = PIQl exp(-elel) (7.38) Therefore, ku Q~ exp[-e~ (el - e2)] (7.39) rl -- k12 -- Q2 Thus the effects of radical activity cancels, and rl can be expressed analytically in terms of parameters independent of the paired interdependence of M1 and M2. (Note that in any ~ven pair of monomers, the monomer cited first is considered as M1and the other as M2.) An expression for r2 is similarly obtained, viz., k~2 Q2 exp[-e2 (e2 - el)] (7.40) r2- k21 -- Q1 Thus rl and r2 can be calculated from Qand e values of monomers forming the pair. Equations (7.39) and (7.40) permit us to calculate Q and e va lu es for single monomers from the values of rl and r2, provided we have one monomerfor which Q and e have been arbitrarily established. Price chose styrene as the standard monomer with the values Q = 1 and e ---- --0.8. One can then calculate the Q and e values of any monomer that has been copolymerized with styrene from the rl and r2 values of the copolymerization of styrene with particular monomer. Knowingthe Q and e values of these various monomers, one can then calculate the Qand e for any monomerthat has been copolymerized with these monomers, that is, where rl and r2 are known. Extensive tabulations of Qand e values for monomersthus exist in the literature. Table 7.4 gives a selection of Q and e values for some of most commonmonomers. As a general rule, monomers with electron-rich double bonds have more negative e values and those that form highly resonance-stabilized radical have higher Q numbers. As noted earlier, Q is a measure for the reactivity of a monomer.However, this reactivity is greater, that is, the readiness of the monomerto undergo reaction forming a radical is greater, the greater the resonance stabilization

Chapter 7 Table 7.4 Q - e Values Monomer e Q Ethyl vinyl ether - 1.17 0.03 Butadiene - 1.05 2.39 Styrene (reference standard) - 0.80 1.00 Vinyl acetate - 0.22 0.03 Ethylene - 0.20 0.01 Vinyl chloride 0.20 0.04 Vinylidene chloride 0.36 0.22 Methyl methacrylate 0.40 0.74 Methyl acrylate 0.60 0.42 Acrylic acid 0.77 1.15 Methacrylonitrile 0.81 1.12 Acrylonitrile 1.20 0.60 Methacrylamide 1.24 1.46 Maleic anhydride 2.25 0.23 Source: Data from R. Z. Greenley, "Q and e Values for Free Radical Copolymerizations of Vinyl Monomersand Telogens," pp 267-274 in Chap. II in Polymer Handbook(J. Brandrup and E. H. Immergut,eds.), 3rd ed., Wiley-Imerscience, NewYork (1989). of this radical. Therefore, Q must also be a measure for the resonance stabilization of the radical formed from the monomer. Using the tabulated Q and e values for any two monomers, one can calculate the ’P1 and r2 values from Eqs. (7.39) and (7.40) for this monomer pair whether or not they have ever been polymerized.

Problem7.10 Calculate the rl and r2 values for the monomerpair styrene(M1)acrylonitrile(M2)

from the tabulated Q and e values.

Answer: From Eq. (7.39), 1.00 r~ = --exp[0.S0(-0.S0 - 1.20)] 0.60 (cf. exptl, rl = 0.29) = 0.336 From Eq. (7.40), 0"60exp[-1.20(1.20 r2 = 1.0----~ = 0.054 (cf.

exptl,

+ 0.8)1 r~ = 0.03)

Until the appearance of the Alfrey-Price necessary to refer relative monomerreactivity

Q -- e scheme, it has been ratios to the particular two-

615

Chain Copolymeriza~ion

component system investigated. By means of two constants Q and e, the Alfrey-Priee scheme enables calculation of reactivity ratios of any vinyl monomer with any other vinyl monomer whose (~ and e parameters are known. The Q--e prediction scheme is not quantitatively reliable. But when regarded as an empirical tool for predicting copolymerization behavior, the Q -- e scheme is of the utmost utility qualitatively, for predicting copolymerization behavior and for obtaining approximate estimates of rl and r2 values.

Problem7.11 Predict the type of copolymerization

behavior that would be expected for the following monomerpairs: (a) Both monomers have low Q values and the e values are of opposite signs (example: ethyl vinyl ether and vinyl chloride). (b) One monomer has a large Q value and the other has a small Q value (example: styrene and vinyl chloride). (c) Both monomers have similar Q values and also similar e values (example: styrene-butadiene). (d) Both monomers have approximately the same Q value and high e values opposite sign (example: styrene-methacrylonitrile). Answer: (a) This condition favors good copolymerization. Both components will occur significant amounts in the copolymer chain. (b) The addition of a radical chain with high resonance stabilization (large Q) monomerwith a small Q value leads from a stable (energy-poor) state to an unstable (energy-rich) state, which on thermodynamic grounds is not very probable. Thus the two monomerswill not copolymerize well. The monomer-polymer composition curves also will deviate widely from the azeotrope line. (c) Since both monomershave similar reactivities and similar polarities, nearly ideal copolymerization will occur. This is also evident from the product of reactivity ratios rx and r 2 expressed by Eqs. (7.38) and (7.39) which give 2 ~exp[-(el - e2)2]. For el --~ e2, rar2 ----- 1 indicating ideal copolymerization. (d) The monomerswill show tendency to alternating addition (inflection point curves). For high e values of opposite signs the value of rlra from Eqs. (7-39) and (7.40) is very small indicating alternating behavior.

SEQUENCE

LENGTH

DISTRIBUTION

The copolymer equation (7.11) describes the copolymer composition only on a macroscopic scale, that is, the overall mole ratio or mole fraction of monomer units in a copolymer sample produced from a comonomer feed. It does not reveal details of molecular level composition or microstructure, that is, the manner in which the monomer units are distributed in the

616

Chapter 7

copolymer. Thus for two monomers M1 and M2, the ratio F1/(1 - F1) gives the overall mole ratio of M1and M2units in the copolymer but no information concerning the average lengths of the -(-M1-)n- and -(-M2-)nsequences in a typical copolymeras illustrated by - M1-M1-M1-M1-M2~-M1-M1-M2-M2-M2-M1-M1-M1

-

where the sequences are underlined. A completely random placement of the two monomerunits along the copolymer chain occurs only for the case rl = r2 = 1. There is a definite trend toward a regular microstructure for all other eases. For example, if r~ > 1, once a propagating species of type M~is formed it will tend to a sequence of M1units. However, there is a random aspect to copolymerization due to the probabilistie nature of chemical reactions. Thus an rl value of 2 does not imply that 100%of all M1units will be found as part of long sequences of M1. Though a very large fraction of M1units will be found in such a sequence, a small fraction of M~units will be randomly distributed. Similar arguments hold for M2 sequences if r2 > 1. The microstructure of a copolymeris defined by the distributions of the various lengths of the M1 and M2sequences, that is, the sequence length distributions. The probabilities or mole fractions nz(M1) and nz(M2) forming M1 and M2 sequences of x are given by (see Problem 7.1 for derivation), nz(M~)

PI F-1Pjt~

nx(M2)

= P~2z-1P~I

(7.41) (7.42)

where the probability (P) values are defined by,

Pll ~-rl[M1] rl[M1] _ "rlfl rill-~ f2 nI- [M2] P12 = rl[M1] [M21 q-[M21_rlflf2 + f2 r2f2 _ P22"- [M~] r2[M2] + ,’2[Mqfl +

(7.43)

P~I = [M]I[M1] + ,’2[M2]_fl fl+

(7.46)

(7.44) (7.45)

Equations (7.41) through (7.46) allow one to calculate the mole fractions different lengths of M~and M~sequences. Such calculations reveal that for a feed composition other than equimolar, thedistribution becomesnarrower for the monomerpresent in lower amount and broader for the monomer present in larger amount. This is a general phenomenonobserved for all sequence distributions.

Chain Copolymerization

617

Problem 7.12 The sequence length distribution for a copolymerization system can be described by the mole fractions of sequences with 1, 2, 3, 4, 5, 6, ...., M1 or M2 units in a copolymer and plotting against sequence length in a bar graph. Describe such distribution for an ideal copolymerization with equimolar feed composition considering two cases: (a) rl = r2 = 1 and (b) x =5, r2 = 0.2. Answer: (a) For the system I =r 2= 1 , fl = 0.5 , Eqs . (7. 43)-(7.46) giv e Pll = P1 P2~ = P~ = 0.50. The values of n~(Ma) calculated from Eq. (7.41) are 0.50, 0.25, 0.125, 0.0625, 0.0313, 0.0156, and 0.0078 for sequences with 1, 2, 3, 4, 5, 6, and 7 M1units, respectively. A bar graph is shown in Fig. 7.7(a). Thus, although the most plentiful sequence is single M~at 50%, there are considerable amounts of other sequences: 25%, 12.5%, 6.25%, 3.13%, 1.56%, and 0.78%, respectively, of dyad, triad, tetrad, pentad, hexad, and heptad sequences. The distra"oution of M2sequences is exactly the same as for M1sequences. (b) For the system r~ = 5, r2= .2 f~ = 0.5. Equations (7.43)-(7.46) PI~ = P21 = 0.8333 and P12 = P220=’ 0.1667. The values of n~(M1) calculated from Eq. (7.41) are 0.167, 0.140, 0.116, 0.097, 0.081, 0.067, and 0.055 for sequences with 1, 2, 3, 4, 5, 5, and 7 M1units, respectively. A bar graph is shownin Fig. 7.7(b). It is seen that the single M1sequences are again the most plentiful but only at 16.7% and the sequence-length distribution is also broader than for rl = r2 = 1 of case (a). There are small amounts of relatively long sequences; 3.2% of 10unit, 1.3% of 15-unit, and 0.4% of 20-unit M1 sequences. The sequence-length distribution for the less reactive M2 monomer,calculated from Eq. (7.42) and shown in Fig. 7.7(b) is seen to be muchnarrower. Single M2units are by far the most plentiful (83.3%) with 13.9% dyads, 2.3% triads, and 0.39% tetrads.

Problem7.13 Consider an alternating

copolymerization with r 1 = ?’2 = 0.1 and fl = 0.5. What percentage of the alternating copolymer structure is made of M~M~ sequence ? Compare the sequence length distribution with that for the ideal copolymer in Problem 7.12(a) which has the identical overall composition. Answer: For the system with rl = r2 = 0.1 and fl = 0.5, Eqs. (7.43)-(7.46) P~I = P22 = 0.0910; P12 = P21 = 0.9090. Therefore, the sequence length distributions for both monomerunits are identical. The single M1and single Mz sequences are overwhelmingly the most plentiful at 90.9% each. Thus the M1Mz sequence comprises 90.9% of the copolymer structure. From Eqs. (7.41) and (7.42) the dyad and triad sequences are 8.3% and 0.75%, respectively, for both and M2. The large difference between this distribution and the distribution in Fig. 7.7(a) for a randomcopolymerhaving identical overall composition clearly indicates the difference between alternating and ideal behavior. The ideal copolymer with an overall composition of F1 = F~ = 0.5- has a microstructure that is very different from that of a predominantly alternating copolYmer.

618

Chapter 7

1.0

1.0

0.8

0.8

nx(M I) ~ nx(M2) ....

c 0.40.2-

2 3 4 5 6 "~ 8 Sequencelength (x)

2 3 4 5 6 7 Sequencelength (x)

(a)

(b)

Figure 7.7 Sequencelength distribution for an ideal copolymerization with (a) r1 = r~ = 1, fl = f~ and (b) ~’1 = 5, r2 = 0.2, fl = f> [In (a) the distribution M2sequencesis not shownas it is the sameas for Mt sequences. In (b) the plots of n~(M2)(---) are shownslightly to the left of the actual sequence length.]

The average sequence lengths ~(M1) and ~(M2) may also be determined from Eqs. (7.41) and (7.42). Wehave already derived these expressions in Eqs. (PT.l.ll) and (P7.1.12): ~(M1) : 1 + ?’I[M1] E(M2) ----

1 -t-

--

r2[M21 [M1]

1 + I ( fl/f2)

(7.47)

_ 1 Jr r2(f2/fl)

(7.48)

The run number, NR of the copolymer is defined as the average number of sequences of either type per 100 monomerunits. Considering, for example, a hypothetical copolymer shown below, in which the sequences are underlined, M2-M1-Mt-M2-M1-M2-M2-M1-Mt-M1-M2-M;-M1-M1-M2-M2-M2-M2-M1-M~ the number of sequences are 10 and there are 20 monomerunits. Hence NR= 50. A larger run number indicates a greater tendency toward alternation. For a perfectly alternating polymer NRis 100.

Chain Copolymerization

619

The rate of sequence formation, dS/dt, regardless of length, is simply the rate at which sequences are ended. Assumingthat the molecular weight of the polymer is fairly large so that chain termination can be neglected, this is given by dS -- k12 [M~I[M2] q- k21 [M~I[MI] (7.49) dt The total rate of polymerization is given by d([M1] q-[M2])

= kll [M~][M1] q- k12 [M~][M2] q- k21 [M~][M1] q- k22 [M~[Mg.]

(7.50)

Elimination of dt by combination of Eqs. (7.49) and (7.50) and use of steady-state approximation [cf. Eq. (7.6)] yields d([Ml] q- [M2]) kll[M1] q- k12[M2] ]~22 [M2] q- /~21 [M1] = + dS 2k12 [M2] 2k21 [M1]

(7.51)

or

d([M1 q-[M21) = 1 q- rl[Ml_ ] q- r2 [M2] dS 2 [M2]2 [M1]

(7.52)

Since the run number is the average number of sequences per 100 monomer units, this maynowbe written as NR = 100 --d([Ml] which after substitution yields

+ [M~])

of Eqs. (7.52) followed by Eqs. (7.47) and (7.48) 2OO

2 q- rl([M1]/[M2])

2O0 -b r2([M2]/[M1])

~(M1) q-

(7.54) Equations (7.47), (7.48), and (7.54) indicate that a knowledgeof [M1] fl), [M2] (or f2), rl, and r~ enables a prediction to be made, not only of the average copolymer composition [from Eqs. (7.11) or (7.18)], also of the average sequence length of each monomer and the average number of sequences of monomerunits per unit length of copolymer. As many significant properties of the copolymers depend on the distribution of monomeralong the chains, the ability to make such predictions from a relatively small amountof experimental data can be very useful indeed.

620

Chapter 7

Problem7.14 Vinyl acetate (3.0 M) is copolymerizedwith vinyl chloride (1.5 M) in benzene solution by adding azobisisobutyronitrile to a concentration of 0.1 Mand heating to 60°C. Calculate for the copolymerinitially formed(a) the probability of formingvinyl acetate and vinyl chloride sequencesthat are 3 units long, (b) the average sequencelengths of vinyl acetate and vinyl chloride in the copolymer, and (c) the run numberof the copolymer. 1 = 0.24, r2 = 1.80] Answer: (a) f~ = (3.0 M)/(3.0 M + 1.5 M) = f2 = 1-0.67 = 0.33 From Eqs. (7.43)-(7.45): Pll = 0.3244, P12 = 0.6756, P~2 = 0.4736, P~I = 0.5264. From Eq. (7.41): n3(M1) = (0.3244)3-~(0.6756) From Eq. (7.42):

nz(M2) = (0.4736)z-~(0.5264)

(b) FromEqs. (7.47) and (7.48): 5(M~) = 1 (0.24)(0.67/0.33) = 1. 48 5(M~) = 1 + (1.80)(0.33/0.67) (c) From Eq. (7.54): Nn = 200/(1.48 + 1.90) =

RATE

OF FREE-RADICAL

COPOLYMERIZATION

In the derivation of copolymer composition equation, Eq. (7.11), considered only the rates of the four possible propagation steps in a binary system. However, the overall rate of copolymerization depends also on the rates of initiation and termination. In deriving an expression for the rate of copolymerization in binary systems the following assumptions will be made [25]: (a) rate constants for the reaction of a growing chain depend only upon the monomerunit at the growing end, and not upon chain length or further composition; (b) steady-state conditions apply both to the total radical concentration and to the separate concentrations of the two radicals; (c) chain termination is by bimolecular radical reaction. By assumption (a) the overall rate of monomerdisappearance is given by [cf Eqs. (7.7) and (7.8)I: d ([Mx] + [M2]) = kll [MI’] [MI] q- ]g21 [M2"] [M1] q- ]g12 [MI’] [M21

+ IMp]

(7.55)

621

Chain CopolTmerizafion

where [M1] and [M2] are concentrations of the two monomers, [MI’] and [M2"] are concentrations of chain radicals ending in M1and M2units, respectively, and k~l is the rate constant for attack of a chain radical ending in Mz unit upon monomerM~,etc. By assumptions (b) and (c) and defining an overall termination rate constant kto, two steady-state assumptions may also be written as ]¢12 [MI’] [M2] ~--- k21 [M2"] [M1] Ri -~ Rt 2~--- 2]et~ ([MI’] ~t. [M2.])

(7.56) (7.57)

where R/ is the overall initiating rate and Rt is the overall termination rate. Solving Eqs. (7.56) and (7.57) simultaneously for [MI’] and [M2"] substituting into Eq. (7.55) we obtain d([M1]-a u [M2]) __ /~/:~(rl[M1] 2 2) + 2[M1][M2]q- r2 [M2] dt -- (2]et~)l/2(r, [Mll/kll ~t_ r2[M2]/k22 )

(7.5s) where rl and r2 are the monomerreactivity ratios given by the propagation rate constant ratios kn/k12 and k2~/k2l, respectively. If the cross-termination rate constant, that is, the rate constant for termination of radical MI" with radical M2-is kfl2, then the steady-state for the total concentration of radicals can also be written as 1~ = Rt = 2~11 [MI’] 2 W 2kt12 [MI’] [M2"] q- 2~22 [M2"] 2 (7.59) If we define mole fractions of the respective radicals as/~ and ~2, that is, ~1

=

[MI’] and ---- i -- = [M2"] [MI’]q-[M2"] [MI’] -F[M2"] (7.60)

then from Eqs. (7.57), (7.59) and (7.60)

R,

- 2 (~11~12

([MI’]-1-[M2"])

2 --

-t-

kt12~1~2

q- k~t22~)

(7.61)

whence ]¢,to

= /gt11~L12 nu ]gt12~l~2 J~- ]¢,t22~/-~

Nowwe define a cross-termination

(7.62)

factor as

¢ = ktl2 / 2(ktll

1/2 k,~22)

(7.63)

622

Chapter 7

to represent the rate constant for cross-terminationrelative to the geometric meanof the rate constants for chain termination of each monomer alone. Its value > 1 is thus indicative of the preference for cross-termination over homotermination.Equation(7.58) then assumesthe following familiar form of the so-called "chemicalcontrol" model: (d[Ml] q- d [M2I) d~:

(rl[Ml] 2 + 2[M~I[M~I+ r2[M212)Pt~/2 (9"1~ [M1]2 1/2 q- 2~brlr2~l~[M1][M2] q- ~2~22[M212)

(7.64) where~t and ~2 represent the termination-propagationrate constant ratios given by ~1 (2k,,t11/"’11)

, ~2 = (2k,

t22

/ k222) 1/2

(7.65)

Equation(7.64) can also be derived from Eq. (7.55) by eliminating radical ¯ concentrationswith the help of twosteady-state assumptionswritten as Eqs. (7.56) and (7.59). Equation (7.64) represents a one-parameter model the copolymerization rate [8] containingthe parameter~b. Statistically, ~b is expectedto equal unity.

Problem7.15 Show that if reaction probabilities

of radicals depend only on encounter rates and are independent of the nature of the radicals, ~ should equal

unity. A rlsw~3,F :

For simplicity consider a reaction mixture in which [MI"] = [M2"]. Since MI" ° radicals with equal frequency and since radicals will collide with M2"and M1 the same applies to M~"radicals, the frequency of M~" - M~’encounters is twice that of MI° -- M~" or M2"-M2"collisions. If the reaction probabilities’depend only on encounter rates, P~12

(/~n P’-tzz) 1/2 -- 2 where

= 2&n [Mx’] -~n ~ /~2 2= 2/~= IMp’]

Thus, ktl2

~ = ~(ktn&~2)~/2 -

Chain Copolymeriza~ion

623

In practice, ¢ can be calculated by inserting experimental copolymerization rates into Eq. (7.64). The values of ¢ thus obtained are frequently greater than unity, and these deviations are ascribed to polar effects that favor cross-termination over homotermination. However, this is not always unambiguous, since the apparent cross-termination factor may vary with monomerfeed composition in a given system [25,26]. It is clear also that termination reactions are at least partially diffusion controlled [27,28]. A dependenceof segmental diffusivity on the structure of macroradicals is to be expected and dependence of diffusion controlled termination on copolymer composition seems reasonable. It is therefore plausible that the value of the overall termination rate constant kt0 in copolymerizations should be functions of fractions (Ft and F2) of the comonomersincorporated in the copolymer. An empirical expression for kt0 has thus been proposed [27]: kt0 = F1]¢,~11 Jr F2/c, t22 (7.66) that represents kto as the average homotermination rate constants weighted on th~ basis of mole fractions F1 and /w2 of the respective monomersin the copolymer; F1 and /w2 can be calculated from the feed composition by the copolymer composition equation [Eq. (7.18)]. Equation (7.58) combination with Eq. (7.66) provides a model for diffusion-controlled reactions with no adjustable parameters (such as ¢ in chemical controlled model). No strong theoretical case can be made for the relationship in Eq. (7.66). It fits some, but not all, of the copolymerization systems to which has been applied. A better fit is often provided by a combined model [29] which uses the parameter ¢ of the chemical control model in combination with an empirical formulation for ~0 related to copolymer composition. The empirical formulation is derived by substituting mole fractions of each monomerfor the radical mole fractions/Zl and h2 in Eq. (7.62) 2 tz2F2 kto = ktl~,Fx 2 + k,t~2F~,F2 + k, (7.67) Combination with Eq. (7.63) then yields ]¢to ---- ktllF12 Jr 2¢(]et11~22)l/2Fl-b-’2 Jr ]¢t22-b-’2 2 (7.68) Equations (7.58) and (7.68) yield the rate of copolymerization, and ¢ be taken from previous studies of the chemical control model, or from an empirical correlation between this parameter and the rlr2 product [27] which is based on the fact that ¢ more or less parallels 1/rxr~ as a measure of the increased preference for cross-termination over homotermination. Direct measurements of ¢ have been obtained [26] by measuring the absolute values of the rates of propagation and termination in pure monomersand in mixtures of various compositions. In the case of styrene-p-methoxystyrene, ¢ = 1, indicating that no polar or other influences favor cross-termination. In most cases, however, cross-termination is

624

Chapter 7

markedly favored, with ¢ varying over a wide range. For styrene-methyl methacrylate, for example, ¢ is 15, while for styrene-butyl acrylate ¢ is 150.

Problem7.16 Bulk polymerization

of styrene in the presence of 1 g/L of AIBNinitiator at 60°C gave a measured polymerization rate of 5.92 mol/Ls. Predict the rate of copolymerization at 60°C of a mixture of styrene (M1) and methyl methacrylate (M2) with 0.579 mole fraction styrene and the same initial concentration of the initiator as in the homopolymerization case. Comparethe rates predicted from chemical control, diffusion control, and combined models with the experimental value of 4.8x10-~ mol/L-s [25]. Use relevant kp and kn values for homopolymerization from Table 6.7 and assume ¢ = 15. [Other data: rl = 0.52, r2 = 0.46; monomerdensity = 0.90 g/cma.] Answer: kll = (kr,)styrene = 165 L mo1-1 -1 ktll --~ (kt)styrene = 6xl0r L tool -1 s-I k2~ = (kp)MMA -1 = 515 Lmo1-1 s kt~ = (kt)MMA = 2.55X107 L mo1-1 -1 For bulk styrene, For styrene -1 mol L

[M] = (1000 cma L-l)(0.90 -1) (104 g tool

g -a)

(M1)-MMA(M2) feed of 0.579 mole fraction

= 8.70

mol L_ 1

styrene,

[M~] =

For styrene homopolymerization [cf. Eq. (6.25)],

1~ =

[M]2 2(5.34 × 10-s mol L-1 s-l) 2 (6 × 107 L mo1-1 -1) (165 L tool -~ s-l) 2 2(8.70 tool L-l)

=

1.66 × 10-7 mol L-1 -1 s

Chemical Control Model For copolymerization, written as

Eq.

d([M1] + [Mz])

dt

(7.64) for chemical control model is conveniently

’

(r1612a~ + 2~rlr~¢S182c~+ r2822

(P7.16.1)

where a = mole ratio of M1 and M2 in feed = 0.579/0.421 = 1.375, rl = 0.52, rz =0.46, /~i = 1-66 x10-7 molL-is-~, ~b= 15

625

Chain Copolymerization 1/2 I2(6 x 107 L tool-’ s-l)] ~1

~ ["

’/ ~ ~ =

TX~g

[

~"

~Ol--~’i"

~l~~i

"~-1~

J

J

1/2 ~--

oi-’ = 13.87

66.39

mol

tool

sl/2 L_1/2 Sill L_,/1

Substituting the values in Eq. (7.16.1), - d( [M,] + [M~] )/dt = 5.~ x 10-~ mol L-1 s-’ Diffusion Control Model Substituting appropriate values in Eq. (7.18), F1 = 0.5624, F1 = 0.4376. From Eq. (7.66), ~o = (0.5624)(6 x 107 L tool -1 s-i) + (0.4376)(2.55 x 107 L mo1-1-1) -1 = 4.49 x 107 L mo1-1 s Equation(7.58) can be conveniently written in the form n112(rl~2 --[- 20~+ "2) [M1] d( [M1] + [M2]) (v7.16.e) (2~)~/~ ~ (r~lk~l + ~1~)~ where a is the mole ratio of M1and M2= 1.375. Substituting appropriate values in Eq. (P7.16.2), - d( [M,] + [M~] )/dt = 12.7 mol L-’ Combined Mode! From Eq. (7.68) with ¢ = 15, ~o = 31.26x107 L mo1-1-1. FromEq. (P7.16.2), - d( [Ml] + [M2] )/dt = 4.8 x 10-~ mol L-1 -i s Comparison (P~)copolym× 105 Chemical control model Diffusion control model Combined model with ¢ = 15 (no parameter) with ~ = 15 Exptl: 4.8

5.4

12.7

4.8

MULTICOMPONENT COPOLYMERIZATION There is considerable interest in multicomponent copolymer systems. An example of terpolymerization, that is, copolymerization of three monomers,

626

Chapter 7

is the radical polymerization of styrene with acrylonitrile and butadiene whichallows muchgreater degree of variation in properties than what can be achievedby binary copolymerizationof styrene with acrylonitrile or with butadiene. Manyother commercialuses of terpolymerization exist. In most of these the terpolymerusually has two monomers present in major amounts to obtain the gross properties desired, with the third monomer in a minor amountfor modification of a special property. Thus ethylene-propylene elastomers have minor amountsof a diene as a third componentto allow the product to be subsequentlycross-linked. The quantitative treatment of terpolymerization is morecomplexthan two-componentcopolymerization though the method is similar. In the copolymerization of three monomersM1, M2, and M3there are three different types of growingchain ends v,~vMl", v, wvM2", and v~wM3". Each of these can react with any of the three monomers of the system, and hence there are nine different chain propagation reactions, as shownbelow: Reaction VWMl"-1- M1 ~ vwMl" VWMl’-}- M2 ~ vwM2" vwM1" q- M3 ~ vwM3" vwM2"~ MI ~ WVMl" wvM2" ~ M2 ~ vwM2" vwM2" q- M3 ~ vwM3" wvM3" nt- M1 ~ vwM1" ~vM3"-~- M2 ~ wvM2" vwM3" q- M3 ~ vwM3"

Rate d[M1]/dt

d

= /ell

[M,’I

d M3]/dt =

d M1]/dt = ]~21 - d ~]/dt = -d iM~]/dt=

[MI’] [M3] [M2"] [MI] [M2"] [M2]

(7.69)

[M2"][Mal

-- d M1]/dt = ~1 - d M~]/dt =

- d M~]/dt=

and six monomer reactivity ratios, r12

w

r21

--

~31 ~

kll k12 k22

k2~ k3~ k~

r13 = ~ kl--~ k22 r23 w r32

~

k23’ k33 k~2 ’

(7.70)

are involved(as well as six terminationreactions). The equations for the rates of monomerconsumptionsare

-d[M1]/dt= kn[MI"][M~] + k21[M2"][M1] + ka[M3"][Mx] (7.71) -d[M2]/dt

= kl2[Ml’][M2]-]-

]g22[M2"][M2]

~-/~32[M3"][M2]

(7.72)

--d[M3]/dt

= kl3[MI’][Ma]

+ k23[M2"][M3]

+ k33[M3"][M3]

(7.73)

627

Chain Copolymerization

As in the case of binary copolymerization, we do not know the absolute values of the radical concentrations [M]I", [M]2", and [M]3", but we can derive the steady-state relationships for these unknownconcentrations. Thus since in steady state, radicals of the type wwvMl" are formedjust as fast as they are converted to types vwvvM2"and vvwvM3",we may write ~12 [MI’] [M2] -Jr- ~13 [MI’] [M3] = k21 [M2"] [M1] q- /g31 [M3"] [MI] (7.74) and likewise for radicals of types vwwM~and vwwMa" : ]g21 [M2"] [M1] q- ]g23 [M2"] [M3]

k12 [MI’] [M2] q- k32 [M3"] [M21 (7.75)

]¢31 [Ma’] [M1] 4- ]¢a~ [M~’] [M~]

k13 [MI’] [M3] q- k23 [M2"] [M31 (7.76)

Combining Eqs. (7.74)-(7.76) with following composition relations:

(7.71)-(7.73)

we can obtain

d[M~] : d[M~] : d[M~] = [MI]-

:

[M~_~]IMpel [MI]-Jr-

q’-Jrr31r23 ~ V31V21 r21r32

[ r12r31

r13r21

+

r12r32

r23r12

÷

/

r32r13- [J

r13’~23

q’r12

-r13 J

+ ÷ /

~

r21

r23 J

~al

~a2

(~.zz) A simpler expression for the terpolymer composition has been obtNned [30] by expressing the steady state with the relationships:

k12[MI’] [M2] = k21[M2"] [M1] k23 [M2"] [Ma] = ka~[Ma’] [Mel k31[M3"] [MI]= k13[MI"] [M3] instead of Eqs. (7.74)-(7.76). The combination of Eqs. (7.78)-(7.80) Eqs. (7.71)-(7.73) yields the terpolymer composition

(7.78) (7.79) (7.80)

628

Chapter 7

d[M1] : d[M2] : diM3]

The conventional [Eq. (7.77)] and simplified [eq. (7.81)] terpolymerization equations can be used to predict the composition of a terpolymer from the reactivity ratios in the two-component systems M1/M2, MlfM3, and Mz/Ma. The compositions calculated by either of the terpolymerization equations show good agreement with the experimentally observed compositions. Neither equation is found superior to the other in predicting terpolymer compositions. Both equations have been successfully extended to multicomponent copolymerizations of four or more monomers [30,31].

Problem7.17 The ternary copolymerization

of a monomer mixture containing 35.92, 36.03, and 28.05 tool% of styrene, methyl methacrylate (MMA)and acrylonitrile (AN) at 60°C for 3.5 h yielded at 13.6 wt%conversion a polymer product which analyzed C 78.6% and N 4.68% (by wt.) [31]. Calculate the initial ternary copolymer composition to compare with the composition obtained by the analysis. Answer: Let the mol%composition of the copolymer be ~, y, and z for styrene (CsHs), MMA(CsHsO2) and AN (C3H3N), respectively. From C and N mass balance andx q- y + z = 100, one obtains x = 45.0, y = 25.6 andz = 29.4. From Table 7.1, Styrene (M1)/MMA(M2) : r12 = 0.52, r21 = Styrene (M1)/AN (M3) : rla = 0.29, rzl = 0.020 MMA(M2)/AN (M3) : r2z = 1.3, r32 = Rewriting Eq. (7.77) A = fl

as d[M1] : diM2] : d[Ma] = A : B : C, where +

+

’r21~’32 ’Y127"32

Yl

d~ -

’f2

9"12 ~’21

-~-

~

629

Chain Copolymerization

T237"12

/’31

one can then calculate for a given starting feed composition,viz., fl, f2, and the copolymercompositions as F1 = A/(A+B+C); F~ = B/(A+B+C); F3 = C/(A+B+C) ]Equation (7.81) can be similarly written and A, B, C defined accordingly. With fl = 0.3592, f2 = 0.3603, f3 = 0.2805, and r values given above, the copolymercompositionis calculated from both Eqs. (7.77) and (7.81). The results are tabulated below: Feed composition Terpolymer composition (mol %) Monomer tool % Found Calcd. from Eq. (7.77) Calcd. from Eq. (7.81) Styrene 35.92 44.7 46.5 46.7 MMA 36.03 26.1 26.3 27.7 AN 28.05 29.2 26.9 25.8

The knowledge of the monomer reactivity ratios for each pair of monomersalso permits the calculation of the size distribution of sequences of units of one kind. Thus if the monomer mixture is made up of the monomersM1, M2, M3, ".. MN,the resulting initial copolymer will consist of sequences of M1, M2, ¯ ¯., and MNmonomerunits. The fraction of all M1sequences which possess x number of M1units will be given by a distribution function [2]: x-1 na:(M1)

[M2] [M1]-ff -TI2

[M1] [M3] q- -7"13

q-

....[MN] rlN

x

[M1] ) (7.82) [M1] q-" [M2] q- [M3] q’-"’" [MN____j] 7"12 7"13 rlN Note that with [MI] = 0 for i = 3,4,.-. (7.41) for the binary system.

N, Eq. (7.82) reduces to Eq.

630

Chapter 7

Problem7.18 Consider the initial fractions units.

of styrene,

terpolymer in Problem 7.17. Calculate the MMA,and AN sequences containing 2 or more monomer

ADsvcer: For a terpolymer Eq. (7.82) becomes

fl fa+--

1+--

?’12

Forx

?’13

r12

~13

= 1 =

1-

=

1-

:1+(:2/?’,~) +(y31,’,~) 0.3592

0.3592 + (0.3603/0.52) + (0.2805/0.29)

= 0.822

Therefore, fraction of styrene sequences with 2 or more styrene units = 1 - 0.822 = 0.178. Similarly, hi(M2)

= 1-

= 0.7345 f~ + f2 + fz r23

Therefore, 0.265.

fraction

of MMAsequences with 2 or more units

= 1 - 0.7345

Again, nl(M3)

= 1-

= 0.9865

s’ +~+S~ ?’32

Fraction

of ANsequences with 2 or more ANunits = 1 - 0.9865 = 0.013.

DEVIATIONS FROMTERMINAL COPOLYMERIZATIONMODEL Thederivation of the copolymercompositionequation [Eq. (7.11)] rests on two importantassumptions-oneof a kinetic nature and the other of a thermodynamic nature. Thefirst is that the reactivity of the propagating species is determinedby the end monomer unit (first-order Markov)and

631

Chain Copolymerizafion

is independent of the identity of the monomerunit which precedes the terminal unit. The second is the irreversibility of the various propagation reactions. Deviations observed from the copolymer composition equation under certain conditions have been ascribed to the failure of one or the other of these two assumptions or the formation of a monomercomplex which undergoes propagation. Penultimate

Effect

The copolymerization behavior of somesystems indicates that the reactivity of the propagating species is affected by the penultimate (next-to-last) unit. The behavior, referred to as the penultimate or second-order Markov behavior, often manifests itself by giving inconsistent values of the monomer reactivity ratios for different monomerfeed compositions. This has been observed in many radical copolymerizations where the monomerscontain highly bulky or polar substituents. Thus in the copolymerization of styrene and fumaronitrile [32], propagating chains rich in fumaronitrile and having styrene as the end unit showgreatly reduced reactivity with fumaronitrile monomer.The effect is explained by steric and polar repulsions between the penultimate fumaronitrile unit in the propagating chain and the incoming fumaronitrile monomer. The mathematical treatment of the penultimate effect [33,34] in a binary copolymerization system involves the use of eight propagating reactions: wwvM1Ml"

+ M1 ~ vwwM1M1Ml"

vwwM1Ml"

+ M2 ~ vvvwM1MiM2"

vwwM~M2" + M1 -~

wwvM2M2Ml"

vvvwM2M2" + M2 ~ vvvvvM~M2M~wwvM2Ml"

+ M1 ~ vvvwM2M1M1"

wvwM2Ml"

+ M2 ~ vvwvM2MiM2"

vvvwM1M2 ¯

+ M~ ~ wwvMiM2Ml"

vvvwM1M2"

+ M2 ~ wwvM~M~M2"

(7.83)

with the four reactivity ratios

(7.84)

632

Chapter 7

Each monomeris thus characterized by two monomerreactivity ratios. One of these represents the propagating species in which the penultimate and terminal monomer units are the same, while the other represents the propagating species in which the penultimate and terminal units differ. The latter monomerreactivity ratios are denoted by prime notations (r~, r~). Following a procedure similar to that used in deriving Eq. (7.11), the copolymer composition equation with a kinetic penultimate effect present is obtained as

(fly + 1) d[M1] diM2]

1 + (r~Y q- 1) -- 1 + r~(r2 + Y) Y(r~ + Y)

(7.85)

where Y = [M~]/[M2]. For the styrene(M1)-fumaronitrile(M2) system, fumaronitrile is incapable of self-propagation (r~ = r~ = 0) and Eq. (7.85) simplifies d[M1] diM2]

_ 1 + r~Y (r~Y + 1)

(7.86) . (r~Y + 1) The experimental copolymer composition data for styrene(M~)-fumaronitrile(M2) give a good fit to Eq. (7.86) with rl ----- 0.072 and r~ : [33], but deviate markedly from the behavior predicted by the first-order Markov model with rl = 0.23. Penultimate effects have been observed in a number of other systems. Amongthese are the radical copolymerizations of ethyl methacrylate-styrene, methyl methacrylate-4-vinyl pyridine, methyl acrylate-l,3-butadiene, and other monomerpairs. The copolymerization data for the styrene(M1)-fumaronitrile(M2) system indicate that there are also effects due to remote monomerunits preceding the penultimate unit. The effect of remote units has been treated by further expansion of the copolymer composition equation by the use of greater number of monomerreactivity ratios for each monomer[34]. However, the utility of the resulting expression is limited due to the large number of variables involved. It should be mentioned that deviations from the terminal (first-order Markov) copolymer equation have also been treated from a thermodynamic viewpoint [35], where the deviations are accounted for in terms of the tendency of one of the monomers(M2) to depropagate (see p. 514). if the concentration of the monomerfalls below its equilibrium value [M2]e at the particular reaction temperature, terminal M2units will be prone to depropagate. The result would be a decrease in the amount of this monomerin the copolymer.

633

Chain Copolymerization COPOLYMERIZATION

AND CROSS-LINKING

If one of the monomersin a copolymerization is a divinyl compoundor any other olefinic entity with functionality greater than 2, a branched polymer can be formedand it is possible for the growing branches to interconnect to form an infinite cross-linked network knownas "gel." Copolymerization of such systems is thus generally analogous to step polymerizations invoMngtriand tetra-functional reactants (see Chapter 5). The cross-linking reactions are not capable in themselves of producing highly branched, finite molecules since gelation occurs before sufficient cross-linking can take place to produce such molecules. At this point the most complex of the molecular species are removedas the insoluble gel fraction. It is useful to be able to predict the conditions under which such gel formation will occur. The criteria for this condition are applicable also to cross-linking of preformed polymers which occurs in radiation induced cross-linking, to vulcanization with addition of other reagents, and to chain-growth and step-growth polymerizations of polyfunctional monomers.

Problem7.19 Consider a sample containing discrete polymer molecules that can be interconnectedeither during further polymerizationor by a separate reaction on the macromolecules.Supposesomeof the molecules are cross-linked by linkages formed between randomly selected monomerunits in them. Showby probability considerations that a sufficient condition for gelation occurs whenthe polymer samplecontains one cross-linked unit per weightaverage moleculeor one cross-link per two polymer chains. Answer: Wechoose a cross-link at random.The probability that the monomer unit on this cross-link resides in a primary molecule containing x monomerunits equals the fraction of all monomer units that are in x-mers. That is, this probability Px is P~----

(P7.19.1) E~%1 where N~ is the numberof x-mer molecules in the sample. However,if Mois the meanformula weight of monomericunits in this sample, -- w~ (P7.19.2) Mo ~1co N~ x wherew~is the weight fraction of x-mersin the sample. If a fraction ’u of all monomer units in the sampleformsparts of cross-links and !hese cross-links are randomlyplaced, then an additional u(x - 1) monomer units mthe x-merare also cross-linked, on the average. In other words,the probability that an arbitrarily selected cross-link is attached to a primarychain whichcontains x monomerunits is w~, and it is expected that u(x - 1) of these, x monomer units are also cross-linked. It maythus be said that the initial, randomlychosen cross-link leads through the primary moleculeand other cross-links to wxu(x- 1) additional primary molecules. P~

~h~p~er7

634

Since x can have any positive non-zerovalue, the expectednumberof additional cross-links y in a moleculethat already contains one arbitrarily chosencross-link is {20 y = u~w,(x-1) = ~[DP~,-~,-w.] = u(DP~,-1) (P7.19.3) whereDP~is the weight-average degree of polymerization of the primary chains in the sample. This molecule can be part of an infinite networkonly if y is at least 1. Thecritical value uc of u for this condition, whichis the condition for gelation, is then given by 1 1 DP~, - 1 DP~, since DP~,>> 1. Since gel~tion occurs whena fraction 1/DP’~of all monomer units is crosslinked, a completelygelled polymersamplecontains at least one cross-linked unit per weight average molecule or one cross-link per two polymerchains.

Cross-linking occurs early or late in the copolymerization depending on the relative reactivities of the two double bonds of the divinyl compound or diene. By the proper choice of a monomerwith two double bonds it is possible to reduce the reactivity of one double bond just enough so that it will not enter polymerization under the same conditions as the other, but can be made to react under more drastic conditions. This leads to postpolymerization cross-linking reactions of which vulcanization reactions are an example. If the two double bonds are well separated in the monomer,the reactivity of one is not affected by the polymerization of the other. Ethylene glycol dimethacrylate and allyl acrylate are examples of divinyl monomersof this type. If, on the other hand, the two double bonds are close enough together that the polymerization of one can shield the other sterically, or if they are conjugated as in 1,3-dienes, a difference in reactivity can be expected. The most notable case in which there is a large drop in reactivity of one group on reaction of the other is in the copolymerization of 1,3-dienes where 1,4-polymerization leads to residual 2,3-double bonds which have lowered reactivity. They are subsequently used to bring about cross-linking as in vulcanization. Divinyl benzene is in the intermediate category with regard to dependency of reactivity of one double bond on the other being reacted. Several different cases can thus be distinguished depending on the type of divinyl monomerin copolymerization. Vinyl and Divinyl Equal Reactivity

Monomers of

Consider the copolymerization of vinyl monomerA with divinyl monomer BBwhere all of the vinyl groups (i.e., the A group and both B groups) have

Chain Copolymerization

635

the same reactivity. Methyl methacrylate (MMA)-ethyleneglycoldimethacrylate (EGDMA),vinyl acetate-divinyl adipate (DVA), and to an extent styrene-p- or rn-divinylbenzene (DVB)are examples of this type of polymerization system. In MMA-EGDMA system, ~H~

CHs

CHs

COOCHa

~OOCH=CH~O~O

the unsaturated groups of EGDMA may be assumed with confidence to be equal in reactivity with the identical group of the MMA monomer. Thus the reactivity of one unsaturated group in EGDMA should not depend perceptibly on whether the other is reacted or not. (DVB,by contrast, is knownto be more reactive than styrene.) Let the initial molar concentrations of vinyl monomerand monomericB groups be [A]0 and [B]0, respectively, and that of divinyl BBbe [BB]0. Thus, [B]0 = 2[BB]0. Since the A and B double bonds are equally reactive, i.e., rl = 9"2 = 1, one obtains from the copolymer equation [Eq. (7.11)], /71 = fl. Thus the molar ratio of B and A groups in the copolymer is simply equal to [B]/[A]. At the extent of reaction p (defined as the fraction of A and B groups reacted), the relative number of various monomericspecies can be listed as follows : Unreacted A~s : [A]0 (1 -- p) Reacted A’s : [A]0 p Unreacted BB’s: [BB]0 (1 -Singly reacted BB’s : 2[BB]0 (1 -- p)p 2Doubly reacted BB’s : [BB]op The number of cross-links is simply the number of BB molecules in which both B groups are reacted and the number of polymer chains is derived in terms of the degree of polymerization DP: Number of polymer chains Total numberof reacted A and B grc~ups = ([A]op

DP + [BloP) /DP

(7.87)

Therefore, Numberof cross-links per chain

p[BBIoDP

([A]o p[B]oDP 2([A]o + [B]o)

(7.88)

636

Chapter 7

At the critical extent of reaction Pc for the onset of gelation, the number of cross-links per chain is ½ (see Problem7.19), whencethe critical extent of reaction at the gel point Pc is obtained as

pc ([A]0 = 2 [BB]0

DP~

(7.89),

in which "DP is now replaced by DP~o since, as shown in Problem 7.19, the appropriate average of DPfor the aforesaid gel condition is DP~o.It is the weight-average degree of polymerization of "primary molecules." The term "primary molecule" is used to designate the linear molecule that would exist if all the cross-links were severed, that is, the polymerchains formed before any crosslinking reactions occurred, In the above problem it may be taken approximately as the weight-average degree of polymerization that would be observed in the homopolymerization of monomer A under the particular reaction conditions. For [A]o ~>> [B]o, Eq. (7.89)reduces Pc ~

[A]o 2[BB]0 DP~

(7.90)

Equation (7.89) predicts that extensive cross-linking occurs during copolymerization of A and BB(see Problem 7.20). The equation holds best for systems containing low concentrations of the monomerBB, that is, at higher gel point conversions where the distribution of cross-links is random; its predictive utility decreases as the concentration of BBincreases. With increasing concentration of BB, Eq. (7.89) predicts gel points at conversions that are increasingly lower than those found experimentally. This behavior has been attributed to the wastage of the BBmonomerdue to intramolecular crosslinking which mayoccur at the expense of cross-linking involving two polymer molecules.

Problem 7.20 Bulk polymerization of methyl methacrylate (MMA)at 60°C with 0.9 g/L of benzoyl peroxide yielded a polymerwith a weight average degree of polymerization of 8600 at low conversions. Predict the conversions of MMA at which gelation would be observed if it is copolymerized with 0.05 mol%of eth}lene glycol dimethacrylate (EGDMA) at the same temperature and initiator concentration as in the homopolymerization case. Answer: Let MMAbe denoted as monomer A and EGDMA as monomer BB. Then [BB]o -- 0.0005

[A]0+[BB]; ’[A]o / [BB]o~ 2000

637

Chain Copolymerization

FromEq. (7.90), with DP-~-~(assumedto be the same as that of the homopolymer) = 8600, ~000 = 0.12 Pc ~-2(8600) So gelation wouldbe observed at about12%conversion of MMA. Problem 7.21 Calculate the conversion at which gelation should be observed in styrene containing 0.14 tool% p-divinylbenzeue (DVB)and 0.04 mol/L benzoyl peroxide initiator at 60°C. Assumefor this calculation that the vinyl groups in both styrene and DVB are equally active and that chain termination occurs solely by coupling. [Data at 60°C : kd = 2.4 x 10-6 s-l; f = 0.4; k~/kn for styrene = -4 L tool -1 s-a.] 4.54×10 Answer: [M] for bulk styrene = 8.65 -1 tool L Kinetic chain length = 8.65

(4.54X 10-4)1/22(0.4 x 2.41 × 10-6)1/2 (0.04)1/2 = 470 D/5~ = 2×470 = 940. For chain termination solely by coupling, -DPw/DP~= (2 + p)/2 (see p. 527). For high polymer, p ~ 1. Therefore, DP~= 1410. Let A ~- Styrene, BB---- DVB.Then [BB]o = 0.0014 [A]o + [BB]o [A]o/[BB]o = 713 713 From Eq. (7.90), p~ -- 2(1410) = 0.253 (~ 25.3%).

Equation (7.89) indicates that gelation can be delayed, that is, the extent of reaction at which gelation occurs can be increased by reducing the concentration of divinyl monomer, by reducing the weight-average chain length (increase initiator concentration or add chain transfer agents), or using a divinyl monomerin which one or both the vinyl groups are less reactive than those in the monovinyl monomer(see later).

Problem 7.22 Howwould the percentage conversion at the gel point change if the styrene-divinyl benzenemixture of Problem7.21 contained additionally a mercaptan chain regulator (Cs = 21) at a concentration of 2x10-4 tool/L?

638

Chapter 7

Answer: In the presence of chain regulator (neglecting other transfer reactions), 1 1 (2 × 10 -4 mol/L) -3 = 1.5x10 + DP~ - 940 (21) (8.65 tool/L) D/5,~ = 645 To determine the proportions of unimolecular (chain transfer) and bimolecular (coupling) termination, let y fraction of molecules be terminated by the former mechanism.Therefore, (1 - y)(940) + y(940/2) = 645, or y ~--ff~/~-ffn is given by a combinationof (1 + p) and (2 p)/2 weighted in proportion to the amountsof umimolecularand bimolecular termination, respectively (see pp. 525-527),that is, DP~/DP,, = 0.63(1+p) +0.37(2 +p)/2 = 1.81 forp~l DP~, = 1.81x645 = 1170 From Eq. (7.90), Pc = 713/(2)(1170) = 0.30 (----30% conversion)

Vinyl and Divinyl Monomers of Different Reactivity A second case in vinyl-divinyl copolymerization is the copolymerization of A and BBin which the reactivities of the vinyl groups A and B are not equal, while the two B groups are equally reactive. If the B groups are r times as reactive as the A groups, they enter the copolymer r times as rapidly and hence the ratio of B and A groups in the copolymer, d [B]/d [A], is d[B]/diAl

= r[B]/[A]

(7.91)

where [A] and [B] are the concentrations of the A and B groups in the monomermixture at any instant. Thus, at the extent of reaction p of A groups, [A] = (1- p)[A]0 and [B] = (1- rp)[B]0, where the subscript denotes initial concentrations. By a derivation similar to the case of vinyl-divinyl monomersof equal reactivity considered above, one then obtains an expression for the critical extent of reaction at gelation as [A]o -t- r [B]o Pc = r2[B]oDP~o

(7.92)

For [B]o << [A]o, Eq. (7.92) reduces p~

[A]o = r2[B]oDP~

(7.93)

ChainCopolymerizafion

639

An alternative expression to Eq. (7.92) for the gel point conversion can be derived in terms of the reactivity ratios of the two types of vinyl groups in A and BB. Let rl represent the relative reactivity of an A monomerand a B monomericgroup, when reacting with a free radical of type A, and the relative reactivity of a B group and an A group, when reacting with a B-typeradical, i.e., ?’1 : ~gAA//gAB, r2 "- ]gBB/~I3A The composition equation for initial copotymer in terms of monomergroup concentration [I31o becomes[cf. Eq. (7.11)]: d[B]

[B]o (r2 [S]o + [Alo)/([B]o + rl [A]o)

[A]o

If we ignore the drift of residual monomercomposition with conversion and assume a randomdistribution of crosslinks, we may predict gelation to occur at a conversion Pc given by 2(rl [A]02 + 2 [A]0[B]0 + r2[B]02) 2Pc -- DP~o[B]0( [A]0 q- [B]o)(r2[B]0 -t-[A]0)

(7.94)

Thus when the double bonds of the divinyl monomerare more reactive than that of the vinyl monomer (r2 > rl), gelation occurs at lower conversions. Gelation is delayed until the later stages if rl > In the case in which componentA is present in considerable excess ([A]0 >> [B]0), Eq. (7.94) reduces to [cf. Eq. (7.93)]: r~ [A]0 Pc = [B]oDPw

(7.95)

Problem7.23 Predict the extent of reaction at which gelation wouldoccur in (a) vinyl acetate-ethylene glycol dimethacrylateand (b) methylmethacrylate-divinyl adipate systems, both containing 15 mol%of the divinyl compound.Assumethat the reaction conditions for the two systemsare such as to yield the sameDP,,, of 1000for the uncrosslinked polymer.Takerl and r2 values from Table 7.1 for the respective analogousvinyl-vinyl copolymerizations. Answer: (a) Vinyl acetate (A) - ethylene glycol dimethacrylate (BB): r 1 ~--- 0.015, r 2 = 20 (for vinyl acetate-methylmethacrylate) Denoting mole fractions of monomergroups A and B by fA and 85 -- 0.74, fB = 0.26 fA = 85-1-2 x 15

640

Chapter 7

Equation (7.94) in terms of fA and fB becomes .Pc (lO00)[fB/fA

2] (~’2fB/fA

+ (fB/fA)

+ 21)

(r7.23.1)

Substituting the values, Pc = 0.3 x 10-3 (--= 0.03%) (b) Methylmethacrylate (A) - divinyl adipate (BB): rl = 20, r 2 = 0.015 (for methyl methacrylate-vinyl acetate) fA = 0.74, fB = 0.26 Substituting in Eq. (P7.23.1) above, pc = 0.89 (= 89%)

One Group of Divinyl

Monomer Having

Lower Reactivity

A third case in vinyl-divinyl copolymerization is the copolymerization of a vinyl monomerA with the divinyl monomerBC where groups A and B have equal reactivities but group C has a muchlower reactivity. An example of such a case would be methyl methacrylate-allyl methacrylate, where A and B groups are the two methacrylate groups and C is the allyl group. For such a system the copolymer will consist of copolymerized A and B groups with pendant unreacted C groups until later stages of reaction. Let p be the fraction of A and B which have polymerized, and r be the reactivity ratio between C and B groups (r = kAC/kAB). At conversion P (< Pc) the concentrations of unreacted A and B have been reduced to [A]0(1- p) and [B]0(1- p), respectively, while the concentration unreacted C groups is still essentially [C]0. During a small increase in p, given by dp, the following changes in concentrations of A, B, and C groups will occur : --d[A]

= [A]o dp

(7.96)

-d[B]

-- [B]o dp

(7.97)

--d[C]

= [A]0dp

]0

O__p)

r

= [cl0r~p-~

The decrease in C concentration up to the conversion p is given by the integral

tcl0,/o’_ tcl0,o( The fraction of C groups which are reacted at the conversion p is therefore

641

Chain Copolymerization

The average number of reacted B groups per chain is closely given by fBcDP where fBc represents the mole fraction of BC in the original monomermixture and DP is the average degree of polymerization. Since the fraction r ln[1/(1 --p)] of these are reacted at the C group, the average numberof cross-links per chain is Numberofcross-linksperchain Whenthis reaches the critical gelation occurs:

= fBcDP rln(11~_p) value of ~1 (with DP replaced

(7.101) by DPw),

1 fBc DP~ r ln (~ l pc) = -~ In(1

- pc)

- 2rfBC

1

DP~

Pc = 1 - exp[-1/(2rfBcDP~)]

(7.102)

Thus gelation is delayed for a lower value of r.

BLOCK

AND

GRAFT

COPOLYMERIZATION

Block and graft copolymerizations involve initiating polymerization reactions through active sites bound on the parent polymer molecule. Block copolymerization involves terminal active sites, whereas graft copolymerization involves active sites attached either to the backboneor to pendant side groups. Copolymerizations only by free-radical processes are discussed in this section; those involving ionic mechanismsare described in Chapter 8. Block and graft copolymerizations by free-radical mechanismare usually conducted in a mixture of the parent polymer, the monomer(s) to be grown on the parent polymer, and fresh initiator. However, the product obtained in this case is likely to be a mixture. Thus, in addition to the desired block or graft copolymer, it may contain homopolymerof fresh monomerand parent tiomopolymer molecules that did not take part in the copolymerization.

Block Copolymerization Block copolymerization by free-radical processes calls for generation of terminal radicals (that is, a radical center at the end of the chain). Terminal radicals can be produced by: (a) Decomposition of peroxide groups that are introduced into the polymer by special means. These peroxide groups may

642

Chapter 7

form an internal part of the chain backbone, or they maybe incorporated as stable end groups; (b) Mechanical cleaving of polymer chains (as cold milling or mastication). A few examples, of these methods are given below. Producing

Internal

Peroxide

Sequences

(a) Copolymerization of small amounts of oxygen with olefinic monomersintroduces peroxide linkages. For example, oxygen copolymerizes with styrene, methyl methacrylate, and vinyl acetate to produce peroxide linkages which can then be decomposedto generate free radicals to initiate polymerization of another monomer:

nCH2=CH +0 2 I X

Initiator

CH2- CH-O-O-CH2 -CH ~ I I X X Heat M Monomer[ -~-~CH2- CH-O-M- M- M-,-~ I X

(b) Initiation of the parent polymer with polymeric phthaloyl peroxide leads to polymer molecules with peroxide linkages. Phthaloyl polyperoxide has the structure HOOCC6H4-(--OOC-C6H 4-COO--)w-C6H4COOH It decomposes by random cleavage to form shorter diradical species, which become incorporated into the parent polymer backbone. Whenfresh monomeris added and additional heating is supplied, the remaining peroxide linkages in the polymer backbone decomposeto form terminal radicals and polymerization resumes leading to a block copolymer. Introducing

Peroxide

End Groups

(a) Initiation with m-diisopropyl benzene monohydroperoxideproduces polymer with isopropylbenzene end groups which can be easily converted into hydroperoxide by reaction with oxygen to initiate polymerization of a second monomer :

643

Chain Copolymerization

H3C~/CH3

H3C~/CH3 /

nM +

~ /O-O-H ~C ~CH3 \CH~

~

CH 3 \CH 3

Oxidation

CH3 H3C~cI ~OOH

I~

O-M-M C/~CH 3 \CH 3

~

(b) An initiator such as azobiscyanopentanoic acid decomposesto produce free radical species with a carboxyl group at one end: O II HO-C-CH2- CH2-

CH CH O 3 3 I / II C-N=N-C- CH2- CH2- C-OH I i CN CN Heat

~

0 CH I1 I 2HO-C-CH2~CH2-C" + N 2 I CN Using such an initiator to produce a polymer, followed by conversion of the carboxylic acid end group to the acid chloride and then reaction of this with t-butyl hydroperoxide results in polymer molecules containing t-butyl perester end groups to initiate polymerization of a second monomer. Mechanical

Cleaving

of

Polymer

Chains

Generation of radical centers at the ends of the polymer chains can be achieved by breaking chemical bonds in the polymer backbone by mastication

644

~hap~er 7

(mixing) of the polymer. If the shearing forces are sufficiently high during mastication, their concentration at individual bonds in the polymer chain results in bond rupture, generating radical chain ends. Block copolymers are obtained by the mastication of either a mixture of two homopolymers:

or a mixture of a polymer and a monomer: vvvwM1Mlwv w m~g vwvvMl" ~ vwwM1M2wwv The former gives a mixture of block copolymer with two homopolymers since the polymer radicals combinerandomly, while the latter yields a block copolymer along with the homopolymer of Mr. Graft

Copolymerization

Weshall consider here graft copolymerization only by free-radical processes. There are three main techniques for preparing graft copolymers via a free-radical mechanism.All of them involve the generation of active sites along the backbone of the polymer chain. These include (i) chain transfer both saturated and unsaturated backbone or pendant groups; (ii)radiative or photochemicalactivation; and (iii) activation of pendant peroxide groups. Chain

7"ranzfer

Methoda

Reasonable efficiencies of graft copolymerization by these methods require rather high chain transfer coefficients. From the values of chain transfer coefficients of representative model compoundsit appears that polymers containing carbon-halogen or sulfur-hydrogen bonds might be susceptible to chain transfer and graft polymerization. The carbon-hydrogen bonds in the c~-position to a carbonyl group, as in poly(vinyl acetate), and the carbonhydrogen bond adjacent to the double bond in unsaturated polymers, such as polybutadiene and polyisoprene, are also suitable for transfer growth. Unsaturated rubber polymers are especially important grafting vehicles. Consider, for example, the polymerization of styrene in the presence of 1,4-poly(1,3-butadiene). A method consists of dissolving the polybutadiene rubber (about 5 to 10%) in monomericstyrene containing benzoyl peroxide initiator and applying heat. Polymer radicals, formed by (a) chain transfer between the propagating radical and polymer or (b) addition (copolymerization) of the propagating radical to the .double bonds of the polymer,

645

Chain Copolymeriza~ion

initiate

graft polymerization of styrene at the active center on the chain:

The relative amounts of the .two processes (a) and (b) depend on identity of the double bond. Grafting via chain transfer (a) predominates for 1,4-poly-l,3-dienes containing relatively unreactive 1,2-disubstituted double bonds. Grafting via copolymerization (b) predominates when the double bond is more reactive, such as polybutadienes with high contents of pendant vinyl groups produced by 1,2-polymerization. Although this methodyields a mixture of homopolymerand graft copolymer, and probably also ungrafted backbone polymer, some of the systems have found highly successful commercial application. These are high-impact polystyrene (HIPS), made polymerizing styrene in the presence of poly(1,3-butadiene), and ABSand MBS,made by copolymerizing styrene/acrylonitrile and methyl methacrylate/styrene, respectively, in the presence of either poly(1,3-butadiene) styrene-butadiene rubber. Ionizing

Radiation

Polymer radicals, that have radical centers on the backbone chain to initiate grafting, can also be produced by irradiation of a polymer-monomer mixture with ionizing radiation. Most radiation graft polymerizations are carried out as heterogeneous reactions. The typical reaction system involves equilibration (swelling) of polymer with monomerfollowed by irradiation of the monomer-swollen polymer while immersed in excess monomer. For example, to produce poly(ethylene-graft-styrene), wwvCH2CH2wwv vwwCH2~Hw~v

Radiation

~ CH2-~Hwvw

+ CH2=CH¢

+ H"

~ vwwCH2--~H-wvw CH2--C. H~b

Most graft polymerizations, irrespective of the initiation process, yield mixtures of the graft copolymer, ungrafled backbone copolymer, and homopolymer of the monomer. The relative amounts of the three species depend on the monomer-polymercombination and the initiation process. Whenhigh-energy radiation is used, the most important factor affecting the grafting efficiency is the radiation sensitivity of the monomerto be

646

Chapter 7

Table 7.5 Relative Sensitivities ofMonomers and Polymersto Ionizing Radiation ~ value Monomers G Styrene Acrylonitrile Methyl methacrylate Vinyl acetate Vinyl chloride

0.66 5.0 6.1 9.6 ,-~ 10

~ Polymers G value Polystyrene 1.5-3 Polyisoprene 2--4 Polyethylene 6-8 Poly(methyl methacrylate) 6--12 Poly(vinylacetate) 6-12 Poly(vinylchloride) 10-15 "Number of radicals formedper 100 eVabsorbed.

grafted relative to the parent polymer. Efficient grafting can be obtained only if the monomeris less sensitive than the polymer. In such a case, active sites will be generated primarily on the polymer backbone. An indication of the relative sensitivities of monomerand polymer is obtained from a comparison of their (~ values, which is the number of radicals formed per 100 eV absorbed (see Table 7.5). Evidently, styrene can be grafted efficiently onto poly(vinyl chloride). This is indeed found to be so and polystyrene homopolymeris virtually absent from such graftings. The total radiation dose determines the number of grafted chains and the dose rate determines their length. This is because the dbse rate controls the rate of initiation, which, in turn determines the kinetic chain length and hence the molecular weight. Graft polymerization can also be achieved by irradiation with ultraviolet radiation, often in the presence of a photosensitizer, such as benzophenone or benzoin. Most UVinitiations for grafting involve the photolysis of polymers having either pendant carbonyl groups or pendant halogen atoms, for these are easily activated by UVradiation. Examples include grafting of vinyl acetate, acrylonitrile or methyl methacrylate onto poly(methyl vinyl ketone) and grafting of styrene or methyl methacrylate on brominated polystyrene. Photolytic grafting is similar to radiation grafting except that the depth of penetration by UVis far less than ionizing radiation.

Chain Copolymeriza~ion

647

1. W. Ring, I. Mita, A. D. Jenkins and N. M. Bikales, Pure and Appl. Chem., 57, 1427 (1985). 2. T. Alfrey, Jr. and G. Goldfinger, J. Chem. Phys., 12, 115, 205, 332 (1944). 3. E R. Mayoand E M. Lewis, J. Am. Chem. Soc., 66, 1594 (1944). 4. E T. Wall, J. Am. Chem. Soc., 66, 2050 (1944). 5. C. Walling, Free Radicals in Solution, Chap. 4, Wiley, NewYork (1957). 6. M. Farina, Makromol. Chem., 191, 2795 (1990). 7. G. Goldfinger and T. Kane, J. Polym. Sci., 3, 462 (1948). 8. H. W. Melville, B. Noble and W. E Watson, J. Polym. Sci., 2, 229 (1947). 9. I. Skeist, J. Am. Chem.Soc., 68, 1781 (1946). 10. V. E. Meyer and G. G. Lowry, J. Polym. Sci., A3, 2843 (1965). 11. R. K. S. Chan and V. E. Meyer, J. Polym. Sci., C25, 11 (1968). 12. I. Capek, V. Juranicova and J. Bartoh, Makromol.Chem., 184, 1597 (1983). 13. K. E O’Driscoll, L. T. Kale, L. H. Garcia-Rubio and P. M. Reilly, J. Polym. Sci. Polym. Chem. Ed., 22, 2777 (1984). 14. V. E. Meyer, J. Po!ym. Sci., 4A, 2819 (1966). 15. M. Finernan and S.D. Ross, J. Polym. Sci., 5, 259 (1950). 16. P. W. Tidwell and G. A. Mortimer, J. Polym. Sci., A3, 369 (1965). 17. T. Kelen and E Tudos, J. Macromol. Sci. Chem., Ag, 1 (1975); MakromoL Chem., 191, 1863 (1990). 18. K. E O’Driscoll and P. M. Reilly, Makromol. Chem. Macromol. Syrup., 10/11,355 (1987). 19. E L. M. Hautus, H. N. Linssen and A. L. German, J. Polym. Sci. Polym. Chem. Ed., 22, 3487, 3661 (1984). 20. P. Bataille and H. Bourassa, J. Polym. ScL Polym. Chem. Ed., 27, 357 (1989). 21. H. P. Plaumann and R. E. Branston, J. Polym. Sci. Polym. Chem. Ed., 27, 2819 (1989). 22. R. Z. Greenley, "Free Radical Copolymerization Reactivity Ratios," pp 153266 in Chap. II in Polymer Handbook (J. Brandrup and E. H. Immergut, eds.), 3rd ed, Wiley Interscience, NewYork (1989). 23. T. Alfrey, Jr. and C. C. Price, J. Potym. Sci., 2, 101 (1947). 24. L. J. Young,J. Polym. Sci., 54, 411 (1961). 25. C. Walling, J. Am. Chem. Soc., 71, 1930 (1949). 26. H. W. Melville and L. Valentine, Proc. Roy. Soc., A(200), 337, 358 (1952). 27. J. N. Atherton and A. M. North, Trans. Faraday Soc., 58, 2049 (1962). 28. K. E O’Driscoll, W. Wertz and A. Husar, J. Polym. Sci., A-1 (5), 2159 (1967). 29. S. S. M. Chiang and A. Rudin, J. Macromol. ScL Chem., A9(2), 237 (1975). 30. A. Valvassori and G. Sartori, Adv. Polym. Sci., 5, 28 (1967). 31. C. Walling and E. R. Briggs, J. Am. Chem. Soc., 67, 1774 (1945). 32. R. G. Fordyce and G. E. Ham, J. Am. Chem. Soc., 73, 1186 (1951). 33. W. G. Barb, J. Polym. Sci., 11, 117 (1953). 34. G. E. Ham,"Theory of Copolymerization," Chap. 1 in Copolyrnerization (G. E. Ham,ed.), Wiley-Interscience, NewYork (1964). 35. G. G. Lowry, J. Potym. Sci., 42, 463 (1960). 36. H. Kruger, J. Bauer and J. Rubner, Makromol. Chem., 188, 2163 (1987). 37. R. Szym,a, nski, Makromol.Chem., 188, 2605 (1987). 38. K. E O Driscoll and E P. Gasparro, J. Macromol. Sci. Chem., A1, 643 (1967). 39. P. J. Flory, J. Am. Chem. Soc., 69, 2893 (1947). 40. P. J. Flory, P~nc~ples of Polymer Chemistry, Cornell Univ. Press, Ithaca, New York (1953).

648

Chapter 7

EXERCISES 7.1. The product obtained in the polymerization of4-methyl-l,6-heptadiene is a linear polymer containing no residual unsaturation. What is its chemical structure ? 7.2. What values of rl and r2 would yield copolymerization diagrams (F1 vs. (a) without inflection points, o) with i nflection p oints ? [Arts. (a) I >1, r 2 < 1 and r 1< 1 , r 2 > 1 ; (b) ’/’1 < 1,r 2 < 1 a rl > 1, r2 > 1 (rare in free-radical copolymerizations, but found in someionic copolymerizations).] 7.3. A monomer pair with rl = 5.0 and r2 = 0.2 is copolymerized beginning with a molar monomerratio [M~]/[M2] = 30/70. Assuming that the copolymer composition within a 10% conversion interval is constant, calculate instantaneous monomer(f~) and copolymer (F1) compositions and cumulative average copolymer compositions at 10 tool% conversion intervals up to 100%conversion. Showthe results graphically as change in composition of the copolymer and the monomer mixture during copolymerization. 7.4. Ferrocenyl acrylate (FMA)and 2-ferrocenylethyl acrylate flEA) were synthesized and copolymerized with styrene (STY), methyl acrylate (MA), vinyl acetate (VA) [C. U. Pittman, Jr., Macromolecules, 4, 298 (1971)]. The following monomerreactivity ratios were obtained:

FMA FMA FMA FEA FEA FEA

STY MA VA STY MA VA

0.020 0.14 1.4 0.41 0.76 3.4

2.3 4.4 0.46 1.06 0.69 0.074

(a) Which of the above comonomerpairs could lead to azeotropic copolymerization? (b) Predict whether FMAor FEA will have higher kp homopolymerization. (c) Is styrene more reactive or less reactive than FMA toward the FMAradical ? By what factor ? (d) List STY, MA, and in order of increasing reactivity toward the FMAradical and toward the FEAradical. (e) List the STY, MA,and VAradicals in order of increasing reactivity toward the FEAmonomer. [Ans. (a) FEA-MA;(b) k;o(FEA) > ka0(FMA);(c) STY50 times more than FMAtoward FMAradical; (d) STY > MA> VAfor both FMAand FEA x > STY~.] radicals; (e) ~ > MA 7,5.

Considering resonance and polarity effects what type of monomers would you choose to copolymerize with vinyl ethers .9

Chain Copolymerizaffon

~49

7.6. On the basis of Q and e values predict the copolymerization behavior of the following pairs of monomers: (a) Vinyl acetate (Q = 0.03, e = -0.22) and ethyl Vinyl ethers (Q = .0.03, e = -1.17). (b) Styrene (Q = 1.00, e = -0.80) and vinyl acetate (Q = 0.03, e = -0.22). (c) Methyl methacrylate (Q = 0.74, e = 0.40) and acrylic acid (Q = e = 0.77). (d) Styrene (Q = 1.00, e = -0.80) and acrylonitrile (Q = 0.60, e = 1.20). 7.7. Using Q and e values in Table 7.4, calculate the monomerreactivity ratios for the comonomerpairs (a) styrene-butadiene and (b) styrene-methyl methacrylate. Comparethe results with the rl and r2 values in Table 7.1. [Ans. (a) rl = 0.51, r2 = 1.84; (b) "~- 0.51, r2 = 0.46.] 7.8. Predict the sequence length distributions for an ideal binary copolymerization with rx = r2 = 1 for (a) fl = 0.5, (b) f~ = 0.8, and (c) 0.2. Comparethe distribution patterns and commenton the results. 7.9. Comparethe sequence-length distribution (by plotting in a bar graph) in the copolymer from the following monomerpairs with and without azeotrope for [Mx]/[M2] = 10/90: (a) rx = r~ = 0.1; (b) = 5- 0, ’r2 = 0. 2. 7.10. For a random copolymer with rx = 1 and a 50/50 composition, plot of n~(Mx)vs.

prepare a

7.11. When0.7 mole fraction styrene (M1) is copolymerized with methacrylonitrile (M2) in a radical reaction, what is the average length of sequence of each monomerin the copolymer ? (rl = 0.37, r2 = 0.44). [Mns. ~(M1) = 1.9; ~(M2) = 1.2.1 7.12. Acrylonitrile monomer(Mx) is copolymerized with 0.25 mole fraction vinylidene chloride (M2). What fraction of the acrylonitrile sequences contain or more acrylonitrile units ? (rl = 0.9, r2 = 0.4).

[~ns.0.53.] 7.13. Styrene (3.0 M) is copolymerized with methacrylonitrile (1.5 M) in benzene solution by adding benzoyl peroxide to a concentration of 0.1 Mand heating to 60°C. Calculate for the polymer initially formed (a) composition the copolymer, (b) probability of forming styrene and methacrylonitrile sequences that are 3 units long, (c) average sequence lengths of styrene and methacrylonitrile in the copolymer, and (d) the run number of the .] copolymer. [Given: rx = rz = 0.25 at 60°C [Ans. (a) 1 57 mol%, M2 43 mol%; ( b) n a(Mx) =0. 0742, ha (M2) = (c) 5(M1) = 1.50, 5(M2) 1.12.] 7.14. The measurement of bulk copolymerization of styrene 0Vlx) and methyl methacrylate (M2) at 30°C in a feed of 0.031 mole fraction styrene with initiation by photosensitized decomposition of benzoyl peroxide gave a value of 7.11×10 -5 mol L-1 s -x, while homopolymerization of styrene under

650

Chapter 7 the same conditions yielded a polymerization rate of 3.02x10 -5 -1 mol L s -1 [H. W. Melville and L. Valentine, Proc. Roy. Soc., A, 200, 337, 358 (1952)]. Calculate the copolymerization rate from (a) chemical control model [Eq. (7.64)] and (b) combined model [Eqs. (7.58) and (7.68)] to with the.experimental value. Use the homopolymerization rate constants at 30°C for styrene as kp = 46 L mo1-1 s -1 and kt = 8.0×106 L mo1-1 s -1 and for MMAas kp = 286 mol L-~ s -~ -x and k~ = 2.44x107 mol L s -1. The reactivity ratios at 30°C are r~ = 0.485 and r 2 = 0.422. Makethe 3.. comparison using ¢ = 10 and ~b = 13. Monomerdensity = 0.90 g/cm Arts. ~. . °, tool L-1 s-1 Rate of copolymenzauon× 10 Chemicalcontrol Combined model Experimental Eq. (7.64) Eqs. (7.58) and (7.68) ff = 10 ff = 13 if= 10 ff = 13 7.11 6.30 5.71 7.85 7.28

7.15. Methyl methacrylate (M1) and vinyl acetate (M2) constitute a system which the nature of polyradical ends has no discernible effect on the overall rate of termination (i.e., ff = 1). The copolymerization rate data measured for this system at 60°C are given below [G. M. Burnett and H. R. Gersmann, J. Polym. Sci., 28, 655 (1958)]: Mole fraction of M2 in feed, f~ 0.915 0.756 0.645 0.548 0.453 0.325

Rate × 105 (mol -1 s -1) 28.0 29.1 42.4 60.7 78.7 103.9

Calculate the copolymerization rate using the diffusion control model [Eqs. (7.58) and (7.66)] and combined model [Eqs. (7.58) and (7.68)] to with the experimental value. Take the reaction with 0.645 mole fraction vinyl acetate in feed as calibration value to normalize on P~. [Kinetic parameters : rl = 28.6; r~ = 0.035; kll = 589, /~ = 2.9 x 107, k22 = 3600, and kt~2 = 2.1 × 108, all in mol L-x s-~; monomerdensity = 0.90 g/cm3.]

651

Chain Copolymeriza~ion Arts.

f2 0.915 0.756 0.645 0.548 0.453 0.325

Rate×lO5, L tool -1 -1 s Diffusion control Combinedmodel 22.2 21.3 30.8 31.5 42.4 42.4 59.2 60.1 82.5 80.1 113.4 108.3

Exptl. 28.0 29.1 42.4 60.7 78.7 103.9

7.16. The ternary copolymerization of a monomermixture containing 31.24, 31.12, and 37.64 mol%of styrene, methyl methacrylate and vinylidene chloride at 60°C for 16 h yielded at 18.2 wt% conversion a polymeric product which analyzed C 68.66% and Cg 12.07% (by wt) [C. Walling and E. R. Briggs, Am. Chem. Soc., 67, 1774 (1945)]. Calculate the initial ternary copotymer composition to compare with the composition obtained by the analysis. Use monomerreactivity ratio data from Table 7.1. Ans. Feed Monomer Mol% Styrene 31.24 MMA 31.12 Vinylidene chloride 37.64

Found 43.4 39.4 17.2

Terpolymercomposition (tool%) Calcd. IEq. (7.81)1 Calcd. [Eq. (7.77)1 44.3 44.3 41.2 42.7 14.5

13.0

7.17. Predict the initial composition of the terpolymer which would be produced from the radical polymerization of a solution containing acrylonitrile (47%), styrene (47%), and 1,3-butadiene 6% (by mol). [Ans. Mt 36.6%, M252.1%, Ma 11.3% (by mol).] 7.18. (a) Calculate the mole fraction composition of the initial terpolymer which would be formed from the radical polymerization of a feed containing 0.414 mole fraction methacrylonitrile (M1), 0.424 mole fraction styrene (M2), 0.162 mole fraction a-methyl styrene (Ma). (b) What fraction of styrene sequences in this copolymer contain 2 or more styrene units ? [Reactivity ratios: M1/M2: rl = 0.44, r2 = 0.37; M1//VI3: rl = 0.38, r2 = 0.53; M~/Ma: rl = 1.124, r2 = 0.627.] [Ans. (a) F1 = 0.443, Fz = 0.403, Fa o= 0.154; (b) 0.23.] 7.19. What should be the concentration of divinyl benzene in styrene to cause gelation at full conversion of the latter, if styrene were being polymerized under conditions such that the degree of polymerization of the polymer being formed were 1000. Assumethat the vinyl groups in divinyl benzene are equally as reactive as those in styrene. [Ans. 0.05 mol%]

652

Chapter 7

7.20. (a) How much of the divinyl monomer, ethylene glycol dimethacrylate (EGDMA),should be added to methyl methacrylate (MMA)to cause set of gelation at 20% conversion when polymerization is carried out at 60°C in the presence of 0.8 g/L benzoyl peroxide. Homopolymerization of MMAunder the same conditions is known to yield polymer with DPw = 1000. M/VIA and EGDMA can be reasonably assumed to be of equal reactivity. (b) Recalculate the amount of EGDMA for the case where of a chain regulator is used to bring down DP~, to 500. [Ans. (a) 0.25 tool% ; (b) 0.5 mol%] 7.21. Consider the styrene-divinylbenzene system of Exercise 7.19. Recalculate the conversion at the gel point taking into consideration the unequal reactivity of styrene and divinylbenzene (r 1 = 0.3, r 2 = 1.0). [Ans. 2.3 tool%] 7.22. Calculate the conversion for onset of gelation in methyl methacrylate (MMA) containing 0.20 tool% ethylene glycol dimethacrylate (EGDMA) when it polymerized at 60°C in the presence of 0.04 moFLAIBNinitiator. Take into account the fact that the chain termination in MMAhomopolymerization occurs both by. disproportionation and coupling, the ratio being 3:1 at 60°C. [Data: k~/G (for MMA)= 1.04x10 -2 L tool -1 s-~; k,~ = 8.45x10 -~ s-l; f = 0.6; monomerdensity = 0.90 g/cm3.] [Ans. Pc = 0.115.] 7.23. Howwould the percentage conversion at the gel point change if the MMAEGDMA mixture of Exercise 7.22 contained additionally an effective chain regulator (Cs = 21 at 60°C) at a concentration of -4 to ol/L? [Ans. Pc = 0.166.] 7.24. Predict the extent of reaction at which gelation would occur in vinyl acetate (VA)-divinyl adipate (DVA)mixture containing 5 tool% DVAand 0.04 mol/L AIBNinitiator at 60°C. Assumedisproportionation to be the predominant mechanism of chain termination. [Data at 60°C: k~/kt for VA= 0.182 L mo1-1 s -1, ka (for AIBN) = 8.45 x -8 s - ~, f 0. 6, mo nomer de nsity = 0.93 g/cm3.] "~.25. Predict the extent of reaction at which gelation would occur in the following two vinyl-divinyl systems, both containing 1 tool% of the dix;inyl component: (a) styrene-ethylene glycol dimethacrylate and (b) methyl methacrylate-divinyl benzene. Assumethat the reaction conditions for the two systems are such as to yield the same D-Pwof 1000 for the uncrosslinked polymer. Take the r~ and ~’z values from Table 7.1 for the analogous vinyl-vinyl copolymerizations. [Ans. (a) Pc = 0.015; (b) Pc = 0.012.]

Chapter Ionic

8

Chain Polymerization

INTRODUCTION Chain or addition polymerization is knownto occur by several mechanisms other than those involving free radicals discussed in Chapter 6. Prominent among these are ionic mechanisms in which the growing chain ends bear a negative charge (carbanion) or a positive charge (carbonium ion). If growing chain end bears a negative charge (wvc¢~), the polymerization knownas anionic polymerization. If the chain end bears a positive charge (ww~), the reaction is cationic po lymerization process. In general, ionic polymerization can be initiated through acidic or basic. compounds. For cationic polymerization complexes of BF3, AICI3, TIC14, and SnCI4 with water, or alcohols, or tertiary oxonium salts have shown themselves to be particularly active. The positive ions are the ones that cause chain initiation. However, also with HC1, H2SO4,and KHSO4one can initiate cationic polymerization. Initiators for anionic polymerization are alkali metals and their organic compoundssuch as phenyllithium, butyllithium, phenyl sodium, sodium naphthalene, and triphenyl methyl potassium. Unlike free-radical reactions which are not selective (as most olefinic monomersundergo radical polymerization), ionic polymerizations are largely selective and are restricted to monomerswhose structures enhance the stability of the ionic species involved in the process. Cationic polymerization is essentially limited to those monomerswith electron-releasing substituents and anionic polymerization takes place with monomerspossessing electronwithdrawing groups. These are elaborated in a later section. The commercial utilization of cationic and anionic pOlymerizations is rather limited because of the high selectivity of ionic polymerizations compared to radical polymerization, as mentioned above. Ionic polymerizations are also most difficult to carry out and require stringent reaction conditions. Thus, unlike in free-radical polymerizations in which the characteris653

654

C,hapt,er 8

tics of the active centers depend only on the nature of the monomerand are generally independent of the reaction medium,in ionic polymerizations the polarity of the solvent strongly influences the mechanismand rate of ionic polymerization. This can be visualized as follows. Ionic polymerizations, as we shall see later, involve successive insertions of monomersbetween a macromolecular ion and a counterion of opposite charge. The macroion and the counterion form an organic salt which may exist in several forms in the reaction medium. The degree and nature of the interaction between the cation and anion of the salt and the solvent -, a (or monomer)can vary considerably. Considering an organic salt A+B continuous spectrum of ionicities ("Winstein spectrum") can be depicted: AB ~ Covalent bonding (I)

A+B ~--~ A+/B ~-~ -A+IIB Contact Solvent Solvated (tight) separated (loose) ion pair ion pair ion pair (II) (III) (IV)

----~A+

+ BFreesolvated ions (V)

One can visualize a range of behavior from one extreme of a completely covalent species (I) to the other of completely free (and highly solvated) ions (V). The intermediate species include the tight or contact ion pair (II) and the solvent-separated or loose ion pair (III). The contact ion pair has counterion (or gegenion) of opposite charge close to the propagating center (unseparated by solvent). The solvent-separated ion pair involves ions that are partially separated by solvent molecules. In cationic polymerization the chain end is cationic and has a negative counterion, while in anionic polymerization the chain end is anionic and has a positive counterion. Most ionic polymerizations involve two types of propagating species-- an ion pair (H-IV) and a free ion (V) - coexisting in equilibrium with other. The relative concentrations of these two types of species and the identity of the ion pair (that is, whether the ion pair is described as species II, III, or IV), depends on the particular reaction conditions, especially the solvent employed. The nature of the solvent has a large effect in ionic polymerization since the different types of propagating species have different reactivities. In general, propagationrates are higher the morethe rnacroionand its counterion are separated. Loose ion pairs are more reactive than tight ion pairs and free ions are significantly more reactive than ion pairs. In general, more polar media favor solvent-separated ion pairs or free solvated ions. Free solvated ions will not exist in hydrocarbon media, where other equilibria mayoccur between ion pairs and clusters of ions. Although solvents of high polarity are desirable to solvate ions, they cannot be employedfor several reasons. The highly polar hydroxylic solvents

Ionic ChainPolymerization

655

(water, alcohols) react with and destroy most ionic initiators and propagating species. Other polar solvents such as ketones prevent initiation of polymerization by forming highly stable complexeswith the initiators. Ionic polymerizations are, therefore~ usually carried out in solvents of low or moderate polarity such as methyl chloride, ethylene dichloride, and pentane. Thoughionic polymerization resembles free-radical polymerization in terms of initiation, propagation, transfer, and termination reactions, the kinetics of ionic polymerizations are significantly different from free-radical polymerizations. In sharp contrast to free-radical polymerizations, the initiation reactions in ionic polymerizations have very low activation energies, chain termination by mutual destruction of growing species is nonexistent, and solvent effects are much more pronounced, as the nature of solvent determines whether the chain centers are ion pairs, free ions, or both. No such solvent role is encountered in free-radical polymerization. The overall result of these features is to makethe kinetics of ionic polymerization much more complex than the kinetics of free-radical polymerization. Rates of ionic polymerization are by and large muchfaster than in freeradical processes. The higher rate of ionic polymerization is mainly because termination by mutual destruction of active centers, which is prevalent in free-radical systems (see Chapter 6), does not occur in ionic systems macroions with the same charge will repel each other and thus concentrations of propagating species can be much higher in ionic than in free-radical systems. Rate constants for ionic propagation reactions vary but some are higher than those in free-radical systems. This is particularly true in media where the ionic active center is free of its counterion. The high reactivity of ionic active centers, which yields fast propagation rates, also results in a greater propensity toward side reactions and interference from trace impurities. Lowtemperatures favor propagation over competing reactions which cause chain termination. Ionic polymerizations are thus performed often at muchcolder temperatures than those used in free-radical processes, which would be impossibly slow under the same conditions. Ionic polymerizations are not as well understood as radical polymerizations because ionic polymerizations are characterized by a wide variety of modes of initiation and termination. The nature of the reaction media in ionic polymerizations is often not clear since heterogeneous inorganic initiators are often involved. Further, it is extremely difficult in most instances to obtain reproducible kinetic data because ionic polymerizations proceed at very high rates and are extremely sensitive to the presence of small concentrations of impurities and other adventitious substances. Because of the high selectivity and sensitivity of ionic reactions, successful ionic polymerizations must be carried out muchmore carefully than

656

Chapter 8

normal free-radical syntheses, consequently, a given polymeric substance will ordinarily not be producedby ionic initiation if a satisfactory product can be made by less expensive free-radical processes. For example, commercial styrene polymers are all free-radical products, though styrene polymerization can be initiated with free radicals as well as with appropriate anions or cations. However, particular ionic processes are used to make research grade polystyrenes with exceptionally narrow molecular-weight distributions and diblock or multiblock copolymers of styrene and other monomers. In this chapter we will review pure ionic polymerizations-first, anionic polymerizations with someof their specific applications and then the polymerization processes which proceed by a cationic mechanism. Coordination polymerizations that are complex polymerizations having partial ionic character and ring opening polymerizations, manyof which proceed by anionic and cationic mechanisms, will be reviewed in subsequent chapters. IONIC

POLYMERIZABILITY

OF

MONOMERS

As we have noted earlier in Chapter 1, a monomermust have a functionality greater than or equal to 2 in order for polymers to be produced from its reactions. For chain growth polymerizations this functionality can be derived from opening of a double bond or opening of a ring. The most important functional groups that participate in chain-growth polymerizations are the carbon-carbon double bond in alkenes and the carbon-oxygen double bond in aldehydes and ketones. The alkene double bond can be polymerized in chain-growth reactions in which the active site is a free-radical, ion, or carbon-metal bond. (Aldehydes and ketones are not activated by free radicals because of the difference in electronegativity of the C and O atoms. Aldehydes and ketones are polymerized only by ionic or heterogeneous catalytic processes.) The processes whereby a given alkene reacts depend on the inductive and resonance characteristics of the substituent X in the vinyl monomerCH~=CHX.Electron-releasing substituents,

R,

RO-,

R--C~C-

and

---((~)

increase the electron density of the double bond and thus facilitate of a cation, H H

H

H /¢,.;

~"~

¢..;

~+ A+B- ~ ACH2--

+... X

B-

addition

657

Ionic ChainPolymerization

Thus monomers like isobutylene (VI), 3-methylbutene-1 (VII), styrene (VIII), and vinyl ethers (IX) all undergo cationic polymerization. CH3 dR (VI)

(VIII)

(VII)

(IX)

Someof these vinyl monomerscan delocalize the positive charge, and this also facilitates reactions with cations, e.g., in the polymerizationof vinyl ethers: A $ + CH_~-----C 6-R

~ A-CH~--C $ ~ A-CH~--~, :O-R 6-R

(s.3)

The alkoxyl substituent thus allows a delocalization of the positive charge over two atoms--the carbon and the oxygen, and this leads to stabilization of the carboniumion. (If the substituent were not present, e.g., in ethylene, the positive charge wouldbe localized on the single a-carbon atom.) Similar delocalization effects occur with phenyl, vinyl, and alkyl substituents, e.g., for styrene polymerization: H vwwCH.~-- C+

H ~

(8.4) vwwCH~5

Electron-withdrawing substituents -C=N,

--~-R,o

--~-OH

or

decrease the electron density of the double bond and thus facilitate of an anionic species on the double bond. H H \ ~ / I /C = C~ + A+B ~ BCH=--C:-... ~ H X ~

attack

H

+ A

(8.5)

The electron-withdrawing substituents may also stabilize the anion formed by delocalization of the charge, e.g., for acrylonitrile polymerization:

658

Chapter 8 H

H

~--N

C=N:-

(8.6)

The stabilization of the propagating carbanion occurs by delocalization of the negative charge over the oz-carbon and the nitrogen of the nitrile group.

Problem 8.1 Contrary to the high selectivity shownin cationic and anionic polymerization,radical initiators can bring about the polymerizationof almost any carbon-carbondouble bond. Explain, giving reasons.

Radical species are neutral and do not have stringent requirements for attacking the ~r-bond. Moreover,resonancestabilization of the propagating radical occurs with almost all substituents, for example, H wvvvCH2--C" ~-~ ~,~A, I C----N

vvCH2--

H C C=N"

H H v~vvCH2--C" ~ vvvvvCH~--Q :CI" ~1

Radical initiation can thus take place with almost any carbon-carbondouble bond. Problem8.2 Explain whyacrylates and vinyl acetate are not cationically polymerizable. Answer: For efficient cationic polymerization of vinyl monomers it is necessary that the carbon-carbondouble bond be the strongest nucleophile (electron donor) in the molecule. If more than one nucleophilic site e.xists in a monomerand the 7r electron system of the double bond does not represent the most nucleophilic site, the other site(s) maycomplexwith the electrophile (proton, cation, Lewis acid). Thus, for methylmethacrylat.e~

Ionic ChainPolymerization

CH3

CH3 CH2~C

659

~

CH2~C

~----O O

CH~ ~

+CH2--C

+C-O-R O

CH 3

CH 3

CH3 ~

~

CH2~

~-O-R O

C-O-R

CH3

CH3

etc.

In addition, with these monomers the substituent not only preferentially complexes the electrophile but mayeven reduce the nucleophilicity of the double bondby electron attraction. Acrylates (and similarly vinyl acetate) thus do not polymerize cationically. (It maybe noted that vinyl acetate is also not polymerizedby anionic initiators as they attack the acetate linkage. Vinylacetate is polymerizedonly by free radicals.)

Phenyl and alkenyl (-CH=CH2)substituents, although electron-pushing inductively, can resonance stabilize the anionic propagating species in the same manner as a cyano group [Eq. (8.6)]. Monomerssuch as styrene and 1,3-butadiene can therefore undergo anionic as well as cationic polymerization. The applicability of various types of initiation mechanismsto the polymeri½ation of commonolefin monomers is summarized in Table 8.1. We see that isobutene can be polymerized only by cationic initiation, whereas monomers,such as vinyl chloride, methyl methacrylate or acrylonitrile with their electronegative substituents will not yield at all to cationic initiation. Vinyl chloride, however, does not respond to anionic initiation as well. Though halogens can withdraw electrons inductively and push electrons by resonance, both effects are relatively weak. Vinyl chloride thus does not undergo either anionic or cationic polymerization. ANIONIC

POLYMERIZATION

Anionic Initiation The overall reaction in anionic polymerization can be divided into initiation, propagation, and termination steps. While anionic initiators are all electron donors of varying base strengths, the initiator type required for a particular polymerization depends on the ease with which an anion can be formed from the monomerwhich acts as an electron acceptor in anionic polymerizations. In general, the strength of the base required to initiate polymerization diminishes with increasing electronegativity of the substituent on the

660

Chapter 8

Table 8.1 Applicability of Various Types of Initiation Mechanismsto the Polyamerization of Olefin Monomers Olefin monomer Ethylene Propylene

Monomerstructure

Free Cationic Anionic Coordiradical nation

CH2 =CH2

q---CH2 =CHC2H5 ---Isobutene CH2=C(CH3)2 -+ --Butadiene-l,3 CH2 =CH--CH=CH2 qStyrene CH2 =CHPh -t-t-tqVinylchloride CH2 =CHCI ---tMethacrylic esters qCH2 =C(CHa)COOCH3. -t--tVinyl ethers CH2 =CHOR -Acrylonitrile CH2 =CH--CN + -+ + aSymbol+ indicates that the monomer can be polymerizedto high molecularweight polymerbythis formof initiation. CH2 =CHCH3

Butene-1

monomer. The electronegativity following order

of some selected substituents

-CN > -COOR > -C6H5 -----

is in the

-CH=CH2 >> -CH3

Thus the relatively weak sodiu m methoxide (NaOCH3)can polymerize acrylonitrile, which has a strong electronegative substituent (-CN). Vinylidene cyanide carries two -CN groups on the same carbon atom and can be polymerized even by weaker bases like water and amines. Polymerization of nonpolar monomerssuch as conjugated olefins, however, requires initiation by very strong bases like metal alkyls. The two principal anionic initiation processes are (a) nucleophilic attack on the monomer which produces one-ended (monofunctiorml) anions addition of the initiator across the double bond of the monomer[see Eq. (8.5)] and (b) electron transfer by alkali metals that leads two-ended (bifunctional) anions (see later). Nucleophilic

Attack

Initiation takes place by a nucleophilic attack which is essentially addition of a negatively charged entity to the monomer. Examples of some reactive bases which can initiate in this manner are n-C4HgLi, C6HsCH2Li, NaNH2, KNH2, C6HsCH2Na, CH3ONa, and EtMgBr. Alkyllithium compounds, probably the most useful of these initiators, are generally low

Ionic ChainPolymerization

661

melting and soluble in inert organic solvents. Organometallic compounds of the higher alkali metals have moreionic character and are generally insoluble; hence they initiate polymerization by a heterogeneous process. Other initiators of this type include organic compoundsof calcium and barium and Grignard reagents, but these are not used as commonlyas the alkali metal compounds. Alkyllithium compoundsare employed commercially in the polymerization of 1,3-butadiene and isoprene. Initiation proceeds by addition of the metal alkyl, e.g., n-butyllithium, to monomer: Y + C4HgLi

+ CH2=CHY

~ C4Hg-CH2--C:

Li

(8.7)

This type of initiation is knownas monofunctionalinitiation as it produces one active (ionic) site for propagation. Propagation takes place by the addition of monomerto the ionic site: Y C4Hg-CH~--6.’-

Li + + n CH2 =CHY ~ C4H9

---(--- CH2CHY--3/w-~ CH~-- C,: H

+ Li

(8.8)

The extensive use of alkyllithium initiators is due to their solubility in hydrocarbon solvents. A commonexample is n-butyllithium which is usually available as a solution in n-hexane. The C-Li bond is not ionic in hydrocarbon media where the initiator molecules exist as aggregates. Initiation is thus fairly slow in hydrocarbon media. Addition of tetrahydrofuran to this solvent increases the concentration of unaggregated initiator (which is more active for initiation) by forming a 1:1 complex with this compound.Alkyls and aryls of the heavier alkali metals, such as Na and K, are poorly soluble in hydrocarbons because of the greater ionic character of the Na-C and K-C bonds. Alkyllithium initiators yield stereoregular polymersof conjugated dienes if the polymerization is carried out in hydrocarbon solvents. Addition of tetrahydrofuran or other more polar solvents changes the microstructure of the polymers that are produced. Alkyl derivatives of the alkaline-earth metals have also been used to initiate anionic polymerization. Organomagnesium compoundsare considerably less active than organolithiums, as a result of the muchless polarized metal-carbon bond. They can only initiate polymerization of monomers more reactive than styrene and 1,3-dienes, such as acrylic and methacrylic esters.

662

Chapter 8

Styrene and other monomers can be polymerized by potassium amide in liquid ammonia.The dielectric constant of the solvent is quite high 22) and this is one of the ionic systems in which the active centers behave kinetically as flee ions. Initiation involves the dissociation of potassium amide followed by addition of amide ion to the first monomerunit: KNH2 ~NH~

+

K+ + NH~ H CH~=C ~ H~N-CH2--C-

(8.9) H (8.10)

Al-so because of the unusual nature of this solvent, chain transfer to solvent is important in this system (see later). Electron

Transfer

Initiation by electron transfer is based on the ability of the alkali metals to supply electrons to the double bonds. This yields an anion radical and a positively charged, alkali-metal counterion. Initiation maybe effected (a) by direct attack of the monomeron the alkali metal, or (b) by attack on the metal through an intermediate compoundsuch as naphthalene. Both result in bifunctional initiation, that is, formation of species with two carbanionic ends. Allah

Metals

Initiation by direct attack of the alkali metal involves transfer of the loosely held s electron from a Group IA metal atom to the monomer. A radical ion is formed:

Li. + CH~=C, --~ .C~--C.. Li" X

(8.11)

The radical ion may dimerize to give a dianion:

7 Li+ --~Li+-~d-CH2-CH2--~. :-Li+ (8.12) 2 "CH:--C. ± k x The eventual result of the initiation process is thus a bifunctional dicarbanion species capable of propagating at both of its ends.

Ionic Chain Polymerization

663

Free metals may be employed as solutions in certain ether solvents, in liquid ammonia,or as fine suspensions in inert solvents. The latter are prepared by heating the solvent and the metal above the melting point of the metal, stirring vigorously to form an emulsion, then cooling to obtain a fine solid dispersion. The metal may also be used as a free-flowing powder coated on an inert support such as alumina. The polymerization process is heterogeneous if the metal is used as a dispersion and homogeneous if it is in solution. [Not all ammoniasolutions of alkali metals initiate by electron transfer; potassium, for example, is believed to form KNt-I2, which initiates polymerization by addition of amide ion, as shownby Eqs. (8.9) lan’d (8.10). [Historically, the most important application of the electron transfer initiation involved the production of stereoregular diene rubbers by lithium metal initiation. The lithium was used as a fine dispersion with a large surface area to speed up the initiation reaction and the process was carried out in hydrocarbon solvents because polar solvents increase the generally undesired vinyl side chain content of the product polymer.] Alkali

Metal Complexes

Polycyclic aromatic compoundscan react with alkali metals in ether solution to produce monomericradical ions [1]. The reaction involves the transfer of an electron from the alkali metal to the aromatic compound.For sodium and naphthalene, for example,

Na,

+

-

Na+ (8.13)

-"

(x) The radical nature of the anion radical (X) has been established from electron spin resonance spectroscopy and the carbanion nature by its reaction with carbon dioxide to form the carboxylic acid derivative. The equilNrium in Eq. (8.13) depends on the electron affinity of the aromatic hydrocarbon and the donor properties of the solvent. Tetrahydrofuran (THF) is a useful solvent for such reactions. This fairly polar solvent (dielectric constant 7.6 at room temperature) promotes transfer of the s electron from the alkali metal to the aromatic compoundand stabilization of the resultant complex, primarily via solvation of the cation. Sodium naphthalenide is

664

Ghapter 8

formed quantitatively in THF, but dilution with hydrocarbons results in precipitation of sodium and regeneration of naphthalene. The naphthalene anion radical (which is colored greenish-blue) transfers an electron to a monomersuch as styrene to form the styryl radical anion (XI) (¢ = --C6H5): H H H H + ¯ -C~.,-,.TC-CI. ’-I I I

Na

¢)H

(8.34) The styryl radical ion (XI) is shown as a resonance hybrid of the forms wherein the anion and radical centers are alternatively on the or-- and /~--carbon atoms. The styryl radical anion dimerizes to form the dicarbanion (XII) ~ : C--C’/Na + ~ Na + + ":"

J

~-CH~-CH~--~.:"

Na

This initiation process is thus similar to alkali metal initiation in (a). That this reaction occurs is shown by electron spin resonance measurements, which indicate the complete disappearance of radicals in the system immediately after the addition of monomer.The monomerin these systems often has a lower electron affinity than the polycyclic hydrocarbon, but dimerization of the monomericradical anion [Eq. (8.15)] drives the equilibrium reaction (8.14) to the right. Dimerizationof radical centers is highly favored by their high concentrations, typically 10-3-10-2 Mand the large rate constants (106-108 L/mol-s) for radical coupling. (Note that the dimerization occurs to form the styryl dicarbanion instead of :CH2CH~bCH~bCH2:, since the former is muchmore stable.) The styryl dianions are colored red (the same as styryl monocarbanionsformed via initiators such as n-butyllithium). Anionic propagation occurs at both carbanion ends of the styryl dianion: H Na :C-CH2-CH2--

’

Na÷ .:-C-CH2-(-

C : Na+ + (n +,m) ~bCH=CH2 ~

cHqb --CH~ ).

( CH2-CH~b--)-~-~ CU~ ’] +

Anionic propagation is generally muchfaster than free-radical reactions.

665

Ionic ChainPolymerization

Problem8.3 Account for the fact that anionic polymerizations are generally muchfaster than free-radical reactions although the kp values are of the same order of magnitudefor addition reactions of radicals and solvated ion pairs (free macroanionsreact muchfaster). Answer: -9The concentrationof radicals in free-radical polymerizationsis usually about 10 10-7 Mwhile that of propagatingion pairs is 10-4-10-2 Mdependinguponinitiator concentrations. As a result, anionic polymerizationsare 10a-107 times as fast as free radical reactions at the sametemperature.

Termination

Reactions

Anionic polymerizations must be carried out in the absence of water, oxygen, carbon dioxide, or any other impurities that may react with the active ionic centers. Glass surfaces carry layers of adsorbed water which react with carbanions. It is thus necessary to take special precautions, such as flaming under vacuum, to remove this adsorbed water in laboratory polymerizations. The monomeritself should be very pure and free from inhibitors. Polymerization Polymerization")

Without

Termination

("Living

Interest in anionic polymerization grew enormously following the work of Michael Szwarc in the mid 1950s. He demonstrated that under carefully controlled conditions carbanionic living polymers could be formed using electron transfer initiation. With ionic polymerization, as we have noted earlier, there is no compulsory chain termination through recombination, because the growing chains carrying same ionic charges cannot react with each other. Thus in anionic polymerization systems, especially of nonpolar monomerssuch as styrene and 1,3-butadiene, initiated by organometallic compoundsand employing perfectly dry inert solvents such as benzene and tetrahydrofuran, termination or transfer is virtually nonexistent and active chain ends have indefinite lifetimes. Such systems are referred to as living po~mers. Propagation occurs with complete consumption of monomerswhile the propagating anionic centers remain intact because transfer of protons or other positive species from the solvent does not occur.

666

Chapter8

The nonterminating character of living anionic polymerization is evident from the fact that if a reaction system is highly purified so that impurities are absent, the color of the carbanions is observed to persist throughout the polymerization and does not disappear or change even at 100%conversion. After 100%conversion is reached, additional polymerization can be effected by adding more monomer,either the same monomeror a different monomer. The added monomeris also polymerized quantitatively, without any new addition of initiator, and the molecular weight of the living polymer is increased. (There is no reason in theory why chain growth should stop if more and more monomeris added and chain terminators are absent. In practice, however, small amounts of terminators are inevitably produced and chain growth also slows downeventually because of the high viscosity of the system or because the chains becomeinsoluble.) Termination of the living chain can be brought about when desired by introducing suitably reactive materials, such as water, alcohol, or ammonia. The aforesaid features offer fascinating possibilities of building model polymer systems such as monodisperse molecular weight polymers (by controlled addition of monomer), structures with defined end groups (by deliberate termination of a living system with appropriate reagents) and. block copolymers (by sequential addition of two or more monomers). These are discussed more fully in a later section. Termination by Impurities 7Yansfer Agents

and Added

Ionic polymerizations, as noted above, are carried out in an inert atmosphere with rigorously purified reagents and cleaned glassware since trace impurities lead to termination. Anymoisture present terminates propagating carbanions by proton transfer: vvwvCH2--~: q- H20 ---* v~wCH2--6H + HO-

x

t

The hydroxide ion is usually not sufficiently nucleophilic to reinitiate polymerization and the kinetic chain is thus broken. Water is an especially effective chain terminating agent. For example, Ctr,s is approximately 10 in the polymerization of styrene at 25°C with sodium naphthalene. Thus the presence of even small concentrations of water can greatly limit the polymer molecular weight and polymerization rate. Ethanol has a C’tr,S value of about 10-:~. Hence its presence in small amounts would not prevent the formation of high polymer because transfer would be slow; the polymer, however, would not be living.

Ionic ChainPolymerization

667

Oxygen and carbon dioxide from the atmosphere add to propagating carbanions to form peroxy and carboxyl anions:

w~wCH2--

(8.18)

+ O2 ~ wvwCH2--C~-O-(~: X

(8.19)

wwvCH2--C: + CO~_ ---~ w~vCH2--C--C--O:

These anions are normally not reactive enough to continue propagation and tile chains are thus effectively terminated. The peroxy and carboxyl anions usually are finally obtained as HOand HOOC groups when a proton donor is subsequently added to the polymerization system. Polymerswith specific end groups can be prepared by deliberately introducing particular reagents that terminate ("kill") living polymers. Telechelic polymers (that is, polymers with reactive end groups), containing one more end groups with the capacity to react with other molecules, can be prepared in this way. Thus, in the anionic polymerization of butadiene with bifunctional initiators, carboxyl end groups are produced by termination with CO2: Na + :-~-CH=CH-CH2vwwwCH2-CH=CH--

_ 9,

Na + + 2CO2 ~

Na ÷ :O--CCH2-CH=CH-CH~w~vwCH~-CH=CH-CH~--

9, -

+ C-O :

Na

(8.20) Reaction with acid then yields carboxyl-terminated polybutadiene (CTPB). Hydroxyl end groups are provided by termination with ethylene oxide: vvwvC: Li + + CH2--CH2

+ wwvCH2-CH~-CH2-O- : Li w~vCH2-CH~-CH2-OH +

Cn_qH LiOCH3

(s.21) Hydroxyl-terminated polybutadiene (HTPB) can be produced by such reactions. Low-molecular-weight (3000-10,000)versions of such elastomers are used as binders and liquid rubbers, which can be shaped more easily than the conventional high-viscosity elastomers. The liquid rubbers can be vulcanized by reactions of their specific end groups. Hydroxyl-ended

668

Chapter 8

polybutadiene can be caused to grow in molecular size and to cross-link, for example, by reaction with isocyanates having functionalities > 2. Spontaneous

Termination

Living polymers do not live forever and even in the absence of terminating agents decays with time [2]. The most stable of all living anionic systems are polystyryl carbanions, as they are stable for weeks in hydrocarbon solvents. The mechanismfor the decay of polystyryl carbanions on aging, referred to as spontaneous termination, is not completely established. The generally accepted mechanismconsists of hydride elimination : wvwCH2CH~CH~--

,~-:

Na+ ~ w~CH~CH~CH=CH~ + HI"

(xni)

+ Na

(8.22)

followed by abstraction of an a[lylic hydrogen from (XlII) by a carbanion center to yield the unreactive 1,3-diphenylallyl anion (XlV) [3]: wvvvCH2~C :

+ wvwCH2CHq~CH=CH¢ ---* wvwCH2-CH2¢

7. ÷ wvwCH2C¢CH=CH¢

(8.23)

(x_w) The sodium hydride eliminated in reaction (8.22) may also take part hydrogen abstraction from (XIII). Problem 8.4 Lower molecular weights and polymerization rates observed in anionic polymerizationsof polar monomers are attributed to the reactivity of the polar substituents towardnucleophiles, leading to termination and side reactions that are competitive with both initiation and propagation. Explain this behavior considering the case of methyl methacrylate monomer. Answer: Several different nucleophilic substitution reactions are possible with MMA. For example, attack of initiator on monomer by the reaction C, H3 ~ CH2 = C~C-OCH3 + R-Li + + ---*

CH3,Oit CH2-----C--C-R

+ CHzC~: Li (P8.4.1) converts active alkyllithium to alkoxide, whichis’a less active initiator, and MMA to i-propenyl alkyl ketone, which forms less active carbanion than MMA. The polymerizationrate is thus decreased.

669

Ionic Chain Polymerization

There may be a nucleophilic attack by a propagating carbanion on the monomer,by the reaction, CH3 l_ vvvwCH~-- C : Li + + ~OOCHz

CH3 O It I C-- C-OCHz -~

CH2 =

C~H3~__ C~ H3 + ~ C=CH2 + CHaO-: vvvvvCH2mC~ COOCH3

Li

(P8.4.2)

whichagain yields a less reactive methoxideinitiator and an i-propenyl keto end group which also forms a less reactive carbanion than MMA. This reaction lowers the polymermolecular weight and decreases the polymerization rate. A nucleophilic attack mayalso take place intramolecularly resulting in chain termination and formationof less active methoxideinitiator

PolymerizationKinetics The kinetics of anionic polymerization is much more complex than the kinetics of free-radical polymerization. Most of the complications arise from the initiation reactions. Th6variety of initiation possibilities often gives the appearance that each anionic polymerization is unique and that no general kinetic treatment is possible. In polymerizations initiated by alkali metals or insoluble organometallics employedas fine dispersions in organic media, the initiation step occurs at a phase interface while subsequent propagation reactions may occur in a homogeneous medium. The overall kinetics of such heterogeneous initiation/homogeneous propagation reactions are often very complexand specific to the particular systems. Useful generalizations mayhowever be provided by systems in which the initiation and propagation processes are both homogeneous.Such polymerizations are discussed next.

670

Chapter 8

Somecases exist in homogeneousanionic polymerization in which the initiator dissociates completely with quantitative transformation into the active ionic form and does so before any significant amount of propagation has occurred (stoichiometric polymerization). This is the case, for example, if one uses, as initiators, alkali organic compounds (e.g., phenyllithium, butyllithium, or sodium naphthalene) in solvents which have unshared electron pairs (Lewis bases). In this case the alkali forms stable positively charged complexions with the Lewis base, so that the organic residue is negatively charged (carbanion) and an ionic polymerization can be initiated by this carbanion [cf. Eqs. (8.13) and (8.14)]: H + C4HgLI + .~.

~ CH3CH2CH2!:-+ [Li-.--.:O~] H

Li {~

"’@ + :NH 3 ~

IL + 3 i -.--

+ 1 :NH

In the above cases, the polymerization kinetics are so simple that it is useful to classify ionic polymerization kinetics according to whether the initiators are quantitatively and instantaneously dissociated or not. (It should be noted that the term "dissociation" does not denote any special kind of charge separation. Probably one has to regard this dissociation as a separation of ion pairs by solvent molecules, but not a separation in the sense of forming single ions which can move independently from the counterion.) Quantitative of Initiator

and Instantaneous

Dissociation

For a kinetic analysis, the process of anionic polymerization has to be divided into at least three main reactions commonto all types of polymerization, viz., initiation, propagation, and termination. Representing the initiator by -) and a terminating agent by X, the reactions can be written CA(or C+A as

Initiation :

+ CA -1-

M ~ AM- C

(8.24)

671

Ionic ChainPolymerization Propagation :

AM- C + + -F

M -~

AMM- C

(8.25)

AMM- C + + M ~ AMMM- C + (8.26) AM~_IM- C+ -]Termination:

AMzM- C+ (q-

M -~ AMxM- C+ (8.27) X) ~-~ AMzM

(8.28)

(Depending upon the solvent, the propagating ion may behave as a free +, or as both.) Living polymerizaion AMzM-or as an ion pair AMzM-C tions are characterized by the absence of termination (as well as transfer) reactions. In most cases it is also possible to find initiators which are reactive enough to give instantaneous initiation, i.e., ki > kv. This implies that no initiation takes place during the polymerization and the number of chain centers to which the monomermolecules may add reaches its maximum value before potymerizalion begins. Moreover, the number of chain centers does not change during the polymerization because there is no termination step. Denoting the total concentration of anions of all degrees of polymerization by [M-] (the positive counterion + i s n ot s hown for s implicity) and assuming that the initiation reaction (8.24) is not only instantaneous but also complete, one can thus write for living polymerization [M-] = [CA]0

(8.29)

Hence the rate of polymerization is given by P~ ----=

- at -- ]~p[M-][M]

= kp[CA]0 [M](8.30)

Here ]~p should more appropriately be called an apparent rate constant or overall rate constant since both undissociated (ion pair) and dissociated (free ions) species usually exist and their propagation rate constants are different (discussed later). Integration of this pseudo-first-order rate equation gives the time dependence of the monomerconcentration as [M] = [M]sp exp (--/¢~[CA]ot)

= ([M]o -- [CA]o)exp (--kp[CA]ot)

(8.31) where the subscript ’0’ indicates initial value. Note that [M]sp is the monomer concentration at the start of propagation [Eq. (8.25)] and thus equals the original monomerconcentration less the concentration of the initiator CA which is quantitatively reacted. From the experimental measurementof [M] at various times after the polymerization is started and knowing[CA]0, that is the concentration of the initiator in the initial reaction mixture, ]~p may

672

Ch~p~er8

be determined either by Eq. (8.30) or Eq. (8.31). The significance of values of kp will be discussed in a later section. Methods Manyanionic living polymerizations proceed too rapidly to be followed by techniques such as dilatometry. The Stopped-flow technique is useful for studying these fast polymerizations. In stopped-flow, rapid-scan spectroscopy [1,4], separate monomerand initiator solutions are rapidly forced through a mixing chamber (where instantaneous mixing occurs) and then into capillary tube located in a spectrophotometer; flow is stopped and the change in absorbance is measured with time. This method allows one to follow the initiation rate (observing the increase in optical density of propagating species) and/or the polymerization rate (by following the loss of monomer). The polymerization rate can be obtained by short-stopping a polymerizing system by means of a highly efficient terminating agent. All propagating centers are quickly terminated with incorporation (into the polymer) of an end group derived from the terminating agent. In a modified apparatus, the rapidly mixed monomer-initiator solution is forced through the flow tube into a solvent containing a terminating agent (Fig. 8.1). A turbulent flow must prevail in this flow tube (Reynolds number > 10,000) since the chains would show growth times of differing duration with a laminar flow. The end groups in the polymer are analyzed after ~eparation of the polymer from the other componentsof the reaction system. The reaction time is given by the ratio of the capillary volume to the flow rate. Short reaction times from 0.005 to 2 sec can be accurately studied in this manner. The conversion, and hence /~, are obtained by analyzing the quenched reaction mixture for either polymer or unreacted monomer.

Problem8.5 Consider the flow tube for rapid polymerization reactions shown schematically in Fig. 8.1. Let ~ be the volumeof the tube (distance between the mixingjets) and V be the volumeof the total liquid flowing through in time t. Denotingthe total concentration (constant) of polymerchain ends by Eq. (8.29) and the concentration of monomer units in polymerby [M]p derive an expression for monomerconversion as a function of t. .~nswer: The effective polymerizationtime is the sameas the residence time ~- given by Vo (P8.5.1)

Ionic Chain Polymerization

673

Integrating the rate of propagation, given by Eq. (8.30) and assuming[M]sp [M]0for high polymers,that is, high monomer to initiator ratio, one obtains -- ln[M] I = k,[CA]0 t I (PS.5.2) where [M]~ = [M]0 - [M]p. Equation(P8.5.2) is transposedwith the aid of Eq. (P8.5.1) t : in [M]o (~) [M]--’-~----- k, [CAIo~" = k, [CA]o where p is the monomerconversion given by tn{1/(1-,)}

V =

(P8.5.3)

[M]o

Average Kinetic

Chain Length

Because there is no termination step in a true "living" polymerization, the kinetic chain growth is ended only when the monomeris completely consumed. The average kinetic chain length in living polymerization with fast and complete dissociation of initiators, will be given by Monomer molecules consumed [M]sp -- [M] Numberof chain centers [CA]o

_

(8.32)

where [M]sp -" [M]0- [CA]0 (8.33) since the initiator is assumed to be instantaneously and quantitatively reacted. Onsubstitution of [M] from Eq. (8.31), ~, is obtained as a function of time, given by [MIsp-~[ {1 cA]°<~ -- e

v- [CA]7

)

(8.34)

In the limit of t --* cx~, or, in other words, at the completionof reaction,

[M]sp [CA]0

(8.35)

Average Degree of Polymerization The average degree of polymerization is given by the number of monomer molecules polymerized per polymer molecule formed. In the case of monofunctional initiation [cf. Eq. (8.7)], the number of polymer molecules

674

Chapter 8

Figure 8.1 Schematic representation of a flow tube for rapid polymerization reactions; Mmonomer; S solvent; I initiator; C chain terminator; X mixingjets; Y collecting container. formed is equal to the number of chain centers (or initiators) number average degree of polymerization then is [cf. Eq. (8.32)] DP --

[M]o- [M] ([Mlsp -[Cn]o

+ [CA]o) --[M] [CA]o

and the

= u + 1 (8.36)

In the case of bifunctional initiation that produces double-ended active species [cf. Eq. (8.12)], the number of polymer molecules formed 1/2 of the number of chain centers or initiators. The average degree of polymerization is then given by DP [M]o-

-

[M] ([M]sp

-~[CAlo 2

-b [CA]o)- [M] : 2(v’ ![CA]o 2

q- 1) (8.37)

Problem 8.6 In an experiment carried out by Szwarc et al [5] styrene (9.2 g) was added to 60 mLof tetrahydrofuran containing 3.3x10-4 tool of sodium naphthalene. Polymerization was carried out at -80°C and after completion (as

Ionic Chain Polymerization determined by constant viscosity) an additional 7.7 g of styrene in 50 mL of tetrahydrofuran was added. The final yield was 16.6 g of polystyrene, i.e. about 100%conversion. Calculate the average molecular weight of the final polymer. Answer: (9.2 + 7.7)g = 0.1625 tool Total moles of styrene -- (104 g mo1-1) Moles of initiator = 3.3 x 10-4 tool Since sodium naphthalene causes bifunctional initiation, Eq. (8.37) is applicable., At complete conversion of monomer, Total moles of monomer [M]0 DP = = 1 [CA] 1 ~ (Total moles of initiator) ~ Since no initiator is added in the second stage of polymerization, Dff = (0.1625 mol) / ~(3.3 x -4 to ol) = 985

Problem8.7 Sodium (1.15 g) and naphthalene (7.0 g) were stirred

together in 50 mL dry tetrahydrofuran to form a dark green solution of sodium naphthalenide. When1.0 mLof this green solution was introduced into a solution of styrene (208 g) in dry tetrahydrofuran by arapid injection technique the latter turned reddish orange. The total final volume of the mixture was 1 L. Assumethat the injection of the initiator resulted in instantaneous homogeneousmixing. After 5 seconds of reaction at 25°C, the styrene concentration was determined to be 1.73 × 10-2 mol/L. After 10 seconds the color of the reaction mixture was quenched by adding a few milliliters of methanol. The polymer was then precipitated and washed with methanol. Calculate (a) Overall propagation rate constant. (b) Initial rate of polymerization. (c) Rate of polymerization after 10 s. (d) Molecular weight of the final polymer. Answer: Sodium= 1.15 g ~ 0.05 tool Naphthalene = 7.0 g = 0.055mol Styrene = 208 g -- 2.0 tool Since naphthalene is in excess, the total number of moles of initiator in 50 mL TI-IF is 0.05 mol. The number of moles transferred in 1 mLof dark green solution = 0.05/50 = 10-3 mol. Hence [CA]0 = 10-3 -1 moi L

676

Chapter 8

-1 [M]o = 2.0 tool L [M]sp = [M]o - [CA]o = (2.0

mol L-1) - (0.001

molL -1)

-1 ~2.0

molL

(a) From Eq. (8.31), kv _ ln([M]sp/[M])

[CA]o t

ln[(2.0 tool -I) / (1.73 x 10-2 mol L - I) (10 .3 mol L-l)(5 -1 -1 950 L tool s (b) From Eq. (8.30), (e~)0

= (950 L mo1-1 -1)(10-3 = 1.90 mol L-1 -1 s

mol L -1)(2.0

t ool L -1)

(c) From Eq. (8.31), at t = [M] = (2.0 mol -1) e xp[-(950 L mo-1 s-l )(10 -a molL-l) (10 s)] --= 1.497 x 10-4 -1 moi L P~

= (950 L mo1-1 s-l)(10 -3 mol L’)(1.497 = 1.42 x 10-4 mol L-1 -1 s

x -4 to ol L - 1)

(d) From Eq. (8.36), DP~

Distribution

=

2(2.0

of the

molL -1 - 1.497x10 -4 -1) molL (10 -3 -1) mol L

4000

Degree of Polymerlzation

The most important difference between a "living" ionic polymerization which has no termination or transfer mechanism and free-radical or ionic processes that do have termination or chain transfer steps is that the distributions of the degrees of polymerization are quite different. The distribution function can be derived by a kinetic approach due to Flory [6], which is analogous to that used earlier for free-radical reactions (see Problem 6.44). However, in the present case with no chain termination the simplifying steady-state approximation cannot be used. Consider the reactions (8.24) to (8.27) for this polymerization. If usual assumption is made that kp is independent of chain length and also that the initiation step [Eq. (8.24)] is instantaneous, application of the law of mass action gives the following set of rate equations : d[AM-] dt

- kv [AM-I[MI

(8.38)

677

Ionic ChainPolymerization d[AMM-]

dt

-- kp [MI([AM-] -- [AMM-])

d[AMMM-]

--

dt

d[AMzM-] dt

kp [M]([AMM-] --

-- kp [M]([AMz-IM-]

[AMMM-])

(8.39) (8.40)

-- [AMzM-]) (8.41)

Substitution for [M] from Eq. (8.31) into Eq. (8.38) yields AM-]__--kp [M]sp / d[AM-]t On integration, [AM-] ----

/ e--kp [CA]0t dt

(8.42)

under the condition that at $ ---- 0, [AM-] = [CA]0, [CA]°exp[CA]0[MIsp

(1 - -kvtCAl0t)]

(8.43)

Combining with Eq. (8.34), this last result may be written in the compact form -u [AM-]----[CA]0e (8.44) To eliminate the time ~, from the remainder of the rate equations (8.39) to (8.41), one may use again Eq. (8.34) for the average kinetic chain length. Differentiation of Eq. (8.34) with respect to t gives du = kp [M]sp e-kv

[CA]t dt

(8.45)

Substitution of Eqs. (8.31), (8.44) and (8.45) into Eq. (8.39), with nation of t, transforms this rate equation for [AMM-]into the differential equation -u (8.46) -b [AMM-] ---- [CA]0e du This is the standard form of a linear, first-order differential equation whose solution is given by -] eU[AMM

---- /eu[CA]oe-Ud~ q- z (8.47) u where e is the integrating factor and z is a constant of integration. After integration of Eq. (8.47) and evaluatio~ of z by the condition that ~ ---- 0 (i.e., at t ---- 0), [AMM-]=0, weobtain

[AMM-] = [CA]0

(S.48)

Chap~er8 The process may now be repeated for Eq. (8.40). Thus, elimination of by substitution of Eqs. (8.31), (8.45), and (8.48) transforms Eq. (8.40) the differential equation d[AMMM-] + [AMMM-] = -v [CA]ore (8.49) dv which may be solved in the same manner as was Eq. (8.46) using the integrating factor ev. This yields 1v [CA]0v:~ e[AMMM-] = ~ (8.50) Repetition of this process soon reveals that the concentration of the anion containing x monomermolecules is given by [AMz-IM-] = [CA]0 (x _ 1)!

(8.51)

and at the completion of polymerization by ,[AMz_IM-]~

= [CA]0 ~-~ --

~

(8.52)

Since each initial anion has produced one polymer species, as there is no termination, the fraction of polymer having degree of polymerization x at the end of reaction is Number of anions containing x monomers Number of anions [AM~-IM-]oo

I~(~o-I ~--v~o

i)!

(s.53)

Substituting for v~ from Eq. (8.35), and using [M]sp = [M]0 for high polymers (i.e., high ratio of initial monomer to initiator), one obtains 1 ([Mlo

- (x - 1)!

~z-X

exp(-[M]0/[CA]0) (8.54)

The number distribution of ~ (or DP) can be predicted from Eq. (8.53) or (8.54) when the average kinetic chain length, v~ (= [M]o/[CA]o), kno/en. Problem8.8 1-Vinylnaphthaleneis polymerizedanionically at 25°Cin a tetrahydrofuran solution containing initially 4 x 10-a MC4HgLi and 0.20 M1-vinylnaphthalene. Showgraphically the numberfraction and weight fraction distributions of

670

Ionic Chain Polymerization

the degree of polymerization. Showfor comparisonthe corresponding distributions for a polymer of the same average degree of polymerization produced by free-radical reaction (assumetermination by disproportionation). Answer: From Eq. (8.33), [M]sp = (0.204 M) -- (4×10-3 M) = 0.20 FromEq. (8.35), (0.20 tool -x) 50 v~o (4 x 10.3 -1) tool L FromEq. (8.52),

(50F-1e-~O

us -- (x - 1)!

lnn~ = (x- 1)ln50- ln(x- 1)! - 50 (P8.8.1) The values of n~ calculated from Eq. (P8.8.1) for different assumedvalues of (50 < x < 50) are plotted against x in Fig. 8.2. The weight fraction distribution w~maybe obtained by multiplication of the numberfraction distribution, Eq. (8.53), by x/’-~--P (= x/v~o). Whenthat is done, one has ~:~--2

(~- 1)! or lnw:= Inx - ln(x - 1)! + (~ - 2)1.~ .~ Figure 8.2 showsa plot of wz vs. x for v~o = 50. For free-radical polymerization, Eq. (P6.44.2) is plotted for u = 50 in Fig. 8.2 showingnumberdistribution (n=) of the degree of polymerization. The weight fraction distribution w=maybe obtained by multiplication of Eq. (P5.44.2) x/’DP(= x/u). This yields w~ --

v~ 1 +

The distribution is showngraphically in Fig. 8.2.

Actually, the observed distributions of DPare usually somewhatbroader than predicted by Eq. (53). This fact is attributed to the existence propagation-depropagation equilibrium: A--M~--M-+

M ~ A--Mx+I--M-

which was not considered in the derivation of Eq. (8.53).

(8.55)

680

Ohap~e~8

(anionic) nx (anionic)

n X (free radical)

1

0

(free radicat)

20

40

60

80

I 100

Degree of polymerization (x) Figure 8.2 Numberfraction and weight fraction distribution of the degree of polymerization with u = 50 for anionic "living" polymers and for free-radical polymerization(with termination by disproportionation). (Problem8.8) Polydispersity Index The general expression [Eq. (8.53)] for the mole fraction of ~:-mer the polymer is Poisson’s distribution formula [6]. From the nature of the problem it is obvious, without the above derivation, that the numbers of molecules of various sizes must be represented by Poisson’s distribution. Since Dt~n = r,~ + 1 at the completion of reaction [cf. Eq. (8.36)], the number average molecular weight will be given by

=

+

(8.5o)

where M0is the molecular weight of the monomerunit. The weight fraction distribution can be derived easily by multiplication of the mole fraction distribution [Eq. (8.53)] by z/DP.. Since DI:’. = ~oo + i,

681

Ionic Chain Polymerization

x-1 e, -u~ 12oo

X Wx

--

(~,~+~) (~_

uoo xe-~ (~ + I)

~x--2

(x-

(S.57)

The weight-averagemolecular weight can be defined by

~

=

E~M~

(s.5s)

Substituting ~q. (857) in Eq. (8.~8) and placing M~= xMo,

= The summationrea~anges to

~ (~ ~2)~ + ~ (~ - ~)t (~~ 3)~ 2 1 (s.~0) = (.~ + 3 + ~/,~)~ 3

Substituting Eq. (8.60) for the summationin (8.59),

DividingEq. (8.61) by Eq. (8.56), ~ --- = ~ + (~ + ~)~ u~/~

(8.62)

which ~n be appro~mated by M~/~

=

~

+-

1

(S.63)

The ratio approachesuni~ ~ymptotically as u~ increases. Narrowmolecul~weight dist~butions are obtained for systems with f~t initiation and e~cient m~ngin the absence of depropagation, termination and chain transfer reactions. Values of polydispersi~ ~dex (PDI) below 1.1-1.2 are found for manyli~ng pol~erizations. The living polymerimtion technique offers a unique methodof synthesizing standard pol~er samples of knownmolecular weights with na~owdistributions. Molecular weight standards are nowco~ercially availabl~ for a numberof polymers - polys~r¢ne, polyisoprene, poly(a-me~yls~rene), poly(me~yl me~ac~late), polfisobu~lene, and poly(tetrahydrofuran). ~1 except the l~t

Chapter 8

682

polymers are synthesized by living anionic polymerizations. The last two are obtained by living cationic polymerization with ring opening polymerization used for poly(tetrahydrofuran). The occurrence of any transfer, termination or side reactions result in broadening of the molecular weight distribution. The termination reactions in methyl methacrylate polymerizations and depropagation in o~-methylstyrene polymerizations broaden PDI. It should be kept in mind that the simple kinetics and the narrow distributions of DPshownabove are not a general characteristic of anionic polymerization, nor are they unique to it. They arise simply because the entire initiation process is completed before propagation starts and because no termination process is present to destroy the chain centers. Effects

of Reaction Media

The use of initiators that dissociate quantitatively before propagation occurs and the absence of termination reactions permits the determination of kp for anionic polymerization in a simple manner using Eqs. (8.30) and (8.31) measured extent of reaction or rate of polymerization. The rate constants obtained in this way are, however, found to be dramatically affected by the nature of both the solvent and the counterion. Polymerization is much faster in more polar solvents. The data in Table 8.2, for example, show the -3 pronounced effect of solvent in the polymerization of styrene by 3 × 10 Msodium naphthalene at 25°C. The overall propagation rate constants in tetrahydrofuran (THF) and 1,2-dimethoxyethane (DME)are, respectively, two and three orders of magnitude higher than the corresponding values in benzene and dioxane. That the dielectric constant is not a quantitative measure of the solvating power is shown by the higher rate in DME compared to THE The faster rate observed in DMEmay be due to a specific solvation effect arising from the presence of two ether groups in the same molecule. Table 8.2 Effect of Solvent on AnionicPolymerizationof Styrene by 3 × 10-a MSodiumNaphthalenide Dielectric constant kp (L tool-x s-x) Solvent at 25°C Benzene 2.2 2 Dioxane 2.2 5 550 Tetrahydrofuran 7.6 3,800 1,2-Dimethoxyethane 5.5 Source:Datafrom M.Szwarcand J. Staid, "TheKinetics of Propagationof AnionicPolymerization and Copolymerization", chap. 5 in Progressin Reaction Kinetics, vol. 2, G. Porter, ed., Pergamon Press, Oxford(1964).

683

Ionic ChainPolymerization

The increase in kp with increasing solvent power of the reaction medium is due mainly to the increased fraction of free ions present relative to ion pairs. It would thus be useful to determine the individual propagation rate constants for the free ions and ion pairs as well as the relative amountsof the two types of propagating species. The simplest situation considered below for illustration corresponds to an equilibrium between free ions and ion pairs, propagating at different rates. (It is likely that various kinds of ion pairs exist [cf. Eq. (8.1)] these complexities can be neglected in the simple treatment.) The reactions involved in the actual propagation step in the polymerization of monomer Mby an initiator CA(C+A-) [cf. Eq. (8.25)] can thus be represented + AM~ C+,--~K

AM~ -]-

C

(8.64)

k~M K ~

C+ AM~’+I Ion pairs

AM~+1 + qFree ions

C

Here k~ and k~ are rate constants for ion-pair and free-ion propagation, respectively, and/( is the equilibrium constant for dissociation of ion pairs into solvated free ions. K is given by = [Zz AM;] [C +] = 2[~z AM;] lea AMZ C +] +] Ez AM; C +] since [C must equal [Ez AM~]for electrical neutrality, -. source of either ion other than C+A Equation (8.65) gives /r~

(8.65) whenthere is no

If there is little dissociation of polymeric ion pairs, ~2z AM~C+] can be ~et equal to the total concentration of living ends of polymerchain, so that Eq. (8.66) becomes

If the polymer molecular weight is high, the consumption of monomerin initiation reactions will be negligible and the overall rate of reaction can be expressed as

684

Chapter 8 Using the approximation of Eq. (8.67) this can be written

The concentration of polymeric ion pairs is given by Ix ~AM; C+] -- [M-] -- ~x AM; -~ [M-] -- g1/2[M-]1/2 Combining Eqs. (8.69) and (8.70) and rearranging (note that [CA]0), one obtains .R ( k ~ - k ~v ) K1/ 2 v [MI[CA]0 -- k~ q[M_]X/2

(8.69) (8.70) [M-] (8.71)

In view of Eq. (8.30) for living polymerization, Eq. (8.71) can also written as ’/2 (k; - k~)K kp = k~ + [M_11/2 (8.72) where kp is the overall or apparent rate constant for propagation. Note that kp is dependent on the living chain ends or initiator concentration. A plot of the left side of Eq. (8.71) against 1/[CA]1/2 yields a straight line with intercept k~ and slope (k~ - k~)K1/2. Since K can be determined independently by measuring the conductivity of solutions of low molecular weight living polymers, k~ can be estimated from the slope. Such measurementsshow[7] that k~- values are of the order of 104 -- 10s L/mol-s compared to k~ magnitudes "~ 102 L/mol-s (see Problem 8.10) which are of the same order as free-radical kp values. The concentration of free ions that can also be determined by this method (see Problem 8.12) is found be only about 10-3 that of the corresponding ion pairs. However, because of the much larger values of k~- compared to k~ as described abdve, a significant portion of polymerization involves free ions.

Problem 8.9 A solution of styrene (1.5 M) in tetrahydrofuran is polymerized at 25°C by sodium naphthalene at a concentration 6f 3.2×10-~ M. Calculate the initial polymerization .rate and the average degree of polymerization at the completionof the reaction. Whatfractions of the polymerization rate are due to free ions and ion pairs, respectively ? Howwill these Valueschangeif the sodium naphthalene concentration is increased to 3.2x10-~ -r M? [Data: Ka = 1.5×10 mol/L; k~- = 6.5 x 104 L/mol-s; k~ = 80 L/mol-s.]

685

Ionic Chain Polymerization Answer: (a)For3.2× -s IV[ so dium na phthalene [M-] = (Kd [M- C+]) 1/2 [cf. Eq. (8.66)] = [(1.5 × 10-7 mol L-1) (3.2 × 10-5 1/2 mol L-l)] -6 -1 = 2,19x 10 molL From Eq, (8.68), +][ P~ M] = k;[M-][M]

+ k~[M-

C

(2.19 x 10-6 mol L-1) (1.5 tool -1) + (80 L mo1-1 -1) ( 3.2 x 10-s to ol L - 1) (1.5 mol L - 1) = 0.214 tool L-1 s -1 + 0.00384 mol L-1 -1 s = 0.218 mol L-1 -1 s = (6.5 x 104 L mo1-1

S-1)

Fraction of polymerization due to free ions = 0.214/0.218 = 0.98 Fraction of polymerization due to ion pairs = 0.00384/0.218 = 0.02 At complete conversion of monomer[cf. Eq. (8.37)], -1) DP,,- 2[M]= 2x(1.5 molL [M- C+] (8.2 x 10 -s -1) mol L

4=

9.4x10

(b) For 3.2x10 -2 M sodium naphthalene [M-] = [(1.5 × 10-7 tool L-1) (3.2 × 10-2 1/2 tool L-l)] -1 = 6.93x 10 -5 molL P~ = (6.5 x 104 L mol-’ -1) ( 6.93 × 10-5 to ol L- ’) (1 .5 to ol L - 1) + (80 L mo1-1 s-*) (3.2 x -2 mol L - 1) (1 .5 mol L - 1) = 6.76 molL -1 s -1 + 3.84 molL -1 -1 s = 10.60 mol L-1 -1 s Fractions of polymerizations due to free ions and ion pairs are thus 0.64 and 0.36, respectively. DP~ = 2(1.5 molL -1)/(3.2x10-2 molL -1) = 93.8

Problem8.10 The kinetics

of sodium naphthalene initiated anionic polymerization of styrene was studied in a less polar solvent dioxane at 35°C. Using an apparatus which permitted quick mixing of the reaction components in absence of air and moisture, aliquot samples were withdrawn periodically and deactivated quickly with ethyl bromide. From the residual monomerand the average degree of polymerization of the polymer formed the following data were obtained

686

Chapter 8

for four different monomer(M)-initiator(CA) compositions. (Reaction times corrected for the time spent in mixing and deactivation step.) [M]0 = 0.31 M [M]0 = 0.47 M [M]0 = 1.17 M [CA]0 = 4.0x10 -4 M [CA]0 = 7.0x10 -4 M [CA]0 = 11.0x10 -4 M Time Conv. DP Time Cony. DP Time Cony. DP

(s) 101 221

(%) 33 58

(s) 514 900

107 189

(%) 45 65

(s) 625 80 857 177

(%) 45 74

1000 1564

Determine the propagation rate constant of ion pair, k~ Answer: Integrating Eq. (8.30) for constant [M-] (i.e., total concentration of chain centers or active end groups is constant throughout the course of polymerization), In [M]-~2° = k, [M-] t

(P8.10.1)

[M]

where kp is the apparent or overall rate constant. Defining conversion as p = ([M]o - [M])/[M]0, Eq. (P8.10.1) is rearranged kp --

In[1/(1 --p)] [M-] t

For fast initiation [M-]

(P8.10.2)

by quantitative reaction with the initiator

= [CA]

CA (e8.10.3)

To check if this condition is satisfied,

[M-] may be calculated from the equation

[M-] = 2([M]0- [M]) = 2V[___~M]0 DP DP

(P8.10.4)

The factor of 2 is used because the living polymer chains have two active ends. Calculation of [M-] from the given data shows that Eq. (P8.10.3) is satisfied fairly well. In such a case [M-] in Eq. (P8.10.2) may be replaced by [GA]o. Equation (P8.10.2) is used to calculate the overall rate constant kp corresponding to each value of [CA]. These values are plotted against 1/[CA]1/~ in Fig. 8.3. The intercept of the linear plot according to Eq. (8.72) gives k~ = 2.8 tool -1 s -1. Since the value of k~ is of the same order of magnitude as the overall rate constant k~,, it may be concluded that the polymerization in the given solvent mostly involves ion pairs and the contribution of free ions is small.

Effect

of Excess

Counterion

It is useful to consider how the presence of excess counterion due to the addition of a strongly dissociating salt (e.g., sodium tetraphenyl borate,

Ionic Chain Polymerization

687

12

10

2

0

,

I

I

I

I

I

10

20

30

40

50

60

Figure8.3 The overall rate constant kp as a function of [CA] plotted according to Eq. (8.72). (Problem8.10) NaBPh4,to supply excess Na+) can affect the kinetics of a living anionic polymerization.Theretarding effect of a dissociating salt forms the basis of a methodby whichthe dissociation constant of a polymericion pair may be determined. If the polymericion pairs are slightly dissociated, [~z AM~C+] can be set approximatelyequal to the total concentration of living ends [M-] and Eq. (8.65) can be written +] K = [~=AM~] [C (8.73) [M-] If ’the addedsalt is strongly dissociated and a relatively large amountof salt has been added, [C+] representing the concentration of free C+ ions in solution can be approximatedby the concentration of C+ producedby the dissociation of the addedsalt, i.e., [C+] ~" [C+]salt

(8.74)

688 Substituting given by

Chapter8 Eq. (8.74) in Eq. (8.73), the concentration of free ions

(8.75)

[C÷]salt and that of ion pairs by

K[M-] = [M-l [C+]salt

(8.76)

Combination of Eqs. (8.75) and (8.76) with Eq. (8.68) then [M] [M-]

--

kp = k~ n u (k~[C+]salt

k~) K

(8.77)

Equations (8.72) and (8.77) allow one to obtain k~, k~’, and K the experimental values of overall or apparent rate kp obtained in the absence and presence of added commonion. A plot of kp obtained in the absence of added commonion versus 1/[M-] 1/2 yields a straight line whose slope and intercept are (k~ - k~p)K1/2 and k~, respectively. A plot of kp obtained in the presence of added commonion versus [C+]salt yields a straight line whose slope and intercept are (k~ - k~p)K and respectively. The combination of the two slopes and two intercepts allows the individual calculation of k~-, k~, and K (see Problem below).

Problem8.11 Polymerization of styrene with sodiumnaphthalene, initiator was performedat 25°C in tetrahydrofuran (THF)using a static technique [8] that suitable for monitoring fast reactions. The conversion was determined by monitoring the residual styrene monomer spectrophotometrically during polymerization and the concentration of living ends [M-] was determined spectrophotometrically at the end of the experiment. In independentexperimentalseries, the overall rate constant kp was obtained [cf. Eq. (P8.10.2)] both at different concentrations initiator (and hence[M-]) without addition of electrolyte and at different concentrations of sodiumions from externally addedsodiumtetraphenyl borate (NaBPh4) salt and constant concentration of initiator. The data are given below:

/on/c Chain Polymerization

689

(a) Without electrolyte (b) With electrolyte ~ s[M-lxl0 [M-] xl0 kp [NaBPh4]x 105 L/mol-s mol/L mol/L mol/L 1.97 5420 7.1 132.0 6.22 3620 16.9 55.5 15.1 58.6 11.0 2450 16.5 2050 18.5 29.5 22.0 100 900 11.5 13.5 490 538 16.7 1000 396 31.0 7.0 Source: Data from Ref. 8.

kp L/mol-s 114 127 137 163 184 210 280

Determine the propagation rate constants k~ and k~- and the overall dissociation constant K for polystyryl sodium in THFat 25°C. (The dissociation constant of NaBPh4 in THF at 25°C is 8.52x10 -s mol/L.) Answer: (a) A plot of the observed k~ vs. 1/[M-]1/2 is shown in Fig. 8.4(a). The slope the line, according to Eq. (8.72), gives (k; - k~)K~/2 = 25.2 L1/2 tool -1/~ s-1 and its intercept leads to k~ ,-, 150 L tool -t s-1. (b) Let x = [Na+]/[NaBPh4], where [NanPh4] denotes c, the total concentration of the salt initially added. Denoting the dissociation constant of NaBPh4salt by Ks and noting that [Na+] ~-- [BPh~] if the polymeric ion pairs are slightly dissociated as compared to salt, ~2

Ks ~- (1 - x) Solving the quadratic, x.c

1 = [Na +] = ~[(K~

+4Ksc) V~-

Ks]

This equation is used to calculate [Na+] corresponding to a given [NaBPh4]0. A plot of the observed k, vs. 1/[Na ÷] is shown in Fig. 8.4(b). The intercept the line, according to Eq. (8.77), gives kff = 80 L tool -1 s -~ and its slope (k; - k~)K = 9.8×10 -s S -1. Since from (a) (k; - k~)KV2 = 25.2 LV~ mol-V2 s -1, by combining these data one obtains k~ = 6.5×104 L tool -x s -1 and K = 1.52×10 -7 -~. mol L lit may be noted that the value of k~ is given by the intercept of line obtained from each set of kinetic data (a) and (b), in the absence and presence of NaBPh4,respectively. However, the results obtained in the absence of the salt produces a steep line with a small intercept, and, therefore, the k~ value derived experimentally is not too accurate. The other method based on the effect of the salt on the observed /~ gives a muchmore reliable value of k~ which should be reported.J

690

Chapter 8

Problem8.12 Calculate the concentrations of polymeric free ions and ion pairs both in the absence of electrolyte and in the presence of electrolyte NaBPh4, considering two polymerization systems of Problem 8.11 that have nearly equal concentrations of living chain ends. What percentage of the free Na+ ions comes from the initiator whenthe electrolyte is present ? Answer: (a) Without e, lectrolyte. The concentrations of polymeric free ions, [~ AM;], and that of polymeric ion pairs [~ AM~-C+] can be calculated from Eqs. (8.67) and (8.70). For [M-] = ll.0xl0 -5 -1, tool L [~A.M~]

=

1/2 K1/2 [M-]

=

(1.52 x 10-7 tool L-l) 1/~ (11.0 x 10-5 V~ mol L-l) 0.41 x 10-5 -~ tool L

=

= [M-] - K1/2 V2 [M-] = (11.0 × 10-5 tool L-1) - 0.41 × 10-5 -1 tool L = 10.6 × 10 -5 -I tool L (b) With electrolyte. Consider the system with [M-] = 11.5×10-5 mol/L and [NaBPh4]o = c = 22.0x 10-5 mol/L; then 1 [(K~ + 4Ksc) I/2- Ks] [Na+]salt = 1.01 × 10-4 -~ mol L Frbm Eq. (8.75), (1.52 x 10-7 mol L-1) (11.5 x 10-5 -1) mol L -1) (1.01 x lO-~mol = 1.7 x 10-~ -1 tool L From Eq. (8.76), +] [~AM~ Na

= (11.5 x 10-5 tool L-1) - (1.7 x 10-7 -1) mol L ~-- 11.5 x 10-5 -1 tool L

(c) Free + io ns co ming fr om the in itiator = 1. 7×10-7 to ol L - 1 Fraction of free Na+ ions coming from the initiator (1.7× 10-7 -1) mol L = ---=1.68x 10-3 (~ 0.17%) (1.01x10-4 + 1.7x -I 10-7) moil

Ionic Chedn Polymerization

691

400 320

~ ~6o 80 I

0

40

I

I

80 120 I/[M;]’t2 (Ll/2

,I

160

I

200

I

240

4

I

O

I

12

I

16

2~0

Figure 8.4 The overall rate constant kr for polystyryl sodium in THFat 25°C as a function of [M-] and iNa+] plotted according to Eqs. (8.72) and (8.77). (Problem 8.11) ~: It should be noted that the observed ion pair propagation

constant

k

is, in fact, an apparentrate constant as it is a compositeof rate constants for the contact ion pair and the solvent-separated ion pair (see Problem below).

Problem8.13 Consider propagation by polystyryl

sodium ion pairs in a 1 M styrene solution in tetrahydrofuran at 20°C. For an ion pair concentration of 2.0x 10-3 M, calculate the relative contributions of contact and solvent-separated ion pairs to the propagation process. [Data: At 20°C, rate constant for contact ion pair, kc = 24 L/mol-s; rate constant for solvent-separated ion pair, ks = 5.5x104 L/tool-s; equilibrium constant for interconversion between contact ion pair and solvent separated ion pair, Kcs = 2.57 x 10-3.] Answer: The observed ion pair propagation constant k~ is a composite of the rate constants k¢ and ks according to k, ~ = ~k, + (1-~)~o 0’8.13.1)

692

Chapter 8

or

k~ = k~ + ksKcs (P8.13.2) I+K~ whereo~ and(1 - ~) arethefractions of solvent-separated andcontact ionpairs, respectively, andK~,is theequilibrium constant forinterconversion between the two types of ion pairs:

+ AM;c

+ AM;//C

(P8.13.3)

Therefore, +] [AM;//C

[AM;c*] + [AM;//C*]

1 +K~

since K= << 1,

[AM; / /C*]=

K~s ([AM; C+] + [AM;//C*]) = (2.57 x 10-3 ) (2.0 x -3 to ol L- ) = 5.14 x 10-6 -1 mol L

k~ [AM; C*] [M] + ks[AM~-//C +] [M] (24 L mo1-1 -I) ( 2.0 × 10-3 moi L - 1) (1 .0 to ol L - 1) + (5.5 x 104 L tool -1 s -1) (5.14 x 10-6 mol L-1) (1.0 tool -1) 0.048 ÷ 0.283 or 0.331 mol L-1 s-1 Fractions of propagation carried by solvent-separated thus 0.85 and 0.15, respectively.

Incomplete

Dissociation

of

and contact ion pairs are

Initiator

The simple kinetics of anionic polymerization described earlier exists because the initiator is converted completely from the inactive form, CA, to the active form, C+A- (or C++A-) before any propagation reactions take place. However, some initiators (e.g., lithium alkyls and aryls) maintain an equilibrium between the active form and the inactive form. Moreover, this equilibrium may extend to the growing anionic chains also. In such a situation we must write the initiation steps as

and the propagation

CA ~ C + A- ~ C + + A-

(8.78)

C+ A- + M ----~ steps as

(8.79)

AM- C+ + M --~

AMn-IM-C

+ -t-

AM- C+ ~ AMC

AMM- C- +.

M~

~ AM2C

(8.80)

AMnM- C + ~ AMn+IC

(8.81)

693

Ionic ChainPolymerization

Wehave included here propagation by ion pairs only. However,propagation b~ free anions must also be considered in the complete scheme[cf. Eq. (8.64)]. Furthermore, the equilibria mayinvolve solvation contributions the solvent, although this is not shownexplicitly in the mechanism.Finally, as a further complicating feature, the monomermay also affect the equilibria. In such a case, the initiation rate would depend on the nature of the monomer,even though all other factors, such as solvent, temperature, concentrations, initiator, and so on were kept the same. The aforesaid complexities makeit virtually impossible to write explicit general equations for the rate of polymerization, kinetic chain length, average degree of polymerization as has been done above for a completely dissociated ionic initiator. Withthe exception of those simple cases discussed above, each system in anionic polymerization represents a kinetically unique problem and must be solved separately. Simultaneous

Propagation

and

Termination

The polymerization rate for an anionic polymerization where termination occurs simultaneously with propagation follows in a manner similar to free-radical polymerizations. An example is the potassium amide initiated polymerization in liquid ammonia.This is one of the few anionic systems in whichall active centers behavekinetically as free ions. In this system, initiation involves dissociation followed by addition of amide ion to monomer: K KNH2 ~---

K+ + H2N:-

(8.82) H

H2N:-

q-

CH2 =CHR -~

H2N-CH2-C:-

(8.83) R

where K is the dissociation

constant given by

K_ [K+]

[KNH2I

Because the second step is slow relative to the first, is given by R4 = ki [NHf] [M] Propagation proceeds according to k~ H2N--M~ + M -~. H2N--MnM-

(8.84) the rate of initiation (8.85)

(8.86)

with a rate given by P~ = kp [M-’]

[M]

(8.87)

694

Chapter 8

where [M-] represents the total concentration of the propagating anionic centers. Extensive chain transfer to solvent occurs by the reaction H2N--M~ -+- NH3 kS-~ 3 H2N--Mn--H q- NH~-

(8.88)

with a rate given by l~tr,Nn3

= ktr,

NH3 [M-] [NH3]

(8.89)

Assuming a steady state whereby R/ = Rt,.,

(8.90) (8.91) (8.92) WhenKNH2is used as the initiator and no external K+ is added, the + concentrations of K and NH~- are equal. Therefore, from Eq. (8.84), K = [NH~-]2/[KNH2], and hence

p,~ = k¢ k~ K1/2 [MI2 1/: [KNH~I

(8.93)

~r,NH~ [NH3]

Problem8.14 Justify the steady-state assumption of Eq. (8.90) in view of the fact that the chain transfer reaction [Eq. (8.88)] is truly not a terminationreaction since the amideion is regenerated. Answer: The second step of initiation [Eq. (8.83)], being slower than the first [Eq. (8.82)}, is rate-determiningfor initiation (unlike in the case of free-radical chain polymerization) and so though the amide ion produced upon chain transfer to ammoniacan initiate polymerization it is but only at a rate controlled by the rate constant, k~, for initiation. Therefore, this chain transfer reaction maybe considered as a true kinetic-chain termination step and the application of steadystate conditiongives Eq. (8.90).

The average kinetic chain length (P) or degree of polymerization

(DP)

Ionic ChainPolymerization

695

is expressed as DP -= V = kp [M] [M-] _ [M] /¢’tr,NH3 [M-] [NH3] CNHa[NH3]

(8.94)

Examination of Eqs. (8.92) and (8.94) for the effects of temperature yields equations [cf. Eqs. (6.180) and (6.185)] dln(_~) dT dln(DP,~) dT

Ei

+ EpRT 2

(s.95)

- E~ -- v RT ~

(8.96)

Reducing the reaction temperature decreases Pqo because Ei + Ep Err ~- +38 kJ mol-I, but increases DP,~ because Ep - Err ~- -17 kJ mo1-1. Due to small magnitude of Ep - Err and high concentration of ammonia,chain transfer to ammoniais highly competitive with propagation and only low molecular weight polymer is formed even at low temperatures. Taking into account also the contribution to termination by moisture, the number-average degree of polymerization is given by 1

CNH~ [NH3]

CH, O [H20]

(8.97)

DPn -[M] + [M] Amide-initiated polymerizations in liquid ammoniaare about the only anionic polymerizations studied that proceed with termination. COPOLYMERIZATION For the copolymerization of two monomersby an anionic mechanism, the copolymer composition equation (7.11) or (7.18), derived in Chapter applicable with the monomerreactivity ratios defined in the same way as the ratios of rate constants: rl = kn/kl~, and r2 = k2~/k2l, where kll and k22 are the rate constants for the homopropagation reactions: wvwM~- + M1 -~

wvwM~-

(8.98)

wvwM~" + M2 -~ vvvvvM~"

(8.99)

and k12 and k21 are rate constants for cross-propagation reactions

696

Chapter8

The reactivity ratios r~ and r2 can be determined from the composition of the copolymer product. However, a serious complication exists because the propagation rate constants, kij, are composite rate constant, being composed of free-ion contributions and ion-pair contributions, and hence the reactivity ratios also will be composite quantities, having contributions from both ion pairs and free ions. Because the relative abundances of free ions and ion pairs are strongly dependent on the reaction conditions, the reactivity ratios will also dependon these conditions and they Can be applied only to systems identical to those for which they were determined. Therefore the utility of such ratios is muchmore limited in anionic than in free-radical polymerization. Relatively few reactivity ratios have been determined for anionic copolymerization processes because of the complicating effects of counterion and solvent. Typical reactivity ratios for the anionic copolymerization of styrene with several monomersare shown in Table 8.3. More dramatic effect of solvent is seen with styrene-butadiene copolymerization. A change from the nonpolar hexane to the highly solvating THFreverses the order of reactivity when the soluble n-butyllithium is used as initiator. Most of the values shown in Table 8.3 were determined from the copolymer composition equation [Eq. (7.11) or (7.18)] using measured copolymer compositions a function of monomerconcentrations, as discussed in Chapter 7. The use of "living copolymerization," in which one monomeris first polymerized and the second monomerthen added, permits cross-propagation reactions (8.100) and (8.101) to be studied independently and allows the respective rate constants k12 and k21 to be measured directly. Since k~l and k~9,, on the other hand, are known from the polymerization of pure monomers, in some anionic systems which undergo living polymerization the reactivity ratios can be determined from measured individual rate constants. The re~tctivity ratios for the polymerizations of styrene with ot-methylstyrene and p-methylstyrene in Table 8.3 were determined in this manner.

Reactivity Groups A polymer with an active carbanionic end will initiate the polymerization of a different monomerif the initial carbanion is more basic than the anion formed from the second monomer in the particular solvent/counterion environment. Thus o~-methylstyryl carbanion will add methyl methacrylate and initiate the polymerization of this monomerbut the methyl methacrylate carbanion will not initiate polymerization of c~-methylstyrene. The relative ease of anionic copolymerization can be correlated with the base strengths of the respective anions or approximately with the e-values of the Q - e scheme (see Chapter 7) as shown in Table .8.4 which presents a number

Ionic Chain Polymerization

Table 8.3 Representative

697 MonomerReactivity

Ratios in Anionic Copolymeriza-

tion Monomer 1

Monomer 2

Initiator

Solvent

Temp.

rl

r2

(°c) Styrene

Acrylonitrile

~ C~HsMgBr

Cyclohexane

Ether Methyl metha- CtHsMgBr ~ C_~HsMgBr Ether crylate C_~H~MgBr Ether

a Butadiene

-45

0.05

15.0

-30 -78 +20

0.05 0.02 0.30

14.0 20.0 2.0

p-Methoxyb styrene

n-BuLi Li

Toluene THF

0 0

10.9 2.9

0.05 0.23

c~-Methylc styrene

Na-K alloy

THF

+25

35

0.003

aAcrylonitrile

RLi

None

-

0.12

12.5

aVinyl acetate

Na

NH5

-

0.01

0.01

aButadiene

n-BuLi n-BuLl n-BuLi n-BuLi

Hexane Hexane THF Tt-~

25 50 25 -78

0.03 0.04 4.0 11.0

12.5 11.8 0.3 0.4

aIsoprene

n-BuLi

Cyclohexane

40 0.046

Isoprene

n-BuLl

Hexane

50

3.38

0.47

NaNI-I2 RLi

NH3 None

-

0.25 0.34

7.9 6.7

NaNH2

NI-I3

-

3.2

0.4

Methyl metha- Acrylonitrile acrylate Vinyl acetate

16.6

bE Dawansand G. Smets, Makromol.. Chem., 59, 163 (1963). V. Tobolskvand R. J. Boudreau, Z PolvmSci., 51, 553 (1961) ~C. L. Lee, J. Sm~d,and M. Szwarc, Z Am. Chem.Soc., 83, 2 961 (1961). "M. Morton,Anionic Polymerization: Principles and Practice, Acaderaie Press, New York (1983)

698

Chapter 8

Table 8.4 Initiation of Anionic Polymerization by Living Polymer Ends Reactivity group Monomertypes e values 1 a-Methylstyrene -1.2 Conjugated dienes -1.0 Styrene -0.8 4-Vinyl pyridine -0.2 2 Acrylate and methacuclate esters 0.4 3 1,2-Epoxies Formaldehyde 4 Methacrylonitrile 0.8 Acrylonitrile 1.2 5 Nitroalkenes 2.6 Vinylidene cyanide Source:L. J. Fetters, J. Pobyrn.Sci. PartC 26, 1 (1969). monomersclassified as to relative reactivity [9]. Polymerization of a monomer in a given reactivity group can generally be initiated by a carbanion from any monomerin a group with a lower number, but the reverse does not occur. Thus the monomersin group 5 generate the weakest (less basic) anions and cannot initiate monomersin any of the preceding group. Generally a monomerin a given group will react with the carbanion generated from any other monomerwithin the same group, but there are exceptions to this guideline. Thus, the initiation of styrene polymerization by a-methylstyrene carbanion or that of isoprene by styryl carbanion is faster than the reverse initiation. Monomersin a given group all have about the same e values and similar abilities to spread out a negative charge through resonance stabilization.

Problem 8.15 What types of polymers will be obtained from anionic copolymerization of the following monomerpairs: (a) styrene/butadiene, and (b) styrene/methyl methacrylate ? Answer: (a) Copolymerizationsanalogousto free radical reactions occur betweenmonomers whichare within the samereactivity group in Table 8.4, whereeach carbanion can initiate either monomer.Since both styrene and butadiene belong to the same reactivity group, they will undergocopolymerization, the copolymercomposition depending on the reactivity ratios. From Table 8.3, rl = 4.0, r2 = 0.3 for

699

Ionic ChainPolymerization

n-butyllithium initiated polymerization in THFat 25°C.Therefore ideal copolymerizationbehaviorwill be observedand a statistical copolymercontaininga larger proportion of styrene in randomplacementwill be obtained. The simple copolymer equation [Eq. (7.11)] can be applied to calculate the copolymercomposition. (b) The styrene-MMApair contains monomersdrawn from different relative reactivity groupsin Table8.4. Polystyryicarbanionwill initiate the polymerization of MMA, but the carbanion of the latter monomer is not sufficiently nucleophilic (basic) to cross-initiate the polymerizationof styrene. This is reflected in the rl, r2 values (see Table 8.3). For example, the styrene/MMA pair has r I = 0.02, r2 = 20.0 when initiated by C4HsMgBr in ether at -78°C. Thusthe polymerizationof the mixturein this case will cause homopolymerizationof MMA followed by that of styrene. However, excess MMA will add to living polystyrene producinga block copolymerof the two monomers.

Block

Copolymers

Anionic polymerization utilizing the living polymer technique is particularly well suited to preparing block copolymers. The simplest vinyl-type block copolymer is a two-segment molecule illustrated by (AAAAA.........

A--)-(--BBBBBB.........

Other commonblock copolymer structures (AAAA......

A-)-(-BBBB ......

AAAA......

A-)-(--BBBB......

ABblock copolymer

B)

are shown below:

B-~--AAAA......

A)

B--)--(--CCCC...... C)

ABAbloqk copmymer ABCblock copmymer

(AA.... A-)-(-B S .... B-)-(--AA.... A--~-BB .... B-)- etc. (AAAAAA......... Isotacticsequence

A--~-AAAAAAA ......... Syndiotactie sequence A)

The properties of block copolymers differ from those of a blend of the correponding homopolymers or a random copolymer (Chapter 7) with the same overall composition. An important practical example is the ABA-type styrene/butadiene/styrene triblock copolymer. These behave as thermoplastic elastomers. Ordinary elastomers are cross-linked by covalent bonds, e.g., vulcanization (see Chapter 2) to impart elastic recovery property, as without this there will be permanent deformation. Such cross-linked rubbers are thermosets and so cannot be softened and reshaped by molding. However, solid thermoplastic styrene/butadiene/styrene triblock elastomers can be resoftened and remolded. This can be explained as follows. At room temperature, the triblock elastomers consist of glassy, rigid, polystyrene domains

700

Chapter 8

(Tg of polystyrene = 100°C) linked together by rubbery polybutadiene segments (see Fig. 8.5). The polystyrene domains thus serve as effective cross-links and stabilize the structure against moderate stresses. They can be softened sufficiently at T > 100°C to enable molding and when cooled the rigid polystyrene domains reform. Sequential

Monomer

addition

Most interesting from the standpoint of commercial development is the formation of block copolymers by the living polymer method. Sequential addition of monomers to a living anionic polymerization system is at present the most useful method of synthesizing well-defined block copolymers. Depending on whether monofunctional or difunctional initiators are used, one or both chain ends remain active after monomerA has completely reacted. MonomerB is then added, and its polymerization is initiated by the living polymeric carbanion of polymer A. This method of sequential monomer addition can be used to produce block copolymers of several different types. Monofunctional

Initiators

With monofunctional initiators (e.g., n-butyllithium), AB, ABA,and multiblock copolymers can be formed. For example, the synthesis of an ABblock copolymer can be shown schematically as A ~ RwvwAAA- Li + ~ RvwwAAAvwwBBBRwwvAAAwwvBBBH + LiOH

Li + -~ (8.102)

The order of monomeraddition is important. For example, to prepare an ABtype block copolymer of styrene and methyl methacrylate, styrene must be polymerized first using a monofunctional initiator and when styrene is completely reacted, the other monomer MMAmust be added. The copolymer would not form if MMA were polymerized first, because living poly(methyl methacrylate) is not basic enough to add to styrene. The length of each block is determined by the amount of corresponding monomer which was provided. To produce ABAtype copolymer by monofunctional initiation, B can be added when A is consumed, and A added again when B is consumed. This procedure is possible if the anion of each monomer sequence can initiate polymerization of the other monomer. Mulfiblock copolymers can also be made in this way.

Ionic ChainPolymerization

701

-- Po[ybutad[ene chain segments

Polystyrene domains

Figure 8.5 Schematic representation of the polystyrene domain structure in styrene-butadiene-s~renetriblock copolymers.

BifunctionaJ Initiators ABAtriblock copolymers cannot be produced by monofunctional initiation when the A anion is not sufficiently basic to initiate polymerization of B monomers.In such cases the bifunctional initiators like alkali metal complexes of polycyclic aromatic compounds(e.g., naphthalene and biphenyl) can be used to produce ABAtriblock copolymers. In these cases polymerization would be started with monomer B to produce a polymeric dianion which would initiate polymerization of the A monomerthat is added later. The process is illustrated below for the commercially important styrene-butadiene-styrene (SBS) triblock copolymer: B ~ ¯ B:- Na+ --~ Na+-:BB: - Na+ ~ Na+-:BBB .... + -~s Na+-:SSSSBBB .... BBBSSSS:- Na

+ BBB:- Na

(8.103) Typically thermoplastic elastomers of this type have molecular weights of about 50,000 to 70,000 for the polybutadiene blocks and 10,000 to 15,000 for the polystyrene bolcks.

702 Coupling

Chapter8 Reactions

In this technique, a living ABblock copolymer is made by monofunctional initiation and is then terminated with a bifunctional coupling agent like a dihaloalkane. Other coupling agents such as phosgene or dichloromethylsilane are equally effective. For example, an ABAblock copolymer can be made as follows: + + Br-(-CH2-~Br 2(AAA--.--)~BBB.-.--)~-B:-Li --~ (AAA...-)e-(--BBB-..--)g+I-(-CH2--)r(-BBB.. "-)~+I-(-AAA"" If the presence of a linking group, like the --(--CH2--)-~-reaction is undesirable, a coupling agent like I2 can be used.

~f2 + 2LiBr (8.104) in the above

Problem 8.16 In a living polymerization experiment, 7.0 cm3 of 0.16 mol/L solution of s-butyllithium in toluene was addedto a solution of 16.8 g styrene in 400 cm3 toluene. After complete conversion of the styrene, 56 g isoprene was added. Whenthe isoprene had completely reacted, the polymerization was terminated by addition of 5.6 cma of a 0.10 mol/Lsolution of dichloromethanein toluene. Write downthe reactions occurring in each stage of this reaction sequence, and evaluate the molecular weight of the final polymer. Answer: The quantities of the individual reactants are : 8BuLl = (0.16 tool -1) (7.0 c 3) (1 ‘-3 L crn-3) = 1.1 2×10-a mol Styrene = (16.8 g)/(104 g mo1-1)= 0.16 Isoprene = (56 g)/(68 g tool -1) = 0.82 mol Dichloroethane = (0.10 tool -1) (5.6 c a) (1 -a L cm- 3) = 0.5 6x10-3 mol The reactions taking place are as follows: SBu-Li+ + riCH2 =CHub~ SBu-(CH2--CH~b~-)-er_I-CH2-CH~b +

(A) (A) + mCH2=CMe-CH=CH2 + SBu-(-CH2-CH(b-)n-(CH2-CMe=CH-CH2--)m_I-CH2CMe=CHCH2Li

(B)

Ionic Chain Polymerization (B)

+ CICH2CI

703

--2LiCI

SBu---(--CH2-CH¢--)-a--(

CH~-CMe=CH-CH2---)m~CH27

~t-- CH2- CH=CMe- CH~--)m--( - CH-CH2---)h---SBu This is an example of the preparation of ABA-typethermoplastic elastomer. Styrene is polymerized first since styryi initiation of isoprene is faster than the reverse reaction. The reaction is carried out in a nonpolar solvent with Li+ as the counteflon to enable predominantly c/s-l,4-polyisoprene to be formed in the second growth stage. The living polystyrene-block-polyisoprene AB di-block copolymer resulting from the second stage is then coupled by a double nucleophilic displacement of CI- ions from a stoichiometric equivalent of dichloromethane to give a polystyrene-block-polyisoprene-block-polystyrene triblock copolymer. Polystyrene end-blocks : Mass of styrene polymerized Moles of living polymer molecules = (16.8 g)/(1.12×10 -a tool) = 15,000

(Mn)ps-----

gmol-’

Polyisoprene center block : The molecular weight of the polyisoprene block in the final triblock copolymer is double that of the polyisoprene block in the di-block living copolymer prior to coupling. Hence, (M,~)p I =

= 2x \mo-~es-o~iv~ng~-~y~er~es] ( mass of isoprene polymerized 2 ×

(56.0 g) -- 100,000 g mo1-1 (1.12 x 10-a mol)

Molecular weight of the final polymer = 1.3 x 105

Star-shaped block copolymers can be made by using polyfunctional linking agents, like methyltrichlorosilane or silicon tetrachloride, to produce tribranch or tetrabranch polymers: 4"x.m,’x~-:- -b SIC14

"J’J’r" --~S

(8.105)

A major advantage of star-block copolymers is that they exhibit much lower melt viscosities, even at very high molecular weights, than their linear counterparts.

704 CATIONIC

Chapter 8 POLYMERIZATION

As we have seen earlier, the propagating species in cationic chain polymerization is a positively charged carbon species. The older term for this trivalent, trigonal, positively charged carbon ion is carboniumion which we have used up to this point. Olah [10] proposed that the term carbenium ion be used instead of carboniumion, the latter being reserved for pentavalent, charged carbon ions, and the term carbocation for both carbonium and carbenium ions. Since the term carbenium ion is not universally followed, to avoid the controversy we will henceforth refer to the propagating species as carbocations. Most text and journal references use the term carbocation and the term carbocation polymerization is used synonymouslywith cationic polymerization in the literature. Suitability of monomersfor cationic polymerization was discussed in the section on polymerizability of monomersand compared in Table 8.1. In short, olefinic monomerswith electron-releasing substituents on the ~carbon, that is monomersthat behave as a nucleophile (electron donor), can undergocationic polymerization and the possibility of resonance stabilization of the carbocationic species increases the reactivity of the monomer.The most active vinyl monomersare thus those with electron-releasing substituents or conjugated double bonds in which the positive charge can be delocalized (see Problems 8.17 and 8.18).

Problem8.17 Comparethe cationic polymerizability of (a) ethylene, propylene, and isobutylene (b) styrene, a-methyistyrene,p-methoxystyrene,and p-chlorostyrene. Answer: (a) The electron-releasing inductive influences of alkyl groups causes isebutylene to polymerize very quickly at low temperatures (yielding high polymer) while propylene reacts inefficiently, yielding only low-molecularweight polymer, and ethylene is practically inert. Also from a consideration of carbocation stability (tertiary most stable, followed by secondaryand primary) the polymerizability in the order ~ / C ~ CH2 > CH~" CH3

CHa % /C ~ CH2 > H

H ~C ~ CH2 H

(b) Conjugationwith the carbocation center helps to disperse the positive charge and tends to increase monomer reactivity. The "effect is particularly strong when

Ionic Chain Polymerization

705

conjugation and electron-donating groups cooperate. On the other hand, substitution of an electron-withdrawing halogen for an ortho or para hydrogen, decreases the monomerreactivity. Thus the reactivity sequence of the styrene derivatives is as follows: H\c//CH2

H\c#CH2

CH3\c//CH2

H\C

OCH 3

~CH2

Cl

Electron-releasing substituents, such as RO-, RS-, and aryl at ortho or para position increase the monomerreactivity.

Problem8.18 (a) Consider the cationic polymerizability of vinyl ethers, cyclic ethers (like tetrahydrofuran), cyclic acetals (like trioxane), and N-vinyl bazole. (b) Whydo these monomersnot copolymerize cationically with olefins like styrene or isobutene ? Answer: (a) In these monomersthe heteroatom is bonded directly to the electron-deficient atom and the respective carboxonium ion and immoniumion are more stable than the corresponding carbocations. These monomersare thus suitable for cationic polymerization : H ’ CH2=C-OR Vinyl ether

__CH2_!

~ H2C=C I

H

H ~

+ O-R

~ ~ --CH2-

+O-R

H --CH2-i+

H --CH2-~

II

(b) As noted above, carboxonium and immonium ions are more stable than the corresponding carbocations. Because of this difference in stability, the above monomerswill not copolymerize cationically with olefins like styrene and isobutene.

706

~hap~er 8

Cationic Initiation Various initiators can be used to bring about cationic polymerization of monomerswhich have electron-releasing substituents. Initiation is brought about by addition of an electrophile to a monomer molecule [cf. Eq. (8.2)]. Compounds used more commonly are the various protonic and Lewis acids. Protonic

Acids

Protonic (Br6nsted) acids initiate cationic polymerization by protonation the olefin : R Rt + H+ A- + CH2 = ~--~ CH3--C A(8.106) However, for polymerization to occur the anion A should not be a strong nucleophile; otherwise it will react with the carbocation to form the nonpropagating .covalent compound CH 3 -- C ~A Thus hydrogen halides are ineffective as initiators of cationic polymerization because of the highly nucleophilic character of halide ions. (Hydrogen iodide, however, shows some tendency to polymerize vinyl ethers, but the polymer yields and molecular weights are low.) Other strong acids with larger and less nucleophilic anions such as sulfuric, perchloric, phosphoric, fluoro- and chlorosulfonic, methanesulfonic and trifluoromethanesulfonic (triflic) acids, initiate cationic polymerization. However,the polymermolecular weights rarely exceed a few thousand. The low molecular weights are because of transfer reactions which are discussed later. Initiation by protonic acids is relatively inexpensive. The major applications of such processes are for reactions of simple olefins like propylene or butenes or olefinic derivatives of aromatics such as coumarone (XV) and indene (XVI). The low polymers of olefins are used as lubricants and fuels while coumarone-indene polymers are employed in coatings and as softeners for rubbers and bitumenes.

707

Ionic Chain Polymerization Lewis Acids

Various Lewis acids are used to initiate cationic polymerization, generally at low temperatures, with the formation of high-molecular-weight polymers in high yield. Lewis acids are halides and alkyl halides of Group III metals and of transition metals in which the d electron shells are incomplete. This is the most generally useful group of initiators and include metal halides such as A1C13,BF3, SnC14, SbC15, ZnC12, TIC14, PC15, and organometallic derivatives, e.g., RA1C12,R2AIC1, R3A1. (Many of the metal halides are familiar to chemists as Fried.el-Crafts catalysts.) A1CI3is the most important Lewis acid for industrial cationic polymerizations. Lewis acids are seldom effective alone; rather they require the presence of trace amounts of water or some other proton donor (protogen) such as hydrogen halide, alcohol, and carboxylic acid, or a carbocation donor (cationogen) such as t-butyl chloride or triphenylmethyl chloride, which, on reaction with the Lewis acid, forms the electrophilic species that initiates polymerization. Thus dry isobutylene is unaffected by dry boron trifluoride but polymerization occurs immediately when trace amounts of water are added. The initiation process for boron trifluoride and water is BFz q- H20 ,~- H+ (BF3OH)(8.107) H+ (BFzOH)- q- (CH3)2C = CHz ~ (CH3)3C + (BF3OH)-

(8.108) Initiation

by aluminumchloride and t-butyl chloride is described by

AICI3 q- (CHa)aCGI ,~(CH3)3C q-

(AIC]4)-

(CHa)3C + (AIC]4)-

(8.109)

--[-

~CH = CHz -~ (CHa)aCCH2CH4(AIC14)(8.110) Because it is the proton or carbocation that initiates the polymerization reaction, the compoundsthat give rise to them are correctly referred to as the initiators, and the Lewis acids as coinitiators [11] (not the other way around, as is commonlydone in the polymer literature). The combination of Lewis acid and proton or cation source is the initiating system. The initiation steps described above can thus be generalized as K L q- IB ~--- -B+(LI) B + (LI)-

q-

M -~

(8.111) +(LI)-

(8.112)

where L, IB, and Mrepresent, respectiVely, the Lewis acid coinitiator, initiator, and monomer.Initiation by the initiating system has the advantage -) that the anion (LI)- is far less over initiation by a Br6nsted acid (H+A nucleophilic than A- [cf. Eq. (8.106)].

708

Chapter8

With certain Lewis acids of higher acid strength such as AICI3 and TIC14autoinitiafion or self-ionization mayoccur. In such cases the initiator and coinitiator are the same, and the initiation is usually represented by 2A1Br3 ~ A1Br~ + (A1Br4)AIBrz + (A1Br4)- + M ~ A1BrzM+ (A1Br4)-

(8.113) (8.114)

However, for polymerizations carried out under most reaction conditions, the moisture content (and/or level of other carbocation or proton donors) is often sufficient for reaction (8.111) so that self-ionization constitutes only a minor propagation of the total initiation process.

Problem8.19 Arrangethe following initiators and coinitiators in the general order of activity for initiating cationic polymerizationof monomers: (a) Initiators: water, acetone, hydrogenchloride, methanol,acetic acid, phenol, and nitroethane. (b) Coinitiators: AICI3, AIR3, AIRCI2and AIR2C1; A1R2CI, AIR~Br, and AIR2I. Answer: The extent of formation of the initiator-coinit/ator complex[i.e., the value of K in Eq. (8.111)] and its rate of addition to monomer [i.e., the value of ki in Eq. (8.112)] generally increase with increasing acidity of the initiator and that of the coinitiator. Thusfor the initiators the general order is HCI > CHzCOOH > O2NC2Hs> C~H~OH> water > methanol > acetone For the coinitiators the order of acidity, and hencethe general order of activity, is AICI 3

> AIRCI2 > AIR2CI > AIR3 and AIR21 > AIR2Br > AIR2CI

It mustbe mentioned,however,that the order of activity of a series of initiators or coinitiators maydiffer dependingon the identity of the other component,monomer, solvent, or the presenceof competingreactions. For example,the activity of boron ha/ides in isobutylene polymerization is observed to be BFz > BCI3> BBr3with water as the initiator, while the order of acidity is BBr3> BCI3> BF3.This is due to hydrolysis of the boronhalides to inactive products increasing in the same order as their acidities.

The effectiveness of a cationogen in initiation depends on carbocation stability in a complex manner. Increased stability of a carbocation formed from the cationogen enhances the initiation, but the carbocation has lower reactivity. Differences in stability of the carbocation formed from the

Ionic ChainPolymerization

709

cationogen compared to the propagating carbocation are also important in determining the effectiveness of a cationogen.

Problem8.20 Give plausible explanation for the following facts. Primary and secondaryalkyl halides are generally ineffective as initiators of cationic polymerization x)f monomerssuch as isobutene and styrene, but t-butyl and cumyi chlorides are effective. Onthe other hand, triphenylmethyl chloride and cycloheptatrienyl (tropylium)chloride are not very efficient in polymerizingisobutylene and styrene but produces rapid polymerization ofp-methoxystyrene,vinyl ethers and N-vinylcarbazole. Answer: Primaryandsecondaryalkyl halides are generallyineffective as initiators of cationic polymerization, because primary and secondaryearbocations are formedtoo slowly and/or in extremelylow concentrations. However,tertiary earbocations such as tbutyl and cumyl(2-phenyl-isopropyl) are sufficiently stable to form but are not morestable than the carbocations derived from their addition to monomers such as isobutene, styrene, or N-vinylcarbazole,so that polymerizationsof these monomers occur. +, and Triphenylmethylhalides and tropyliumhalides ionize to formtrityl, ~3C cycloheptatrienyl (tropylium), CrH~-,carbocations [Eqs. (P8.20.1) and (P8.20.2)], which are too stable to efficiently polymerize less reactive monomerssuch as isobutylene and styrene, but polymerizationofp-methoxystyrene,vinyl ethers and N-vinyl carbazole, whichare morereactive, proceeds rapidly. (C6H5)3CCI ~- (C6H5)3 C+ -1- CI-

(P8.20.I)

+ Ct-

(P8.20.2)

Chlorine, bromine, and iodine act as cationogens in the presence of more active Lewis acids such as trialkyl aluminum., or dialkylaluminumhalide. The initiating species is the haloniumion X+ formed via the equilibrium reaction between Lewis acid and halogen. The polymerization by syncata~ytic systems consisting of an aluminum compound R2AIZ(R = alkyl and Z = alkyl or

710

Chapter 8

halogen) and halogen or an interhalogen compoundare initiated addition of a positive halogen ion to the monomer: ~ X+ R2AIYZ-

XY + R2AIZ

X+ R2A1YZ-

by the (8.115)

R I + CH2 = C ~ X-CH2--CI.

R + R2AIYZ - (8.116)

where X is a halogen atom and Y is either a halogen or a different group. Iodine is unique amongthe halogens in that it initiates polymerization of the more reactive monomerseven in the absence of a Lewis acid. Iodine is not the actual initiator when it is used in the absence of a Lewis acid. Iodine adds to the double bond to form a diiodide that eliminates hydrogen iodide [12]. The hydrogen iodide generated by this process acts as the cationogen with iodine acting as a Lewis acid to form the initiating system. (Hydrogeniodide itself is not an efficient initiator because iodide ion is too nucleophilic) I2

+ CH2~CH ~--I OR

ICH2--cHI I OR

~ ICH=CH I OR

+ HI

(8.117)

(8.118) I2 + HI ~ H+ I~This initiation route is moreefficiently utilized by directly adding a mixture of hydrogen iodide and either iodine or a metal halide such as ZnX2or SnX2to the reaction system. An alternative mechanism has been suggested [13] which postulates formation of ion pair directly from the 1,2-diiodo compoundand/or its reaction with I2: + IICH2-- CHI ~ ICH2--CH=OR

I

(8.119)

OR ICH2~CHI I OR

+ 12 ~

ICH2--CH--OR I~

(8.120)

followed by propagation, + ICH:2CH---OR I~" + CH2

=CHOR ~ + ICH2CH(OR)CH2CH--OR I~ (n =1 or

The halogen initiation

(8.121)

mechanismis not directly confirmed, however.

711

Ionic ChainPolymerization Propagation

of Cationic

Chain

The initiator ion pair (consisting of carbocation and its negative counterion) produced in the initiation step [Eq. (8.112)] proceeds to propagate successive addition of monomermolecules. Considering isobutylene polymerization [cf. Eq. (8.108)], for example, this can be represented + H~CH2C(CH3)2--]~-CH2

C(CH3)2 (BF3OH)- + (CH3)2C=CH2 ÷ (8.122) H-{-CH2C(CH3)2-~+I-CH2-C(CH3) 2 (BF3OH)-

or,ingeneral terms[cf.Eq.(8.116)], HMnM + (LI)- -1- M ~ HMn+I M+ (LI)-

(8.123)

The addition proceeds by insertion of monomerbetweenthe carbocation anditsnegative counterion. The propagation reaction canbe complicated in somecasesdue to the occurrence of intramo]ecular rearrangements causedby hydrideion(H:-) or methide ion(CH3:-) shifts. Polymerizations proceeding withsuchrearrangements are referred to as isomerization polymerizations. The extent of rearrangement during propagation will depend on the relative stabilities of the propagating and rearranged carbocations and the relative rates of propagation and rearrangement. Both these factors favor propagation without rearrangement for monomerssuch as styrene, isobutylene, and vinyl ethers, since these monomerspropagate via reasonably stable carbocations such as benzyl, tertiary, and oxycarbocations, respectively. Extensive rearrangement during propagation, however, occurs for a variety of 1-alkenes. Consider, for example, the polymerization of 3-methyl-l,2-butene. Isomerization occurs by a 1,2-hydride ion shift in the first-formed carbocation (XVII) prior to the addition of the next monomerunit. The rearranged ion (XVIII) is a tertiary carbocation and is more stable than the first-formed carbocation, which is a secondary carbocation. H I + vvvwCH2--C

+ vvvvvCH2-CH2--C(CH3)2

CH(CH3)2 (XVII)

(XVIII)

¯ Thepolymer thuscontains mostly therearranged repeating unit(XIX)but alsosomeof thefirst-formed repeat units(XX)as somenormalpropagation occurs athigher temperatures. Itis observed thattheproduct contains about 70 and100%of (XIX)at polymerization temperatures of -130and -I00°C, respectively.

Chapter8

712 w~wCH2 ~ CH2--

C(CH3)2w~v

vwwCH2--C,

(x x)

H CH(CH3)2

(xx)

Propylene, 1-butene, and higher 1-alkenes yield only oligomers (DP no higher than 10-20) with highly irregular structures due to various combinations of 1,2-hydride and 1,2-methide shifts, proton transfer and elimination, and chain transfer. For example, protonation and ethylation of ethylene are rapidly followed by energetically favorable isomerization: + CH~_~H~ =CH2 ~ CH3-CH2 + HT shift + CHaCH2-CH2-CH2 ---~ CHaCH2-CH-CH~ ~ etc.

H + + CH2

(8.124)

These transformations follow the favorable enthalpy differences between primary to secondary (-22 kcal/mol) and secondary to tertiary (’2_ --33 kcal/mol) carbocations.

Problem8.21 Write equations to show the different

structural units that may result from intramolecular hydride and methide shifts involving only the end unit. Which of the resulting repeating units would be the most abundant? Answer: Five different end units (XXI - XXV)may arise from 1,2-hydride and methide shifts. The first-formed carbocation (XXI) undergoes hydride shifts to form carbocations (XXII), (XXIII), and (XXIV); (XXIII) rearranges to (XXV) methideshift : H + CH2=CH -----,...--.CH2-I CH2CH (CH3) 2

~. I CH2CH (CH3) 2 (XXI)

H.rshift

~

~C

H I+ I CH2CH2CH (CH3) 2 (XXII)

I

H:" shift

CH3 ~

-, CH2CH2CH2]~+H~-shift CH 3

0OClV)

H I ?CH3 shift + CH2CH2-C CH 2 (CH3)

(XXlII)

CH3 H I I + = ~ CH2CH2CH-C CH3

(xxv)

713

Ionic ChainPolyrnerization

The tertiary carbocation (XXIV)is the most stable and hence the repeating units derived from it will be present in the greatest abundance(found 42-51%). The other carbocations are of comparablestability and the repeating units derived from themwill be found in comparablequantities.

Other monomers that undergo isomerization polymerization include 5-methyl-l-hexene, 4,4-dimethyl-l-pentene, 6-methyl-l-heptene, cr-pinene, and vinylcyclopropane. Chain Transfer

and Termination

Termination reactions are harder to define in cationic processes because they are easy to confuse with chain transfer. Termination of chain growth in cationic polymerization may take place in various ways. Many of the reactions that terminate the growth of a propagating chain do not, however, terminate the kinetic chain because a new propagating species is generated in the process. Chain

Transfer

to

Monomer

Chain transfer to monomeris perhaps the most important chain breaking reaction in cationic polymerization. Transfer to monomerinvolves transfer of a fl-proton from the carbocation to a monomermolecule. This results in the formation of terminal unsaturation in the polymer molecule. Since in isobutylene there are two different types of fl-protons, two different unsaturated groups are possible: + H-{--CH2C(CH3)2-]rr-CH2 C(CH3)2 (BF3OH)- + CH2 =C(CH3)2 + (CH3)3C (BF3OH)- + H-[-CH2C(CH3)2-]rrCH2C(CHz)=CH~ + H-{-CH2C(CH3 ) 2 Jl~ CH=C(CH3) 2

(8.125)

Other monomerswhere two different end groups are possible include indene and ot-methylstyrene. For monomerssuch as styrene and ethyl vinyl ether only one type of unsaturated end group is, however, possible. In general terms, chain transfer to monomercan be written + (LI)BMnM

-[-

ktr,M

M ~ Mn+l -t-

+ (L I)-

(8.126)

It should be noted that the kinetic chain is not terminated by this reaction since a new propagating species is generated. Manypolymer molecules are usually produced for each initiator-coiniti~tor species present. The relative rates of transfer and propagation are given by the ratio ktr,M/kp, which is the monomerchain transfer constant C’M. The value of

714

Chapter 8

this constant determines the molecular weight of the polymer if other chain breaking processes are not significant. The larger the value of CMthe lower will be the molecular weight of the polymer [cf. Eq. (6.148)]. Chain transfer to monomeris the principal reaction that limits the polymer molecular weight for most monomers,especially at reaction temperatures higher than about 20°C. Chain transfer, however, is usually suppressed by working at lower temperatures since chain transfer to monomer generally has a higher activation energy. Another type of chain transfer to monomerinvolves hydride ion transfer from monomerto the propagating center [14]: + H-[-CH2C(CHa)2-]n-CH2C(CHa)2(BF3OH)- + CH2=C(CH3)2 CH2 = C(CH3)-CH2(BF3OH)4- H-[-CH2C(CH3)2-]rrCH2CH(CH3)2 (8.127) For isobutylene this reaction is a less likely mode of chain termination compared to proton transfer to monomer[cf. Eq. (8.125)1 since the tertiary carbocation formed by proton transfer is more stable than the allyl carbocation formed by hydride transfer. However, the hydride transfer reaction may account in part for the oligomers obtained in the polymerization of propylene, 1-butene, and other 1-alkenes where the propagation is not highly favorable due to the low stability of the propagating carbocation. The proton transfer and the hydride transfer are kinetically indistinguishable, but one [Eq. (8.125)] results in unsaturated end groups, while the other [Eq. (8.127)] results in saturated end groups. Spontaneous

Termination

Spontaneous termination is a rearrangement of the propagating ion pair and involves regeneration of initiator-coinitiator complexby expulsion from the propagating ion pair; the polymer molecule is thus left with terminal unsaturation. For example, for the system isobutylene/BF~JH20, + H-~CH2C(CH3)2--]w-CH2 C(CH3)2 (BF3OH)H+ (BF3OH)- + H--~-CH2C(CH3)2--}n--n CH2C(CH3)--CH2 (8.128) or, in more general terms, + (LI)BMnM

~t, ~ Mn+l

+

+ B

(LI)-

(8.129)

This type of termination, also referred to as chain transfer to counterion, differs kinetically from chain transfer to monomerin that the latter has a first-order dependence on monomer,whereas chain transfer to counterion is independent of monomerconcentration. Both types of chain transfer have

715

Ionic ChainPolymerization

similarity in not terminating the kinetic chain. The kinetic chain is interrupted, but the initiator-coinitiator complexis regenerated and can initiate new kinetic chains. However, chain transfer to counterion is almost never a dominant termination reaction compared to chain transfer to monomer. Combination

with

Counterion

This process, unlike the above mentioned processes, terminates the kinetic chain. In general terms, the reaction can be represented by BMnM+ (LI)-

~ BMnMLI

(8.130)

This type of termination occurs, for example, in the trifluoroacetic initiated polymerization of styrene [15]: H vvvvvCH2-- C

H

O -;O-CCF3

~

vvvvvCH2--

acid

O

C-O--CCF3 (8.131)

Alternatively, the propagating ion may combine with an anionic fragment from the counterion [16] as in BX3.OH~.(X m halogen) initiated polymerization or terminate by alkylation or hydridation [17,18] when aluminium alkyl-alkyl halide initiating sytems are used.

Problem8.22 Write equations to describe plausible termination reactions in cationic polymerization of isobutylene initiated by (a) BF3.OH2,(b) BCI3.OH2, (c)AlR3/t-butyl chloride, and (d) Al(C2Hs)z/t-butylchloride. Answer: (a) For BF~.H20 initiated polymerization,chain transfer to monomer [Eq. (8.125)] is the major modeof chain breaking with a minor contribution by combination with OH, ÷ H’-[--CH2C(CH3)2--~

CH2

C(CH3)2

(BF3OH)-

H--[--CH2C(CH3)2--]-a--, CH2C(CH3)2OH -b

(P8.22.1)

(b) Termination in the BCI3.H20initiated polymerization of isobutylene (and styrene) occurs almost exclusively by combinationwith chloride, +

H--[---CH2C(CHz)2--]a--~ CH2C(CHz)2 (BCI3OH)H--[--CHzC(CH3)2--]-~--~ CH2~(CH3)2CI+ BCI2OH(P8.22.2)

716

Chapter 8

The differences betweenthe reactions (P8.22.1) and (P8.22.2) are explained the order of bond strengths: B-F > B-O > B-CI. Thus, chloride ion easily transfers from the counterion to terminate the propagatingcenter, while fluoride ion is inactive towardtransfer. (c) Besides chain transfer to monomer,combinationwith counterion is also important whenaluminumalkyl-alkyl halide initiation systems are used. Termination occurs by alkylation, that is, transfer of an alkyl anion to the propagatingcenter: +

vvwvCH2--C(CH3)2 (R3AICI)’~’CH~-C(CHa)2R

+ R2AIC1

(P822.3)

(d) Terminationby hydridation, that is, transfer of a hydride ion from the alkyl anion to the propagatingcenter, occurs in preference to alkylation whenthe alkyl aluminumcontains/3-hydrogens: +

vv~vCH~-C(CHa)~ [(CHaCH2)aAICI]vvvwCHzCH(CHa)2 + CH~=CH~ + (CHaCH2)~AICI

Chain

T~ansfer

to

(P8.22.4)

Polymer

Several chain transfer to polymer reactions are possible in cationic polymerization. Transfer to cationic propagating center can occur either by electrophilic aromatic substitution (as in the polymerization of styrene as well as other aromatic monomers)or hydride transfer. Short chain branching found in the polymerizations of 1-alkenes such as propylene may be attributed to intermolecular hydride transfer to polyr~er. The propagating carbocations are reactive secondary carbocations that can abstract tertiary hydrogens from the polymer:

+ vvvvvCHz--CR

n t-

H I vvwvCHz--Cvww ~

H

R H

vwwCH2-- CR + I H

vvvwCH2-- Cvv~v I R

(8.132)

This reaction and the corresponding intramolecular transfers [Eq. (8.124)] are responsible for the production of only low-molecular products from 1alkenes. For example, reaetion (8.133) produces a relatively stable tertiary carbocation from a more reactive propagating secondary carbocation.

717

Ionic Chain Polymerization H

H vwvvCH2C[[ + (BF3OH)CH~

vvwvCH2---C-CH 2 vvwv ~ CH3 CH~v

vvvvvCH2--CH2-- CH3 + wwvCH~ + (BFaOH)-

(8.133)

CH3

Other

Transfer

and Termination

Reactions

Various transfer agents (denoted here by TAor S) present as a solvent, impurity, or deliberately added to the system can bring about termination of the growing polymer chain by transfer of a negative fragment A-. In general terms, the reaction can be represented by BMnM+ (LI)-

ktr,S

+ TA ~ BM,~MA

+ T+ (LI)-

(8.134)

where the term ktr,s as in Chapter 6, is used to denote the rate constant for chain transfer to solvent or any other transfer agent. Water, alcohols, acids, anhydrides and esters have varying chain transfer properties. When present in sufficient concentrations, these play the dominantrole in termination. Termination by these compoundsinvolves the transfer of HO, RO, or RCOO anion to the propagating carbocation, e.g., + vvvvvCH2--C(CH3)2

(BF3OH)-

w~CH2--C(CH3)2OH

+ H20

+ + ( BFzOH)-

(8.135)

Nucleophilic reagents such as water, alcohol (often with KOH), ammonia, and amines, are often used in excess to quench a cationic polymerization. This is typically carried out after complete or maximum conversion has been reached.

Problem8.23 Explain howthe following substances act as inhibitors or retarders in cationic polymerization: water, tertiary amines, trialkyl phosphines, and pbenzoquinone. Answer: Water,whenpresent in very small concentrations,acts as an initiator and initiates polymerizationin combinationwith a coinitiator (e.g., SnCI4). However,in larger concentrations, it inactivates the coinitiator (such as by hydrolysis of SnCI4) competessuccessfully with monomer for the initiator-coinitiator complexto form

718

Ohapter 8

the oxoniumsalt, whichis too unreactive in protonatingoleflns becausethe basicity of the carbon-carbondouble bondis far less than that of water: SnCI4 + H20 ~- SnCI4.OH2 H,_~ (H30)+ (SnCI4OH)Terminationcan also occur by proton transfer to water, e.g., 4vvwvCH2--C(CHz)2 + H20 ~ vv~vCH=C(CHz)2 + +

(P8.23.1) (P8.23.2)

or OH-transfer to carbocation, e.g., +

~,~¢H~-C(CH3h (BF3OH)- + H~O wvvvCH~C(CHa)~OHq- + ( BFaOH)(P8.23.3) + + H (BFaOH)- + HzO ~ (HaO) (BFaOH)(P8~.3.4) In the presenceof considerable amountsof water, reaction (P8.23.4) takes place preference to addition of monomer.This explains the decrease in polymerization rate causedby water in higher concentrations.Alcoholsand acids similarly function as inhibitors or retarders. Tertiary aminesand trialkyl phosphinesreact with propagatingchains to form stable cations that are unreactive to propagation: + + (LI)BM~M + :NR3 ~ BMnMNRa(LI)(P8.23.5) This causes inhibition or retardation. p-Benzoquinone acts as an inhibitor by receiving proton from the carbocation and/or initiator-coinitiator complex:

+

©

2 vvvwCH2 CH¢ CCl3CO~" + O~ + ~ 2 vvvvvCH=CHq~ + HO~

~O

~

÷ =OH + 2CCIzCO[

(P8.23.6)

Kinetics Unlike free-radical .and homogeneousanionic syntheses of high polymers, cationic polymerizations cannot be fitted into a generally useful kinetic frameworkinvolving fundamental reactions like initiation, propagation, and so on. This is because of the complexity of the cationic initiation, the uncertainty concerning the extent of ion-pair formation [cf. Eq. (8.1)] at the propagating chain end, and the possibility of the reaction being heterogeneous as the initiator is often only partially soluble in the reaction medium.

719

Ionic ChainPolymerization Ions

and Ion

Pairs

Electroneutrality demands that carbocations always be accompanied by counterions. The distance between the carbocation and counterion is important because it largely determines species reactivity. Dependingon the distance between the charged particles, which in turn is determined by the intrinsic properties of the ions and experimental conditions, a continuous spectrum of ionicities exists as depicted earlier by Eq. (8.1). Free earbocations can exist only under the purest conditions in the absence of even traces of moisture. In the absence of moisture, high polymerization rates can be obtained and the energy of activation is close to zero. Free ions exist in ’)’-ray radiation-induced bulk polymerizations of certain olefins, for example, o~-methylstyrene, fl-pinene, cyelopentadiene. However, in the majority of cationic polymerization systems reported in the literature, the propagating species are probably associated ion pairs. Thoughthe accurate definition of system ionicity in terms of actual species in solution and their concentration is almost impossible, an order of magnitude of free-ions and ion-pair concentrations can be obtained provided simplifying assumptions are made (see Problem 8.28). Figure 8.6 illustrates, for example, the complicated equilibria [19] in initiation by alkyl halides which are widely used as initiators in combination with eoinitiator such as aluminum alkyl halides or aluminumhalide Lewis acids. Each carbocation can initiate polymerization or remove an alkyl (ethyl) group from the counterion to produce a saturated hydrocarbon, REt, and a more acidic Lewis acid. The propagating cation can also terminate by the same process to produce ethyl-capped polymers and new Lewis acids. Thus, even though the coinitiator used is diethylaluminum chloride there may be major contributions to the polymerization from ethylaluminum dichloride or aluminumchloride. Both the initiation and propagation processes are, moreover, influenced by equilibria between various degrees of association of the active center and its eounterion. As a minimum,it is necessary to consider the existence of solvent-separated ion pairs, and free solvated ions. A simplified scheme [20] is shownin Fig. 8.7. The existence of contact (associated) ion pairs [as in Eq. (8.1)] is neglected in this scheme because the dielectric constants of the solvents usually used for cationic polymerizations are high enough (9-15) to render concentrations of intimate ion pairs negligible compared to those of solvated ion pairs. The observed kv in these simplified reactions will thus be composedof contributions from the ion pairs and free solvated ions :

=

+

-

±

where o~ is the degree of dissociation of ion pairs into free solvated ions, the respective rate constants being k~ and k~-. In media of low polarity,

720

Chapter 8

RC[ +AIEt2Cl

~ R+(AtEt2C|2) l

RCI+ AlEtCt 2 )~R+(AtEtCI3

I

RCt + AlCt 3 )~ R+CALC[4

- ~ RM+(AtEt2(:;12 REt 2 + AIEtCl I ~M RM+(AIEtC[3

)-

~ Polymer

-’-

)-

~ Polymer

:3 REt + AICI ~ RM+(A[CI.4)

-

M ~ Polymer

R= Atky[ or Aryt groups Et = CH3CH 2 -M = Monomer(styrene) Figure 8.6 Equilibria in initiation by aluminumhalides and alkyl halides. (After Ref. 19.)

Solvent B+//(LI)"+M separated ion pairs Sqlvated free ions

B++ (LI)-+M

~ BM+//{LI)

~ BM++ (LI)-

- ~ ~ M+// M

~ ’-~-M++ M

(LI)’-

(LI)-

~ Po[ymer

~ Polymer

Figure 8.7 A simplified reaction schemefor initiation and propagationin cationic polymerization. (After Ref. 20.)

like bulk monomer,the k~+ values for cationic olefin polymerizations are of the order of 106-109 L/mol-s, as comparedto k~" values of the order of 103-105 L/mol-s for anionic polymerizations under more or less equivalent conditions. (Carbocations are thus significantly more reactive than carban-

721

Ionic ChainPolymerization

ions under similar conditions.) In general, both lop+ and ]e~ will decrease with increasing solvent polarity since more polar solvents tend to stabilize the initial state (monomerplus ion or ion pair) at the expense of the transition complex (in which the monomeris associated with the cation and the charge is dispersed over a larger volume). The /ev+ values are generally at least 100 times as great as the corresponding]ep:t: figures for olefin monomers. Simplitled

Kinetic

Scheme

It is clear from the above discussions that deriving a general kinetic scheme for cationic polymerizations is rather unrealistic. Nevertheless, we shall postulate a conventional polymerization reaction scheme based upon the chemistry given in the earlier sections and show where its inherent assumptions are questionable in cationic systems. An ideal reaction scheme is shown below: Pre-initiation equilibrium: L q- IB ~ B+ (LI)(8.137) Initiation:

B+ (LI)-

Propagation:

+ (LI)BMnM

+ M --~

Termination:

+ (LI)BMnM

-~ BMn+ILI

(8.140)

+ (LI)BMnM

k~r,M + M ~ Mn+l + + (L I)-

(8.141)

Transfer:

+ M --~ + (L I)BMn+IM+ (LI)-

(8.138) (8.139)

where L, IB, and M are as defined earlier on p. 707. The scheme is greatly oversimplified to begin with, because we have ignored the existence of equilibria between free solvated ions and ion pairs of various degrees of intimacy. So all the rate constants that are listed are actually composite values [cf. Eq. (8.136)] which will val~j with the nature of the medium and counterion. The observed ]¢p values, moreover, can vary with the total concentration of reactive species because of the tendency of organic ion pairs to cluster in the more nonpolar environment, and the reactivity of aggregated and individual ion pairs is not generally equal. If we ignore all these important complications we can write the following expressions for the rates of initiation (R4), propagation (R~), termination (/~t), transfer to monomer(/~tr,M): R4 = ki (K [L] [IB])[M] +] P~ = ]~p [M] [M

(8.142) (8.143)

722

Chapter8 +] -]~t = ]¢.t [M

(8.144)

Rtr,M = /¢tr, M[M+] [M] + where [M+] represents ~2n=I[BMn_IM oc (LI)-], that is, it is concentration of propagating chains of all sizes irrespective of of association with counterions. It is necessary to invoke the steady state assumption d [M+]/dt= O) to make the model mathematically tractable. assumption, +1 ----[M

and

4=

Kki [L] [IB] [M] ~ Kkikp 2ILl [IB] [M]

/a

(8.145) the total their state (P~ = Rt, With this (8.146) (8.147)

Thus cationic polymerizations show a first order dependence of/~o on P~/ or initiator concentration in contrast to radical polymerizations which show a one-half-order dependence of _/?.p on R/. The difference is a consequence of their basically different modes of termination. Termination is second order in the propagating species in radical polymerization but only first order in cationic polymerization.

Problem8.24 Derive expressions for cationic polymerization rate for the following three cases: (a) an added chain transfer agent terminates the kinetic chain [Eq. (8.134)]; (b) the rate-determiningstep in the initiation processis the forward reaction in Eq. (8.111); and (c) the coinitiator L or initiator IB is present large excess. Howwouldthe order dependenceof the polymerization rate change if either monomer or any of the componentsof the initiating system is involved in solvating the propagatingspecies. Answer: (a) Reaction (8.134) becomesthe termination reaction when+ (LI)- i s unable t initiate newchains (as in the case of a moreactive agent present in considerable amounts).In that case, at steady state,

/~ = ~+P~r,s whichyields [M+] = Kk, ILl [IBI [M] kt -b ktr,S IS] FromEq. (8.143), Kk, k, 2[L] [IB] [M]

ta +t,,r,s [sl

723

Ionic ChainPolymerization

Applyingsteady-state approximation [M+I-- kl ILl [IBI / k, Hence P~ = kl kp ILl [IB] [M] / k, Thus ~ has one order dependenceon [M] instead of 2 order as in Eq. (8.147). (c) If L is present in large excess P~ = k [IB] [M] +] where k is a composite constant. Since [M

= [m] [M]2 / Thus/~ is zero order in L. Similarly, if IB is in excess, P~ will be zero order in IB. If monomer is involved in solvating the propagatingspecies, the/~ will show higher than 2 order in monomer. If L or IB is involvedin solvating the propagating species, the propagationrate is then dependenton the concentration of L or I]3, while initiation is first order in both L and IB [cf. Eq. (8.142)]~thus resulting higher than first order dependencefor P~ on L or IB.

Even whenconditions are scrupulously controlled, the kinetics of cationic polymerization are rarely simple. Water is highly reactive towards organic cations and if present as initiator, any excess will terminate polymer chains. Excess water may also destroy the coinitiator in some cases, or compete successfully with monomerfor the initiator-coinifiator complex (see later). The kinetic influence of water is thus complicated. In some systems, the initial rate of polymerization increases with concentration of water at low concentrations and becomes independent as this concentration increases. Such behavior has been reported for the polymerization of isobutene in dichloromethane initiated by titanium tetrachloride and water [21]. In other systems, the initial rate of polymerization mayrise to a maximumand then decline with increasing concentrations of water. Such behavior has been observed in the SnC14/H20initiated polymerization of styrene in carbon tetrachloride [22]. Degree

of

Polymerization

If the molecular growth is controlled by chain transfer to monomer,which is more likely than the case in which termination reactions limit the size of the macromolecule, then

D °n = / Rtr,M= /

r,M

(S.14S)

724

Chapter 8

Whenchain breaking involves spontaneous termination [Eq. (8.129)] and chain transfer to chain transfer agent or solvent S [Eq. (8.134)], in addition to chain transfer to monomer [Eq. (8.126)] and combination with the counterion [Eq. (8.130)], the concentration of the propagating species remain unchanged (assuming relatively small amounts of TA such that the coinitiator is not inactivated), and the polymerization rate is still given by Eq. (8.147). However, the degree of polymerization is decreased by these other chain breaking reactions and is given by

DP,~=

Rp

(s.149)

Rt q- Rts q- Rtr,M q- Rtr, s

Substituting expressions for the rates given earlier, and rearranging

1 D P,, -

a, qkv[M] kp

Is] q- CM+ Cs[M]

(8.15o)

where CMand (7 s are the chain transfer constants for monomerand chain transfer agent defined by ktr, M/kv and k, tr,S/k v, respectively. Equation (8.150) is the cationic polymerization equivalent of the previously derived Mayoequation [Eq. (6.148)] for radical polymerization. Problem8.25 In studies of the low temperature polymerization of isobutylene using TiCI4as coinitiator [21] the following results were obtained at -35°Cfor the effect of monomer concentration on the avearage degree of polymerization : 0.667 0.333 0.278 0.145 I 0.059 [C4H8] (mol/L) 6940 4130 2860 DP,, 2350[ 1030 Fromthese data, evaluate the rate constant ratios ktr/kp and kt/kp Answer: Since spontaneoustermimition (chain transfer to counterion) is never a dominant termination reaction comparedto chain transfer to monomer,the second term in Eq. (8.150) can be neglected. Further, ignoring the presenceof any chain transfer agent (S), Eq. (8.150) can be approximated D P. k~ [M l

+CM

Figure 8.8 shows a plot of 1/DPnvs. 1/[C4Hs]. The slope gives kt/k~ = 5.0x10-5 tool L-1. -4. From the intercept, CM= k.t~,M/kp = 1.1xl0 Problem8.26 In studies similar to those in Problem 8.25 but performed over a range of temperatures, the following values were found [21] for the intercepts

Ionic Chain Polymerization

725

of plots of 1/(DP.) vs. 1/[C4H8]: t(°C) 103/Dp~

+18 4.37

-14 --35 0.50 0.098

--48 0.027

Evaluate from these data the difference in activation energy betweenchain propagation and chain transfer to monomer. Answer: Expressingthe rate constants by the Arrheniusexpression Intercept

-- kt, _ (At,’~ e(~_F~)/t~r

where/~ and Nt are the actNation energies for propagation and transNr, r~p~tively; A~and At~ are the respective pre-exponential Nctors. TaNnglogarithms ln(Inter~pt) = ln(At~/~) N - N~ RT Figure 8.9 showsa plot of In(Intercept) vs. lIT. Fromthe slope of the finear plot, ~ - Nt~ = (5.1x10~ °K)(1.987 cal mol-~ °K-~) = 10.1x10a -x. cal tool

10 8

1

0

4

|

1

8

12

I

16

20

-1) 1/[C4H8](L rno~ Figure 8.8 Plot of 1/D_P,,vs. ]/[M]. (Problem8.25.)

Chapter 8

-2

3.6 (1/T) 103 (oh-~) Figure 8.9 Plot of In (intercept) vs. 1/T. (Problem 8.26.)

Validity

of Steady-State

Assumption

The steady-state assumption that is helpful in simplifying the analysis of free-radical kinetics is not valid in many, if not most, cationic polymerizations, which proceed so rapidly that steady-state is not achieved. Some of these reactions (e.g., isobutylene polymerization by A1C13at -100°C) are essentially complete in a matter of seconds or minutes. Even in slower polymerizations, the steady-state may not be achieved if P~/ > Rt. The expressions given above can only be employed if there is assurance that steady-state conditions exist, at least during someportion of the overall reaction. Steady state is implied if/~ is constant with conversion, except for change~ due to decreased monomerand initiator concentrations. A more rapid decline in R:v with time than what is expected or an increase in Pup with time would signify a nonsteady state. Thus manyof the experimental expressions reported in the literature to describe the kinetics of specific cationic polymerizations are not valid since they are based on data where steady-state conditions do not apply. Another consideration in the application of the various kinetic expressions is the uncertainty in some reaction systems about the solubility of the

Ionic ChainPolymerization

727

initiator-coinitiator complex. Thus failure of the usual kinetic expressions to describe a cationic polymerization mayoften be an indication that the reaction system is actually heterogeneous.

Absolute

Rate

Constants

The determination of the various rate constants (ki, kv, kt, kes, ktr) for cationic chain polymerization is muchmore difficult than in radical chain polymerization (or in anionic chain polymerization). It is convenient use P~p data from experiments under steady-state conditions, since the concentration of propagating species is not required. The Rv data from non-steady-state conditions can be used, but only whenthe concentration of the propagating species is known.For example, the value of ]¢p is obtained directly from Eq. (8.143) from a determination of the polymerization rate when [M+] is known. The literature contains too many instances where [M+] is taken equal to the concentration of the initiator, [IB], in order to determine kv from measured Rv. (For two-component initiator-coinitiator systems, [M+] is taken to be the initiator concentration [I]3] when the coinitiator is in excess or the coinitiator concentration [L] whenthe initiator is in excess.) Such an assumption holds only if R~> P~ and the initiator is active, i.e., efficiency is 100%.Using this assumption without experimental verification maythus lead to erroneous results. Because the expressions for D.Pn do not depend on either steadystate reaction conditions or a knowledgeof [M+], it is more convenient to calculate the ratios of various rate constants from D.Pndata than from data. However,the use of DP,~data, like the use of R,p data, does require (if the Mayoequation is used) that one employs data at low conversions where reactant concentrations have not changed appreciably. Most values of kv and other rate constants reported in the literature are questionable for several reasons. First, there is ambiguity, as discussed above, about the concentration of the propagating species. Second, the calculations of various rate constants and kinetic parameters are often carried out without adequate substantiation of reaction kinetics and mechanisms. The kinetics, for example, may deviate from Eq. (8.143) and then there is the big question of howto interpret any obtained rate constants in view of the knownmultiplicity of propagating carbocation species. Thus, the kinetic expressions given above are written in terms of only one type of propagating species-usually shownas ion pair. This is incorrect, since both ion pairs and free ions are simultaneously present in most systems of cationic polymerization, usually in equilibrium with each other (see Fig. 8.7). Thus the correct expression for the ~ate of any step in polymerization (viz., initiation, propagation, termination, and transfer) should include sep-

728

Chapter 8

arate terms for the respective species.

contributions

of the two types of propagating

Problem8.27 The term k~ in Eq. (8.143) is only an apparent or overall propagation rate constant, (k~). Showhow it could be related to the propagation rate constants k~+ and k~: of the free ions and ion pairs. Suggest methods of obtaining individual k~+ and k~ values. AEISWeF :

Taking into account the contributions rate may be written as

of free ions and ion pairs the propagation

Rr = k~+ [M+]free [M] q- k~ [M+ (LI)-] [M] (P8.27.1) where [M+]free and [M+ (LI)-] are the concentrations of free ions and ion pairs, respectively. From Eq. (8.143), P~ = k~ [M+] [M] (P8.27.2) where [M+] is the total concentration of cationic ends (comprising free ions and ion pairs). From Eqs. (P8.27.1) and (P8.27.2), k~ = k~+ [M+]free + k; +[M+ (LI)-] (P8.27.3) [M+]free + [M (LI)-] Individual k~+ and k~ values can be obtained by experimental determination of individual concentrations of free ions and ion pairs by a combination of conductivity and short-stop experiments. While conductivity directly yields the concentration of free ions (that is, only flee ions conduct), short-stop experiments yield the total concentration of ion-pairs and free ions. For mostly aromatic monomersthe total concentration of ion pairs and flee ions may also be obtained by UV-visible spectroscopy, assuming that ion pairs show the same UV-visible absorption as free ions since the ion pairs in cationic systems are loose ion pairs (due to large size of the negative counterions).

One of the most often encountered errors in reported k~. values is their assignment of apparent or overall rate constant k~ as k~:. This can be erroneous since even small concentrations of free ions with their high value of kp+ can have a significant effect on the propagation rate constant. The overall rate constant k~ is related to k~+ and k~ (see Problem 8.28)

k°p

= ak+~ + (1-

a)k~

(8.151)

Ionic Chain Polymerization

729

or k°p = k~ + k~ Kid/2

/[M+] 1/2

(8.152)

where o~ is the degree of dissociation and gd the dissociation constant of the propagating ion pair, and [M+] is the total concentration of propagating cationic ends (both free ions and ion pairs). One can thus safely equate kv with k~ only for systems where [M+]/Ka --" 103 - 104 or larger (see Problem 8.28). Problem8.28 Consider styrene polymerization by triflic (trifluoroethanesulfonic) acid in 1,2-dichloroethane at 20°C where Ka is 4.2x10-z mol/L[23]. For experimentsperformed(using stopped-flowrapid scan spectroscopy) at a styrene concentration of 0.397 Mand acid concentration of 4.7x10-3 Mat 20°C, the maximum concentration of cationic ends (both free ions and ion pairs) was found -4 M, indicating that the initiator efficiency is 0.030. At 20°C, to be 1.4×10 k~= is reported [23] to be 12. k27t (a) Whatis the ratio of free ion and ion pair concentrations (b) Whatwouldbe the relative contributions of free ions and ion pairs to the overall propagationrate ? (c) Howmuchwouldbe the error in assigning the overall rate constant kg as k~ Answer: (a) For ion-pair dissociation equilibrium, vvvvvM+ (LI)- ~ vv~vM+ + (LI)ga = [M+]free[(LI)-] [M+]pair

(P8.28.1)

+] _ a2 [M (1 -- 0

(P8.28.2)

wherea is the degree of dissociation of ion-pair and [M+] is the total concentration of cationic ends ([M+]free÷ [M+]pair) Thus, c~ [M+]free = [M+lfre~ +] [M -- [M+]free = 1 -- a [M+]pair SolvingEq. (P8.28.2) for c~, ~ = (Ka/2[M+]) [(1 + 4[M+]/Kd) 1/2

--

(P8.98.3)

l]

Substituting for c~ in Eq. (P8.28.3) fromEq. (P8.28.4) yields (1 + 4[M+] / gd)1/2 - 1 [M+]fre~

[M+]p~r - 1 + (2[M+]/K.) - (1 +4[M+]/K.)I/2

(P8.28.4)

(P8.28.5)

This equation showsthat the fraction of free ions decreases rapidly at higher values of [!vI+]/Ka (dimensionless). Thusfree ions constitute approximately99,

730

Chapter8

90, 62, 27, 9, 2, 1, and 0.3%of the propagating species at [M+] / Ka values of 0.01, 0.1, 1, 10, 102, 10z, 104, and105, respectively. For the given experimentalcondition, [M+] = 0.030(4.7x10-a tool L-1) = 1.41x 10-4 -a tool L [M+]/Ka= (1.41x10 -4 tool L-1)/(4.2×10 -r tool L-a) = 336 FromEq. (P8.28.5), [M+]free/[M+]pair-= 0.056. That is, free ions constitute 5.6%of the propagating species. (b) To obtain a relation betweenk~, +, and k~, Eq. (P 8.27.1) can bewritten as k~[M+][M] = kv +a[M*][M] +j + IM] k~(1 - a)[M whichyields k~ = o~k~ + + (1(P8.28.6) If a << 1, Eq. (P8.28.2) can be approximated Ka = a 2 [M+] (P8.28.7) +] << 1, Substitution of Eq. (P8.28.7) into Eq. (P8.28.6) gives, for Ka/[M

k; = k; + ~,+K~/2 / [M+]I/2 From Eq. (P8.28.7), for [M+]/Ka= 336, a is approximately 0.054, while Eq. (P8.28.4) yields a = 0.053. FromEq. (8.28.6),

V~

+ = (0.05z)(12)+ (1- 0.053)

= 0.64 + 0.95 --~ 1.6 Hence,flee ions’ contribution to rate =

0.64 = 0.40, 0.64 + 0.95

i.e.,

40%

(c) from (b), ~/~,~~_1.6 So there is a 60%error in the value of k~ obtained by equating k~ with k~.

Similarly if [M+]/Kdis less than 0.1, the system consists of more than 90%free ions and less than 10%less reactive ion pairs (see Problem 8.28) and hence one can safely equate k~ with k~-. Such conditions may be obtained in polymerizations initiated by stable carbocation salts such as hexachloroantimonate (SbCI~-) salts of triphenyl methyl [(C6H5)3C+] and cycloheptatrienyl (CTH~’)carbocations, which are thus useful for evaluating the free-ion propagation rate constant [24]. However,since these cations are stable, their use is limited to the initiation of the more reactive monomers like N-vinylcarbazole and alkyl vinyl ethers. The dissociation constant

Ionic ChainPolymerization

731

value of the propagating ion pair can be obtained most directly from conductivity measurements or, indirectly, from kinetic data for which an useful method is to study the effect of a commonion salt (e.g., tetran-butyl ammonium triflate for the triflic acid initiated polymerization) on the kinetics of polymerization in a procedure analogous to that described for anionic polymerization (cf. Problems 8.11 and 8.12). Polymerizations initiated by ionizing (7) radiation are also useful for measuring free-ion propagation rate constants since ionizing radiation of bulk monomersyields free ions (in the absence of ion pairs) whoseconcentrations can be obtained by conductivity measurements [25], However, the method cannot be used for polymerizations in solution, e~pecially in polar (usually alkyl halide) solvents. Although there are relatively few systems in which reasonably accurate measures of both k~+ and kf are available, it is generally considered that no more than one order of magnitude difference in reactivity exists for free ions and ion pairs in cationic polymerization [26]. In the majority of cationic polymerization systems reported in the literature, the propagating species are probably associated in pairs. Moreover, the overall rate of carbocationic polymerizations is affected in a complex manner by solvation. Whereas initiation rate is enhanced by increasing solvent polarity on account of increasing rate of ion generation by solvent participation, propagation rate maydecrease (see later). In view of the role of moisture, existence of free ions, ion pairs, aggregates, solvation and other complexeffects, comparison of data from different investigators should be done with caution.

Problem 8.29 The problem of reproducibility of rates and the discrepancy betweenresults obtained by various authors is usually attributed to variations in impurity levels, the most important being the traces of water. To determine the minimum concentration levels at whichwater can still be effective, consider a polymerizationwhere0VII = 0.1 tool/L, ki = 3 L/mol-sand the DP,, of the polymer formedis 10s (these values are reasonably dose to a styrene polymerization)and assumethat a rate of 1%conversion per 24 h can be determined with sufficient accuracy. Determinethe concentration of water in the systemcontaining a Lewis acid (L) coinitiator in a relatively large concentration. Answer: (0.1 tool -1) (0.01) P%’ = (24 h)(3600 sh-’) = 1.16x10 -8 molL-1 -1 s I~ / P~ = D P. ~ =I0 Therefore, /~ = 1.16xl0 -n tool L-1 s-1

732

Chapter 8

The equation describing ion generation in a H20/L system having a relatively large concentration of L is displaced to the right, ki

H~O + L~---H

+ + LOH-

so that

+] = [H Ri

-I s-I)[H20](0.1tool-I) k~ [H20][M] = (3 L tool -II toolL-~ s -~) ( 0.1 mol L -~)

x 10 [Ho]-- (3 (1.16 L mo1-1 -~) ----

3.8x 10 -n -1 molL

This water concentration is certainly far below any analytical detection limit and may be present in the equipment, despite most careful drying and baking, via diffusion of moisture from glass or metal walls.

Molecular

Weight

Distribution

The theoretical molecular weight distributions for cationic chain polymerizations (see Problem 8.30) are the same as those described in Chapter 6 for radical chain polymerizations terminating by disproportionation, i.e., where each propagating chain yields one dead polymer molecule. The polydispersity index (PDI = DPw/DP,~) has a limit of 2. Many cationic polymerizations proceed with rapid initiation, which narrows the molecular weight distribution (MDI). In the extreme case where termination and transfer reactions are very slow or nonexistent, this would yield a very narrow MDI with PDI close to one (p. 681).

Problem 8.30 Consider the cationic

polymerization

Initiation:

tM +

I

-~

M

Propagation:

M~+

+ M _~_~ M=+ + 1

scheme given below: (1) (2)

Transfer: spontaneous

M~+ -~

to monomer

M~+ + M ~:~

Termination:

M. + --~

M= + I

M~

M. ÷ M~"

(3) (4) (5)

Derive expressions for numberand weight distributions of chain lengths of cationic polymers for the special case where the monomerconcentration is kept constant

733

Ionic Chain Polymerization (which is approximated to in experiments with very low initiator

concentration).

Answer: In the special case where the monomerconcentration is kept constant, the rates of all processes are at all times proportional only to the concentration of active species. The relative probability of a growth step therefore remains constant throughout the reaction, and the distribution should have the most probable form, as in the case of condensation polymerization (see Chapter 5). This relative probability of growth may be expressed as P

=

Rate of growth step Sumof rates of all steps +] ~ [M] [M +] +] k v[M] [M + kt. [M + ktr,M [M] [M+1 + k.t [M+I kt.

+ kn + (kv + ktr,M)[U]

Thus P is constant for constant [M]. If N chains are started and allowed to grow till number stopped

The number fraction

at length

1 2 z

unit units units

all are stopped, then

= N(1- P) = NP(1- P) = NP*-I(1P)

having length x is

n= = N~/N

= P~-I(1-

P)

giving the number average degree of polymerization DP~

-

~ xN~

-

~, xNP ~-~ (I - P )

= The weight distribution W~ = zN~

is given by

= NzP ~-~(1-

P)

and the weight~action w~ = W~/EW~ = zP~-~(1giving the weight-average degree of polyme~ation

Hence the ratio of weight to number average, is DP~/DP~

= 1

for long chain polymers

+ P ~

~

where P ~ 1

P)~

734

Gh~pter 8

The cumulative weight fraction of all species up to length a~, which is the most convenient function for comparison with an experimental fractionation will be

1

1

Note: Where the monomerconcentration falls during a polymerization, the value of _P will also fall, and the distribution cannot be formulated by these simple arguments. It may, however, be seen qualitatively that the distn’bution will broaden, shifting its maximum to lower chain lengths.

For polymerizationscarried out to high conversions wherethe concentrations of propagatingcenters, monomer, and transfer agent as well as rate constants change, the polydispersity broad molecular weight distributions polymerizations.

Table 8.5 Representative Monomer

~ Isobutyblene

index increases are generally

MonomerReactivity

Monomer

~ Coinitiator

Ratios in Cationic Polymerization aSolvent

2 1,3-Butadiene

Isoprene Cyclopentadiene Styrene

a-Methylstyrene Styrene c a-Methylstyrene

Isobutylene

considerably. Relatively encountered in cationic

Temp.

vl

r2

(°c) AIEtCI.~ AICIa

CHaCI CHaCI

-100 -103

43 115

0 0

AICIa

CHaCI

- 103

2.5

0.4

BFa,OEt~

~bCHa

-78

0.60

4.5

SnCl4 AICIa

EtCl CHaCI

0 -92

1.60 9.02

0.17 1.99

TiCl4 SnCl4 BFa.OEt2

~bCHa EtCl CH2C12

-78 0 -20

1.2 0.05 02-0-~

5.5 2.90 10-14

Toluene n-Hexane

-78 -20

1.20 1.20

1.78 0.54

TiCI4 TiCI4

EtCI -30 to 0 0.8 0.1 Styrene a Isoprene SnCI4 ’~Et =CaHs, ~b = t~henvl bj. p. Kennedyand I~. M’arechal, CarbocationicPolymerization, Wiley-Interscience, New York (1982). ~A. Tsukamotoand O. Vogl, Progn.Polym. Sci., 3, 199 (1971). ~F. E. Lipatova, A. R. Gantmakher,and S. S. Medvedev,Dold. Akad. Nauk_ SSSR, 100, 925 (1955).

Ionic ChainPolymerization

735

CATIONIC COPOLYMERIZATION Cationic copolymerization can be treated in identical manner to anionic copolymerization. The mechanistic scheme for propagation is obtained simply by replacing the negative signs in Eqs. (8.98) to (8.101) by positive signs. The copolymerization equation derived for free-radical initiation [Eqs. (7.11) and (7.18)] mayalso be applied to cationic polymerization to termine ratios, several of which are listed in Table 8.5. The situation is complicated, however, due to counterion effects [11, 27]. Thus, unlike free radical copolymerization, different cationic initiators can cause variations in reactivity ratios. Moreover,solvent polarity mayalso have an effect, because this governs the degree of chain-end ion-pair dissociation. Another significant difference is that there is no apparent tendency for alternating copolymers to form in ionic copolymerization. On the contrary, block copolymers or homopolymerblends are more likely. The most important of the commercial cationic copolymers is butyl rubber prepared from isobutylene and isoprene. Because of its very low air permeability, butyl rubber finds extensive use in tire inner tubes and protective clothing. It is manufactured by low-temperature (- 100°C) copolymerization of about 97%isobutylene and 3% isoprene in chlorocarbon solvents with A1Cla coinitiator (see Table 8.5). More recently, an ozone-resistant copolymer of isobutylene and cyclopentadiene has been marketed.

1. M. Szwarc, Carbanions,Living Polymers, and Electron Transfer Processes, Wiley-Interscience, NewYork (1968). 2. M. Fontanaille, "Carbanionic Polymerization: General Aspects and Initiation," pp 365-386 and "Carbanionic Polymerization: Termination and Functionalization," pp 425-432 in ComprehensivePolymerScience (G. C. Eastmond, A. Ledwith, S. Russo, and E Sigwalt, eds.), vol. 3, Pergamon Press, London(1989). 3. G. Spach, M. Levy, and M. Szwarc, J. Chem.Soc., 355 (1962). 4. M. Sawamotoand T. Higashimura, Macromolecules,i1,328, 501 (1978); 12, 581 (1978); 12, 581 (1979); Macromol. Chem. Macromol. Syrup., 3, 83 (1986); 32, 131 (1990). 5. M. Szwarc, M. Levy and R. Milkovich, J. Am.Chem.Soc., 78, 2656 (1956). 6. E J. Flory, J. Am.Chem.Soc., 62, 1561(1940). 7. M. Szwarc(ed.), Ions and Ion Pairs in OrganicReactions, Wiley, NewYork (1974). 8. D. N. Bhattacharyya, C. L. Lee, J. Smid, and M. Szwarc, J. Phys. Chem., 69, 612 (1965). 9. L. J. Fetters, J. Polym.Sci., Part A C26,1 (1969). 10. G. A. Olah, J. Am. Chem. Soc., 94, 808 (1972); Macromol Chem. Macromol.Symp., 13/14, 1 (1988). 11. J.P. Kennedyand E. Marechal,CarbocationicPolymerization,Wiley-Interscience, NewYork (1982). 12. E Giusti and E Andruzzi, Chim. Ind. (Milano), 48, 435, 442, 1079(1966). 13. A. Ledwithand D. C. Sherrington, Polymer,12, 344 (1971).

7~6

O~apter8

14. J. P. Kennedyand R. G. Squires, Polymer, 6, 579 (1965); MacromoL Sc L Chem., AI, 861 (1967). 15. J. J. Throssell, S. E Sood, M. Szwarc, and V. Stannett, Z Am.Chem.Soc., 78, 1122(1956). 16. J. E Kennedyand S. C. Feinberg, J. Polym. Sci., Polym. Chem.Ed., 16, 2191(1978). 17. J. P. Kennedy,J. Polym.Sci. Syrup., 56, 1 (1976). 18. L. Reibel, J. P. Kennedy,and D. Y. L. Chung,J. Polym. Sci. Polym. Chem. Ed., 17, 2757(1979). 19. J. P. Kennedy,Cationic Polymerizationof Olefins, Wiley(Interscience), New York (1979). 20. A. Ledwithand D. C. Sherrington, in Comprehensive ChemicalKinetics (C. H. Bamfordand C. E H. Tipper, eds.), voi. 15, chap. 2, Elsevier, Amsterdam (1976). 21. R. H. Biddulph,P. H. Plesch and P. P. Rutherford, J. Chem.Soc., 275(1965). 22. R. O. Colcloughand E S. Dainton, Trans. FaradaySoc., 54, 886, 894 (1958). 23. T. Kunitake and K. Takanabe,Macromolecules,12(6), 1061, 1067(1979). 24. E Subira, J. P. Vairon, and P. Sigwalt, Macromolecules, 21, 2339(1988). 25. W.C. Hsieh, H. Kubola,D. R. Squire, and V. Stannett, J. Polym.Sci. Polym. Chem.Ed., 18, 2773(1980). 26. G. Sauvet and P. Sigwalt, "Carbocationie Polymerization: General Aspects and Initiation," chap. 39 in Comprehensive PolymerScience, vol. 3, (G. C. Eastmond,A. Ledwith,S. Russo, and Sigwalt, eds.), PergamonPress, Oxford (1989). 27. G. E. Ham(ed.), Copolymerization,Wiley-Interscience, NewYork (1964).

EXERCISES 8.1. Predict the order of reactivity (and justify your prediction) of the given monomers. (a) Styrene, 2-vinylpyridine, 3-vinylpyridine, and 4-vinyl pyridine in anionic polymerization. (b) Styrene, p-methoxystyrene,p-chlorostyrene, and p-methylstyrenein cationic polymerization. 8.2. Isobutylene undergoes cationic polymerization in the presence of strong Lewisacids like AICIabut it is not polymerizedby free radicals or anionic initiators. Acrylonitrile, on the other hand, is polymerizedcommerciallyby free radical meansand can also be polymerizedby anionic initiators like potassiumamidebut does not respond to cationic initiators. Accountfor the difference in behavior of isobutylene and acrylonitrile in terms of monomer structure. 8.3. Suggestpractical ionic initiators and solvents for the polymerizationof the following monomers: rene ~/ [sop Vinylidene cyanide Tetrahydrofuran Propylene /:ed)/C°umar°ne 8.4. Chlorinated aliphatic solvents are useful-for cationic polymerization but cannot be used in anionic polymerization. Conversely,tetrahydrofuran which is a useful solvent in anionic polymerization cannot be used in cationic polymerization. Why?

737

Ionic Chain Polymerization

8.5. Polymerization of styrene by sodium naphthalene was performed at 35°C in dioxane. From the residual monomerand average degree of polymerization of the polymer formed in accurately measured reaction time the following data were obtained for two different monomer (M) - initiator (CA) compositions : Composition Run A: [M]o = 0.31 M [CA]o = 4.0x10 -~ M Run B : [M]0 = 1.17 M [CA]0 = 11.0x10 -4 M

Time (s) 290

Conversion (%) 69

DP 1063

329

90

2079

Calculate the overall rate constant and the total concentration of chain centers corresponding to the two initiator concentrations. [Ans. (a) kp = 10.1 L tool -1 s -1, [M-I = 4.0×10-4 M; (b) -1 v =7. 0 L to ol s -t, [M-] = 1.0× 10-a M; Note that the overall or apparent rate constant depends on the initiator concentration.] 8.6. The apparent propagation rate constant for polymerization of styrene in THFat 25°C using sodium naphthalene as initiator is 550 L tool -t s -t. If the initial concentration of styrene is 156 g/L and that of sodium naphthalene is 0.03 g/L, calculate the initial rate of polymerization and, for complete conversion of the styrene, the number average molecular weight of the polystyrene formed. Commentupon the expected value of the polydispersity index (M~,/M,~) and the stereoregularity of the polystyrene produced. [Arts. P~ = 0.165 tool L-1 s-l; ~,~ = 1.56×106 g mol-1.] 8.7. Styrene was added to a solution of sodium naphthalene in tetrahydrofuran at 25°C so that the initial concentrations of styrene and sodium naphthalene in the reaction mixture were 0.2 mol/L and 0.001 tool/L, respectively. Five seconds after the addition of styrene its concentration was determined to be 1.73 x 10-3 mol/L. Calculate: (a) Overall propagation rate constant. (b) Initial rate of polymerization. (c) Rate of polymerization after 10 (d) Numberaverage molecular weight of the polymer after 10 [Ans. (a) kv = 950 L mol-t s-l; (b) (P~)0 = 0.190 tool -1 s-l; ( c) ( e~) = 1.422×10-5 tool L-1 s-t; (d) ~ = 20800.] 8.8. Howmuch sodium one needs to prepare 1 kg of polystyrene of molecular weight 300,000 by anionic polymerization ? Howmuch water is sufficient to completely prevent polymerization in this case ? [,,Ins. Sodium0.153 g; water 0.12 g.] 8.9. Onemilliliter initiator solution in scrupulously clean and dry tetrahydrofuran containing 8.6x 10-4 mol of sodium and 9.4× 10-4 tool of naphthalene was injected into a solution of styrene (0.048 mol) in 50 mL dean and dry tetrahydrofuran at -70°C. After a few minutes the reaction was complete. A few milliliters methanol was then added to quench the reaction and the reaction mixture was allowed to warm to room temperature. The polymer

738

Chapter 8 was recovered by precipitation. What is M,~ of the polystyrene formed in the absence of side reactions ? What should M~, of the product be if the polymerization were carried out so that the growth of all macromolecules was started and ended simultaneously ? [Ans. M~= 11,300; Mw= 11,400.]

8.10. Polymerization of styrene by sodium naphthalene was performed at 20°C in 3-methyl tetrahydrofuran (3-Me-THF) using a flow apparatus and technique suitable for fast polymerizations. The conversion was determined gravimetrically after precipitation of the polymer with methanol-water or optically by the determination of the residual monomerand the concentration [M-] of the active centers was calculated from the degree of polymerization at a g.iven conversion [see Eq. (P8.10.4)]. In two independent experimental series, the overall rate constant kp was obtained [el. Eq. (P8.10.2)] both at different concentrations of initiator (and hence [M-]) without addition of electrolyte and at different concentrations of sodium ions from added salt sodium tetraphenyl borate (NaBPh4)and constant concentration of initiator. From the resulting data given below determine k~, k~ and K for polystyryl sodium in 3-Me-THFat 20°C. Without electrolyte With electrolyte [M-] x 103 kv [Na+]’~ s× l0 k~ (tool -1) (L mo1-1 -1 ( mol L -I) (L tool -1 -1) s 5.10 1.83 2.78 93 2.20 255 1.69 97 0.73 381 0.75 135 0.40 479 O.45 188 0.26 600 0.37 215 aCalcuiated from the dissociation constant of NaBPh4in 3-Me-THFdetermined by conductivity measurements (1.12x10 -5 L/mol at 20°C) [Ans. k~ = 66 L tool -~ 8-1; k~ = 1.10xl0 s L tool -1 s-l; K = 6×10-9 tool 8.11. (a) Calculate the concentrations of polymeric free ions and ion pairs in the absence of electrolyte taking the polymerization system of Exercise 8.10 in which [M-] is 0.40 x 10-3 mol/L and no electrolyte has been added. (b) What would be the concentrations of free ions and ion pairs if the polymerization system contained NaBPh4in concentration equal to the total concentration of living ends 9. (c) What percentage of free + io ns co mes fr om the initiator in (b)9. [Ans. (a) 1.55x10-~ mol L-~ and 0.40xlO-3 tool L-1. (b) 1.95x10-s -1 mol L and 0.40x10-3 tool L-1. (c) 0.016%.] 8.12. A polystyrene sample of (number) average degree of polymerization 50 was made by living polymerization in tetrahydrofuran solvent using sodium naphthalenide as the initiator. Showgraphically the number fraction distribution of the degree of polymerization for this polymer and, for comparison, similar distribution for a polystyrene of the same average DP produced by free radical polymerization.

Ionic Chain Polymerization

739

8.13. Write reactions to show the type of end group that will result from the use of each of the following reagents to terminate butyllithium initiated living polymers of styrene in THF: (a)Ethyi benzoate; (b) benzyl chloride; bromoaniline; (d) phthalic anhydride; (e) phosgene; (f) carbon disulfide. 8.14. Write reactions illustrating the synthesis of poly(methyl methacrylate)-blockpolyacrylonitrile by anionic polymerization. Showhow you could synthesize an ABAblock copolymer of the two monomers. 8.15. Showby equations the synthesis of the following types of block copolymers (a) ABAand (b) CABAC where A, B, and C represent styrene, butadiene and isoprene. 8.16. It has been found that the cationic polymerization of 2,6-diphenyl-l,6heptadiene, CH2 =C(¢)-(CH2)3-C(¢)=CH2, yields a polymer whose peat unit consists of six-membered non-aromatic ring structure. Propose a structure of the repeat unit and a mechanismof this polymerization. 8.17. If trifluoroacetic acid (mixed with ethylbenzene) is added dropwise to styrene, no polymerization occurs. However, if styrene is added to the acid, high molecular weight polymer forms rapidly [J. J. Throssell, S. P. Sood, M. Szwarc, and V. Stannett, J. Am. Chem. Soc., 78, 1122 (1956)]. Suggest explanation. 8.18. Account for the fact that 1-butene can be used to control polymer molecular weight in cationic polymerization of isobutylene. 8.19. Account for the fact that polymerization of propylene by the cationic process yields oligomers that have extremely complicated structure with methyl, ethyl, n- and i-propyl, and other groups. Whyonly oligomers are formed by this polymerization ? 8.20. The cationic polymerization of 4-methyl-l-pentene, CH2=CHCH2CH(CH3)2, yields a polymer that is a excellent rubber having a structure identical to a 1:1 alternating copolymer of ethylene and isobutylene. Propose a mechanism for this polymerization. Discuss any side products that you would expect. 8.21. In the cationic polymerization of 3-methyl cyclopentene, the NMP,spectra gave a singlet resonance in the methyl proton region. Propose a structure for the polymer consistent with the NMR.Explain your reasoning for the structure. 8.22. Write equations to show the different structural units that may result from intramolecular hydride and methide shifts (involving the end monomeric unit) in cationic polymerization of 4-methyl-l-pentene. 8.23. The cationic polymerization of 3-methyl-l-butene produced a polymer whose NMRspectra consisted of only two singlets. Propose a structure for the polymer consistent with the NMRand suggest a mechanism for this polymerization. 8.24. Under certain conditions, the rate of polymerization of isobutylene using SnCI4/I-t20 initiating system is found to be first-order in SnC14,first order in water, and second order in isobutylene. The polymer formed initially has the number average molecular weight of 20,000 and contains 3.0x10-5 tool OHgroups per gram but no chlorine. Suggest plausible reaction schemes to fit these data and derive appropriate expressions for the rate and degree of polymerization. Indicate any assumptions made in the derivation.

740

~hapter 8

8.25. Underwhat reaction conditions mighta cationic polymerizationwith SnCIa/H20 initiating systemshowa dependenceof the polymerization rate whichis First order in monomer. Zero order in SnCI4or water. Secondorder in SnCI4or water. 8.26. Consider the eopolymedzationof 1,3-butadiene with the following monomers: n-butyl vinyl ether, methylmethacrylate,methylacrylate, styrene, vinyl acetate, acrylonitrile, maleie anhydride. If the copolymerizationswerecarried out using cationic initiation, what wouldbe expected qualitatively for the eopolymercompositions? List the copolymersin order of their increasing butadiene content. Wouldcopolymers be formed from each of the comonomer pairs ? Explain what wouldbe observedif one used anionic initiation ? 8.27. Discuss the general effects of temperature, solvent, and catalyst on the monomer reactivity ratios in ionic copolymerization. Howdo these compare with the correspondingeffects in radical copolymerization? 8.28. Whatexperimental approaches could be used for determining whether the polymerization of a particular monomer by ionizing radiation proceeds by a radical or ionic mechanism ? 8.29. Accountfor the following facts observedin earbocationic polymerization: (a) Isobutylene polymerization proceeds rapidly in CH3CIbut not in CH3I diluent. (b) Olefins cannot be polymerizedin diethyl ether and the polymerization of vinyl ethers by BFa.OEt2proceeds faster in hexanethan in Et.~O. (e) Propylenecan be polymerizedby I-IBr/AIBra to a very broad molecularweight-distribution product. (d) Whereasthe initiation rate is enhancedby increasing the solvent polarity, the propagation rate maydecrease. (d) Bulk polymrizationof unsaturatedhydrocarbonsinitiated by "),-radiation often showsvery high rate constants (108 to 108 L mo1-1s-l). 8.30. Giveplausible explanationsfor the followingfacts: (a) Molecular weight does not dependon initiator concentration in ionic polymerizationas it do~s in free radical polymerization. (b) Polymerizationrate is moresensitive to solvent effects in ionic polymerization than in free-radical polymerization. (c) In cationic polymerization, DP= ~, but this is not always the case in free radical or anionic polymerization. (d) Ethyl vinyl ether undergoes cationic polymerization faster than /3ehioroethyl ether under the same conditions. (e) Cationic polymerization of methyl a-methyl vinyl ether gives predominantly syndiotactic polymerin toluene, whereasmethyl vinyl ether yields mainlyisotactic polymer.

~/

Chapter

9

Coordination Addition Polymerization INTRODUCTION The field of coordination polymerization came into existence in the mid1950s with the work of Karl Ziegler in Germanyand Giulio Natta in Italy. In the early 1950s, Ziegler discovered that a combination of aluminumalkyls with certain transition metal compoundssuch as TiCI4 or VCI4, generated a system that would polymerize ethylene at low temperatures and pressures to give polyethylene an essentially linear structure. Nowreferred to as highdensity polyethylene (HDPE), the product is denser, tougher, and higher melting than the branched, low-density polyethylene (LDPE) produced high pressures by free-radical polymerization, because the more regular structure results in closer chain, packing and a high degree of crystallinity. Following close on the heels of Ziegler’s remarkable discovery was the recognition by Natta that catalysts of the type described by Ziegler were capable of polymerizing 1-alkenes (or alpha olefins, as they are called in the chemical industry) to yield stereoregular polymers. Natta and his coworkers, as well as scientists in other laboratories, subsequently extended the range of catalysts, both soluble and insoluble, to produce polymers exhibiting a wide range of stereoregular structures, including those derived from dienes and cycloalkenes [1,2]. Many polymers are now manufactured on a. commercial scale using Ziegler-Natta catalysts. Indeed, stereoregular (isotactic) polypropylene high molecular weight, which cannot be made by free-radical or ionic polymerization, has already achieved the status of a commodityplastic. The scientific and practical significance of Ziegler and Natta’s work earned them the joint award of the Nobel Prize in Chemistry in 1963.

741

742

Chapter 9

Ziegler-Natta catalyst systems appear to function by a coordination complex between the catalyst, growing chain, and incoming monomer. Hence this process is referred to as coordination addition polymerization, and the catalysts as coordinationcatalysts. Other types of complexcatalysts that have received attention for stereospecific polymerization are the reduced metal oxides and the alfin catalysts (prepared from compoundsof sodium). All three types are mainly used heterogeneous polymerization, although some homogeneous processes are also commercially important. -ZIEGLER-NATTA Catalyst

CATALYSTS

Composition

The number of compounds and combinations of compounds that fit into the category of Ziegler-Natta catalysts are far too numerous to describe here. However, a Ziegler-Natta catalyst may be simply defined as a combination of two components: (1) a transition metal compoundof an element from groups IVB to VIIIB, and (2) an organometallic compoundof a metal from groups I to III of the periodic table. The transition metal compound is referred to as the catalyst and the organometallic compoundas the cocatalyst. A combination of the catalyst and the cocatalyst is often referred to simply as the catalyst. Most commonly,the catalyst componentconsists of halides or oxyhalides of titanium, vanadium, chromium, molybdenum, or zirconium, and the cocatalyst componentoften consists of an alkyl, aryl, or hydride of metals such as aluminum, lithium, zinc, tin, cadmium, beryllium, and magnesium. The catalyst systems maybe heterogeneous (some titanium-based systems) or soluble (most vanadium-containing species). Perhaps the best knownsystems are those derived from TiCI4 or TIC13 and an aluminum trialkyl. The catalysts which are useful for the preparation of isotactic polymers are heterogeneous, i.e., they are insoluble in the solvent, or diluent, in which they are prepared. Their activity and stereoregulating ability are greatly affected by the components, and method, used for their preparation. The inclusion of ~lectron donors such as Lewis bases (e.g., amines, ethers, ketones, and esters) during preparation of the catalyst also can improve stereospecificity, often but not always with a loss of activity. Catalysts are prepared by mixing the componentsin a dry, inert solvent in the absence of oxygen, usually at a low temperature. They are characterized by having high reactivity toward many nonpolar monomers(e.g., ethylene and alpha olefins) and are usually capable of giving polymers with a high degree of stereoregularity. Catalyst activity usually changes with time, and

CoordinationAddition Polymerization

743

the maximum activity is often reached only after aging periods of one to two hours. The nature of the insoluble Ziegler-Natta catalysts is not well understood and the trial-and-error methodis frequently used in developing new catalysts. Changes in the catalyst system and addition of additives affect the rate of polymerization as well as yield, molecular weight, and the degree of stereoregularity of the polymer. Tables 9.1 to 9.3 show how some of the structural variables of Ziegler-Natta catalysts affect the stereoregularity of polypropylene. The effects of different catalyst-cocatalyst combinations (some with additives) on the stereoregularity of polypropylene are shown in Table 9.1. Catalyst systems consisting of the same aluminumalkyl as the cocatalyst but different transition metal halides as the catalyst affect the extent of stereoregularity as shown by the data for polypropylene in Table 9.2. The organic group of the organo-aluminum cocatalyst also influences the performanceof the catalyst system. Thus, the stereoregularity of polypropylene is found to decrease with increasing size of the organic group attached to aluminum(see Table 9.3). Table 9.1 Stereoregularity of Polypropylene Produced with Different Catalyst Systems b Catalyst system Stereoregularity (%) 35.2 R3AI + TIC14 84.7 R3AI + a-TiCl3 R3AI + fl-TiCl3 45 R3AI + TiCl4 + NaF 97 R3AI + TiCI4 + Compounds of P, As, or Sb 98 R3AI + TiCIz + Amine 81 RzAI+ Ti(O-/-butyl)4 20 0 RaAI + V(acac)3 R3AI + Ti(CsHs)CI2 70-90 R2AIX+ TiCI3 90-99 RAIX2+ 7-TiCI3 + Amine >99 RAIX2 + TiCIz + HPT 97 RNa + TiCI3 90 RNa + TiCI4 90 RLi + TiCI4 90 R~Zn + TiClz 65 R~Zn + TiCI3 + Amine 93 R = alkyl, (acac) = acetylacetonate,X halogen, HPT.= hexamethylphosphorictriamide, CsH~= cyclopentadienyl. Source:Data fromRef. 3.

744

Chapter 9

Table 9.2 Influence of the Transition the Stereoregularity

Metal on

of Polypropylene

bTransition metal compound Stereoregularity (%) TiCI4 48 TiBr4 42 TiCIa, c~, 7, or 6 80-92 TiCI3, fl 40-50 ZrCI~ 55 73 VCI3 VOCI3 32 VCI4 48 CrCIa 36 The organometallic compoundis AI(C2H~)3in each case. Source: Data from Ref. 3.

Table 9.3 Influence on the Stereoregularity

of R group of AIRa of Polypropylene

Stereoregularity R TIC13 C2H5 79.4 n-CaH7 71.8 74.5 i-C4H9 C~H5 65.4 C4H1~ 64.0 59.0 C16Haz Source: Data from Ref. 3.

(%) TiCI4 47.8 50.9 30.0 26.2 16.2

Nature of the Catalyst One first assumed that polymerization with Ziegler-Natta catalysts, such as aluminum-alkyls plus halides, works by a simple ionic mechanism. Since single aluminum alkyls normally cause anionic and titanium halides a cationic chain reaction (Chapter 8), the two components of the initiator should neutralize each other and only the excess one over the other should be active. If this were true, then either one of the components alone should be able to initiate the polymerization of ethylene or propylene, but this is not the case. A simple anionic or cationic mechanism can therefore not explain the polymerization with Ziegler-Natth catalysts. The nature of the Ziegler-Natta catalyst systems is still a subject of debate. One fact, however, does appear to be certain, especially about the

Coordination Addition Polymerization

746

insoluble catalyst systems-the true catalysts are not simple coordination adducts formed from the original metal halide and aluminum alkyl. A critical "aging" period for the catalyst is often needed before it achieves its highest activity, and complexreactions occur during this period. These reactions probably include an initial exchangeof substituent groups between the two metals to form transition metal-carbon bonds by exchange reactions such as those shownin Eqs. (9.1)-(9.3): AIRs + TIC14 ,~- R2AIC1 + RTiCls

(9.1)

R2A1C1 + TiCI4 ~ RA1CI2 + RTiC13

(9.2)

A1R3 + RTiC13 ~ R2AICI + R2TIC12

(9.3)

These organotitanium halides are unstable and can undergo reductive decomposition processes, such as shown in Eqs. (9.4) and (9.5): RTICI~ ~ R" + TiCI~

(9.4)

RzTiClz

(9.5)

~ R" + RTiC12

(Note that TIC13 can also be used as an initial catalyst componentin place of TiCI4.) Further reduction mayoccur yielding TIC12: RTiCI2 ~ TIC12 + R" RTiCl3 --~ TiCl~ d- RCl

(9.6) (9.7)

In addition, TiCl3 may be formed by the equilibrium: TIC14 -I- TIC12 ~ 2TiCla (9.8) For heterogeneous catalysts, the reactions are more complicated than implied by these equations. Radicals formed in these reactions may be removedby combination, disproportionation, or reaction with solvent. While such reactions undoubtedly occur in catalyst formation, it is not known to what extent and the aging process certainly requires more clarification. However, the analogous vanadium-containing systems are soluble and maywell be represented fairly accurately by reactions similar to those shown in Eqs. (9.1)-(9.5). In contrast to the heterogeneouscatalyst, the soluble catalysts appear to have well-defined structures. For example, the soluble catalyst system generated from triethyl aluminumand bis(cyclopentadienyl)titanium dichloride has been shown by elemental and x-ray analysis to have a halogen-bridged

746

Chapter 9 C2H 5

Ct Ti

C5H5

/

~Cl

~C2H 5

structure (I). In all these systems, one of the most important steps is the reduction of the transition metal to a low-valency state in which the metal possesses unfilled ligand sites. These low-valency transition metal species are believed to be the real catalysts or precursors of the real catalysts. Evolution

of the Titanium-Aluminum

System

Since the original discoveries of Ziegler and Natta there have been literally thousands of different combinations of transition and Group I-III metal components, often together with other compoundssuch as electron donors, studied for use in alkene polymerizations. However, the major interest in this chapter will be on the titanium-aluminum systems, more specifically, TiCI3 with AI(C2Hs)2C1 and TiCI4 with Al(C2Hs)3-probably the studied systems. It will be therefore useful to review the evolution of the titanium-aluminum catalyst systems starting with the original system used by Ziegler. The original catalyst used by Ziegler for ethylene polymerization was obtained in situ as a precipitate on mixing the components TIC14 and AI(C2Hs)s in a hydrocarbon solvent. This mixture was then used directly for initiating polymerization. Recognizing that the major product of the reaction of the two catalyst components was ~-TiCIa (brown in color), Natta explored various methods of performing it outside the polymerization system, for example, by reduction of TIC14 with hydrogen, aluminum, and various alkylaluminum compounds, including AI(C~Hs)2C1.The stereospecificity producedby these early catalyst systems was low, yielding polymerswith only about 20-40% isotactic content. There was a dramatic improvement with stereospecificity increasing to 80-95%whenthe or-, ~-, or "/-crystalline form of TiCla (all violet in color) was used directly. The efficiency or activity of these early catalyst systems was, however, low. The term activity, as it is used in most .literature references, refers to the rate of polymerization, and is often expressed in terms of kilograms of polymer formed per gram of catalyst. Because of low activity, substantial amountsof catalyst were needed to achieve acceptable yields of polymer, and

CoordinationAddition Polymerlza~ion

747

the spent catalyst had to be removedfrom the finished product. The catalyst activity was enhanced by various ball-milling and heat treatments of the catalyst componentsbefore and after mixing. (Ball-milling not only increases the surface area but also facilitates reactions between the componentsof the catalyst.) However,the activity was still low in comparison to the present catalyst systems, with considerably less than 1%of the Ti being active in polymerization. Subsequent generations of catalysts involved large increases in activity without sacrificing stereospecificity. The effective surface area of the active component is increased by more than two orders of magnitude by impregnating the catalyst on a solid support such as MgC12or MgO.As an example, a typical TiCla-AIRacatalyst yields about 50 to 200 g of polyethylene per gram of catalyst per hour per atmosphere of ethylene, whereas as muchas 200,000 g of polyethylene and over 40,000 g of polypropylene per gram titanium per hour may be produced using a MgC12-supportedcatalyst; thus eliminating the need for the costly step of removingcatalyst from the product. Suchcatalyst systems are often referred to as high-mileagecatalysts. Stereospecificity of the catalyst is kept high (> 90-98%isotactic dyads) by the presence of electron-donor additives such as ethyl benzoate. Thus, a typical recipe for a present day superactive high-mileage catalyst system involves initial ball-milling (mechanical grinding or mixing) of magnesium chloride (or the alkoxide) and TIC14followed by the addition of AI(C2Hs)3, an organic Lewis base being usually added in each of the steps of catalyst preparation [4]. High activity is very important from the commercial point of view not so muchfor the savings in the amountof initiator required but for eliminating the expensive task of initiator removal from the polymer product.

MECHANISMOF ZIEGLER-NATTA POLYMERIZATION Despite the tremendous amount of research that has been done in this area [1,2,5,6] the true mechanismof Ziegler-Natta polymerization is not entirely clear. It is generally agreed that heterogeneous polymerization occurs at localized active sites on the catalyst surface. The organometallic componentis believed to activate the site by alkylafion of a transition metal atom at the surface. Of the various mechanisms that have been proposed, the two that are most generally accepted are the so-called monometallic and bimetallic mechanisms[2,3], the former being favored in heterogeneous processes. In both processes, the monomeris pictured as being incorporated into a polymer by insertion between a-transition metal atom and the terminal carbon of the coordinated polymer chain. These two mechanisms are separately discussed in later sections.

748

Chapter 9

Coordination catalysts perform two functions. First, they supply the species that initiates the polymerization. Second, the fragment of the catalyst aside from the initiating portion has unique coordinating powers. The catalyst fragment, acting as gcgcnion or countcrion of the propagating species coordinates with both the propagating chain end and the incoming monomerso as to orient the monomerwith respect to the growing chain end and bring about stere0spccific addition. So the polymerization can bc considered as a concerted multicentered reaction. Mechanism of Stereospecific

Placement

Manydifferent mechanismshave been proposed to explain the usual isotactic placement obtained with coordination initiators [7]. Figure 9.1 depicts a general situation for an anionic coordination polymerization with isotactic placement. In this mechanismthe polymer chain end has a partial negative charge and the Catalyst fragment G (gegenion or counterion) has a partial positive charge. (A cationic coordination polymerization wouldinvolve a similar mechanismexcept for reversal of the signs of the partial charges,) The generally accepted mechanismfor stereospecific polymerization of ot-olefins and other nonpolar alkenes is a ~r-complexation of monomerand transition metal in G (utilizing the metal’s d-orbitals), followed by a four-center anionic coordination insertion process in which monomeris inserted into a metal-carbon bond. To elaborate further, the catalyst fragment G (containing transition metal) is coordinated with both the propagating chain end and the incoming monomermolecule. The latter is thus oriented and "held in place" by coordination during addition to the polymer chain. Coordination between the catalyst fragment G and the propagating center is broken simultaneously with the formation of bonds between the propagating center and the incoming monomerunit and between the initiator fragment and the incoming monomerunit. Propagation thus proceeds in the four-center cyclic transition state by the insertion of monomerbetween the catalyst fragment G and the propagating center. The insertion reaction has both cationic and anionic features. There is a concerted nucleophilic attack by the incipient carbanion polymer chain end on the oz-carbon of the double bond of the monomertogether with an electrophilic attack by the cationic counterion (G) on the alkene electrons. The catalyst fragment acts essentially as a template or mold for the orientation and isotactic placement of incoming successive monomer units. Isotactic placement occurs because the initiator fragment forces each monomerunit to approach the propagating center with the same face. This mechanism is referred to as catalyst site control or enantiomorphicsite control.

749

Coordination Addifion Polymerization

.H H x ,H

,H

H, :

"’C /= C

R

%H IH C -- CC’:=a H H C -- C’~’: ..... ~÷ G

Figure 9.1 Mechanism for anionic coordination polymerization with isotactic placement. Bimetallic

and Monometallic

Mechanisms

A numberof structures have been proposedfor the active species (sites) in Ziegler-Natta catalyst systems. The diversity of the proposedspecies arises from the numerous products that have been observed or can be postulated in the interaction of the two componentsof a Ziegler-Natta system[see Eqs. (9.1)-(9.8)]. Theproposedactive species fall into either of two general categories: monometallicand bimetallic [8-10] depending on the numberof metal centers. The two types can be illustrated by the structures (II) and (III) for the active species fromtitanium chloride alkylaluminumcomponentssuch as TIC14or TIC13with A1R 3 or A1R2C1. R

i/Cl

(R)Cl~ ,..,R.. /R(Cl)

Ti.";At

(R)Ct/ "’Ct’"

(n)

\R(Ct)

CI/I

(In)

750

Chapter 9

Structure (II) represents a bimetallic species that is the coordination complex of titanium and aluminum compoundsarising from the interaction of the original catalyst components[cf. Eqs. (9.1)-(9.8)] with exchange of R C1 groups. The placing of R and C1 groups in parentheses indicates that the exact specification of the ligands on Ti and A1 cannot be made. The identity and the number of ligands attached to each metal center may vary from one catalyst to other depending on the components and their relative amounts. Structure (HI) of a typical monometallic species represents active titanium site at the surface of a TiCI3 crystal. The titanium atom shares four chloride ligands with its neighboring titanium atoms and has an alkyl ligand (incorporated through exchange of alkyl from the alkylaluminum chloride) and a vacant orbital Bimetallic

Mechanism

Accordingto the bimetallic theory [8] the truly active catalysts are complexes that have an electron-deficient bond, e.g., Ti-.-C.--AI in (II). The bimetallic mechanisminvolves propagation in which growth occurs at two metal centers of the bridge complex as shown in Fig. 9.2(a). The mechanismis similar to that shownin Fig. 9.1 except for the detailing of the structure of the catalyst fragment G. It is suggested that the nucleophilic olefin forms a 7r-complex [Fig. 9.2(b)] with the ion of the transition metal and, following a partial delocalization of the alkyl bridge, is included in a six-membered ring transition state [Fig. 9.2 (c)]. The monomeris then incorporated into the growing chain between the AI and the C, thereby regenerating the complex. For steric reasons, the =CH2group of the incoming monomer points into the lattice and the CH3group to one side. The chain growth always takes place from the metal end (like hair from its root) and the process thus leads to an isotactic polymer. While a limited amount of experimental evidence does lend support to the bimetallic concept, major objections were voiced by Ziegler, who was of the opinion that like dimeric aluminumalkyls the Ti-A1 complex is not likely to be the effective catalyst agent. Other more recent work also favors the second and simpler alternative, the monometallic mechanism. Monometallic

Mechanism

Majority opinion nowfavors the concept that the d-orbitals in the transition element are the mainsource of catalytic activity and that chain growth occurs at the Ti-alkyl bond, which acts as the polymerization center, the function of the aluminum alkyl being only to alkylate TiCI3. The monometallic mechanism presented below are mainly based on the ideas of Cossee and

CoordinationAddition Pol~rnerization

~C,~ CH3

~ CH ~_.~.~.’G 3

6"CH 2

.,,.

(a)

751

>+i’"""

"’""At<

>÷~++ ""’At<(b)

¯ °ol31. CH3 I CH2 I CH--CH 3 I .CH2

m)>T~: i"

"’-At< "°(:t°° i Ano+ther sequence I I o’f monomer Qddi’kion

CH3 I ..~.....CH ,CH 2 :

"CH2-CH 3

(c) oO

o° °O°¢l.¯

CHIn I CH 2 I

CH-CH 3I CH2 I CH-CH3 I ..CH 2 (e)

>Ti:i"’"

"".’At<

CH-CH 3

I

.. CH2

>T~"" ""’:~,~<(#)

"’Ct""

Figure 9.2 Bimetallic mechanismfor stereospecific polymerization. (After Ref.

8.)

752

Chapter 9

C,~5

/zH5

C2H5 [’-l,~ C| + AI.(C2Hs)

3 ~

CIAI(C2H5)2 Ct

Figure9.3 Interaction

of aluminum alkyl with an octahedral vacancy around Ti stage of monometallic mechanism. (After Ref. 11.)

in the first

CH3 H

C2H5

Ct--ri--[~] Jr CH3--CH=CH 2 -----i. Cl--Ti -- II

/ I

ct/I Ct

Ct Ct Activecenter’

CH2--CH--C2H 5 Cl--Ti~

[]

,~

Migration

H H

[

Complex

F" []

C2H5

/ ~cl I

ct--’~

_c:.-.

CH

C~Hs3\ /

I

/

~1 ct--’n.

I~)ai .....c" I

Cl Newactive center I Another sequence I of steps

Transition state

CH2--C,H--CH2-C, H-- CzH S

I

ct/I Fibre 9.4 Ref. 11.)

Monometallic

mechanism for stereospecific

polymerization.

(After

Coordination Addition Polymerization

Z53

Arlman [10]. The monometallic mechanism was put on a sound theoretical basis by the quantum theory developed by Cossee [11]. The first stage is the formation of the active center. Cossee suggested that the active center formation was due to the interaction of aluminumalkyl with an octahedral vacancy around Ti. Considering o~-TiC13as catalyst this can be represented as shown in Fig. 9.3. The five-coordinated Ti3+ on the surface has a vacant d-orbital, represented by--U. Following chemisorption of the aluminumalkyl on the surface of the TiClz crystal, alkylation of the Ti 3+ ion takes place by an exchange mechanism to form TiRCI4-D, and the vacant site can then accommodate the incoming monomerunit. The monomerforms a 7r-complex with the titanium at the vacant d-orbital and is then inserted into the Ti-alkyl bond. Using propylene as the monomer, the mechanism can be developed as Shownin Fig. 9.4. After insertion of the monomerbetween the Ti-alkyl bond, the polymer chain migrates back into its original position and consequently the vacant site also migrates to its original position ready for a further complexing reaction. This migration is necessary, else an alternating position is offered to the monomerand a syndiotactic polymer would thus result. The main features of the monometallic mechanismare: (1) an octahedral vacancy on the Ti3+ is available to complex the olefin; (2) the presence of an alkyl to transition metal bond at this site is required; and (3) the growing polymer chain is always attached to the transition metal. Problem 9.1 From the accurate kinetic data that have been obtained [12] for the polymerization of C3I-I6 with a-TiCl3 and Al(C2.Hs)3it appears that the steady state the rate is strictly proportional to the pressure of C~I-I6. The polymerizationrate is also proportional to the amountof a-TiClz and independent of the concentration of AI(C~Hs)3.Suggest a kinetic schemein conformity with these observations. A qualitative use maybe madeof the fact that an activation energy of 11-14 kcal/mol has been observed for this polymerization and that no stable complexbetweena-olefins and Ti has been found. Answer: The complexformation and the rearrangement can be described schematically as kl

s + M ~ SM

(P9.1.1)

SM ~ S’ (P9.1.2) whereS is the vacant site, S~ is the new va6ant site with alkyl group being one unit longer, SMis the complexbetween vacant site and monomer,and Mis the monomer.The following relations hold: a[M]/at = - ~I[S] [r~] + ~[sr~] 0’9.~.~)

754

Chapter 9 d[SM]/dt = - k3[SM] + k,[S] [M] - k2[SM]

(P9.1.4)

[SM] + IS] =[C]

0’9.1.5)

where[C] is the total numberof active sites (vacant + filled). Understeady-state conditions, d[SM]/dt= 0. After a few eliminations one thus obtains diM] klk3[C] [M] (P9.1.6) dt - kl[M] + k2 + k3 Accordingto Eq. (P9.1.5), strict proportionality of the polymerizationrate the monomerpressure requires that either k2 >> k~[M] or k3 >> kl[M]. When k3 >> kl[M], one has the situation where every moleculeentering the vacant site would react immediately. The rate-determining step wouldthen be the complex formation. This, however,seemsnot very likely, if it is assumedthat diffusion of the monomerthrough the growing polymer is not a limiting factor. It is also difficult to visualize that putting a neutral moleculeinto a vacant position wouldrequire an activation energy of 11-14 kcal/mol that has been observed. This activation energy wouldpreferably be attributed largely to the rearrangement,and the conclusion maythus be drawnthat k2 >> kl[M]. This also fits the observation that no stable complexesbetweena-oleflns and Ti have been found. The polymerization rate nowbecomes klk~ (P9.1.7) /~ -- k: [CI [M] and the measuredactivation energy is AE = AEn - AHc (P9.1.8) where AERis the activation energy for the rearrangement and AHcis the heat of complexformation.Since the latter is probablyvery small, the activation energy for the rearrangementof the complexwill be of the order of 10 kcal/mol.

Stereoregulatlon The most important characteristic of Ziegler-Natta catalysts is their ability to produce stereoregular polymers. On the basis of the monometallic mechanism of Cossee and Adman, described above, stereoregulation of propylene polymerization can be explained as follows. To obtain a stereoregular polymer, the chemisorption of the monomer on the catalyst surface must be controlled so that the orientation of the incoming monomeris always the same. Examination of models reveals that a molecule such as propylene will fit into the catalyst surface in only one way if a position of closest approach of the double bond to the Ti3+ ion is to be achieved (Fig. 9.5). This places the =CH2group of the incoming monomer pointing into the lattice and for steric reasons the orientation of the -CH3 group to one side is preferred. This determines the configuration of the monomerduring the complexingstage, and it is always the same. Orientation

755

CoordinationAddition Polymerization

Figure 9.5 Cross-sectional diagram of the propylene-catalyst complexthrough the z - z plane of the octahedral structure. of successive monomermolecules in this way, prior to the incorporation into the polymer chain, leads to an isotactic polymer. For isospecific polymerization by the Cossee-Arlman mechanism, migration of the vacant site back to its original position is necessary, as otherwise an alternating position is offered to the incoming monomerand a syndiotactic polymer wouldresult. This implies that the tacticity of the polymer formed depends essentially on the rates of both the alkyl shift and the migration. Since both these processes slow down at lower temperatures, syndiotactic polymer would be formed when the temperature is decreased. In fact, syndiotactic polypropylene can be obtained at -70°C. KINETICS Typical

OF ZIEGLER-NATTA

Shapes of Kinetic

POLYMERIZATION

Curves

The kinetics of Ziegler-Natta polymerization, like the mechanismof the reaction, are complex. The relatively few polymerizations that are homogeneous behave in a manner generally similar to noncoordination ionic polymerizations (Chapter 8). However, as has been pointed out earlier, the heterogeneity of the reaction system is more of a rule in Ziegler-Natta polymerization than an exception. The heterogeneous systems usually exhibit complicated behavior [2,5,7], as can be seen from sometypical kinetic rate-time profiles, types (a)-(f) in Fig. 9.6. Suchprofiles are significant

756

Chapter 9

(b)

Time --~

Time --~ Cc)

Time

(d)

Time --~ (e)

Time--,.

(f)

Time--,.

Figure 9.6 Sometypical kinetic rate-time profiles. (a) Rate increases in initial acceleration or settling period to reach a moreor less steady value. (b) Rate increases in an initial settling period to reach a maximum and then decreases. (c) Noinitial settling period and rate decreases rapidly frombeginning.(d) Rate rises very rapidly to a maximum value and then decreases rapidly. (e) Nosettling period and rate decreases slowly. (f) Nosettling period and rate remainsconstant. (After Ref. 7.) Ziegler-Natta polymerizations in that their particular shape maybe characteristic of a particular catalyst or catalyst-monomersystem. In general, such profiles can be considered to consist of three periods, viz., an acceleration period, a stationary period, and a decay period. Somecatalyst systems show all three types. Type (a) behavior is shown by many first generation catalyst systems, e.g., a-TiCl3, VCI:3, etc., when used with dialkylaluminum halides as co-

CoordinafionAddifion Polymerization catalysts for the polymerization of propylene in hydrocarbon media. These polymerizations show an acceleration period during which the rate increases to reach a more or less steady value. The duration of this acceleration period is of the order of 20-60 minutes for manypropylene polymerizations when carried out at 1 atm pressure in the temperature range 50-70°C. The nature of this acceleration period was a subject of debate for a long time. Early research by Natta and Pasquon [12] using a-TiCl3 catalysts established that breakdown of the ot-TiCl3 matrix took place due to the mechanical pressure of the growingpolymer chains in the early stages of the polymerization, thus exposing fresh Ti atoms and leading to the formation of new active centers with consequent increase in the polymerization rate. After breakdownof the ot-TiCla matrix to primary crystallites has taken place, a steady rate is observed for a significant period of time. Such attainment of a steady-rate behavior permits accurate measurementof polymerization kinetics. Type (b) behavior, in which the rate of polymerization increases in the acceleration period to reach a maximumand then decreases, is often observed when trialkylaluminum compoundusually produces a more active but less stable catalyst. Type (c) or type (d) behavior in which the rate may either start at a maximumvalue or rise very rapidly to a maximumvalue and then decrease rapidly with time is exhibited by manyhigh-activity supported catalyst systems, e.g., MgC12/ethylbenzoate/TiCl4-A1Et3system when used for either ethylene or propylene polymerization. Type (c) behavior is also shown by many homogeneouscatalyst systems e.g., Cp2TiEtC1-A1EtC12 in ethylene polymerization (Cp = cyclopentadiene) and by modified catalyst systems, such as VOC12.2THF-AIRain vinyl chloride polymerization. Type (e) behavior is attributed to almost instantaneous breakdown of porous catalyst particles on treatment with the cocatalyst so that the acceleration or settlement period is practically eliminated. This behavior is shownby ether-treated highly porous catalysts in propylene polymerization [7]. The polymerization rate decreases very gradually with time and the catalyst system showsgood stability. Type (39 behavior, for which the polymerization rate shows no settling or adjustment period but whose rate remains constant with time, is not often found in practice. One example of such behavior is the polymerization of 4-methyl-l-pentene on MgC12-supportedcatalysts containing phthalate esters. Here the polymerization rate remains almost completely constant with time [13]. Diffusion control of the propagation reaction has also been postulated. Diffusion of the monomerthrough the formed polymer to the propagation centers may becomerate-determining at higher conversions. Even at

Chapter 9 low conversions, diffusion maybe rate-controlling at a low degree of agitation in heterogeneous systems. This has been substantiated in somesystems where the polymerization rate increases with increased rate of stirring [5].

Effect of CatalystParticle Size Natta and Pasquon [12] were the first to study the effect of the catalyst particle size on the rate of polymerization. For a constant concentration of the monomer, they found that the rate of polymerization changed with time. With ground TiO2 (particle size _< 2#) the rate of propylene polymerization quickly reached a maximumand then decreased gradually to an asymptotic stationary value. In the case of ungroundparticles (size up to 10/z), however, there were no maximabut the rate accelerated to approach the same asymptotic stationary value. The former behavior is referred to as the decay type, whereas the latter is knownas the build-up or acceleration type. Figure 9.7 shows typical rate curves of these types and the different zones, termed build-up, decay, and stationary periods, into which they can be classified.

--

I

Figure 9.7 Typical kinetic curves obtained during propylene polymerization by TiCla. (A: decay type; B: build-up or acceleration type; I: build-up period; II: decayperiod; III: stationary period.)

CoordinationAdditionPolymerization

7~9

As mentionedearlier, the acceleration type behavior is explained by an increase in surface due to breakupof catalyst particles subjected to mechanical pressure of growing polymer chains anchored to the catalyst active centers. The smaller the particle size, the greater the mechanical energy required for further size reduction, and so the particle size-and hence the specific surface area-would reach someasymptotic value. The stationary polymerization rate wouldcorrespondto this catalyst particle Size. Thedecay type behaviorobservedin somecases maybe due to active site destruction. This can be due to thermal deactivation or further reduction of the transition metal by the group I-III metal component.The decay type kinetics is explainedlater while discussing the mechanism of polymerization in terms of the deactivation of someof the active sites. Chain Termination Ziegler-Natta polymerizationshave the characteristics of living polymerization with regard to catalyst active sites but not individual propagating chains. Thusthe propagatingchains have lifetimes of secondsor minutesat most, while active sites havelifetimes of the order of hours or days. Each active site producesmanypolymermolecules. The termination of a polymer chain growingat an active center mayoccur by various reactions, as shown below with propylene as an example. 1. ChainTransfer to monomer: ¯ ---Ti-CH2--CH-~v~v

ktr,M

+ CH3CH=CH2 ~ *----Ti-CH2CH2CH3

CH3

+ CH2----C--wvw (9.9) I CH3

and ~---Ti-CH2--CH-~wv+ CH3CH--CH2 I CH3

ktr,M

-~ ~---Ti-CH=CH-CH3

+ CH3~CH--~ww (9.10) I CH3 where~---Ti represents the transition metal active center on the catalyst site at whichpropagationoccurs. 2. Chaintransfer to the GroupI-III metalalkyl : ~--Ti-CH2--CH--wwv CH3

+ Al(Cg.H5)3

~ ~Ti-CH2CH3

(C2H5}2AI-CH2--CH--,,w~ I CHa

(9.11)

Chapter 9

760 3. Spontaneous intramolecular r-hydride transfer: k, ¯ ---Ti-CH2-CH--ww ~ ~Ti-H + CH2= CH---wvw CH3

CH3

4. Chain transfer to an active hydrogencompoundsuch as molecular hydrogen (external agenO ~---Ti-CH2--C~ I CH3

+ Hg.

~

~---Ti-H

+ CH3--CH--,vww I CH3

(9.13) Noneof the above reactions terminates the kinetic chain. All are treated as chain transfer reactions since there is reinitiation of new propagating chains. The relative extents of the various termination reactions depend on the monomer,identity and concentrations of the initiator components, temperature, and other reaction conditions. There are considerable differences in the efficiencies of chain transfer to different Group I-III metal components; for example, diethylzinc is muchmore effective in chain transfer compared to triethylaluminum. Molecular hydrogen is a highly effective chain-transfer agent and is commonlyused for molecular weight control in the industrial production of polypropylene.

Problem9.2 Termination reactions are more complexthan described above. Thus polyethylene producedby Ziegler-Natta polymerization contains vinyl (CH2=CHR), vinylidene (CH2=CRR’),and trans-vinylene (RCH=CHR’) end groups in amounts of approximately60, 20, and 20%,respectively. Accountfor the formation of such groups. Answer: Vinyl groups are formed by chain transfer to monomerand ~-hydride transfer, that is, the equivalent of Eqs. (9.9)-(9.11). Theformationof vinylidene transvinylidene end groups is less clear. Onepossibility is participation of vinyl end groups in chain transfer analogous to Eq. (9.10). Transfer to the methyleneand methineprotons of vinyl end groups wouldgenerate trans-vinylene and vinylidene end groups, respectively.

KINETIC MODELS POLYMERIZATION

FOR

ZIEGLER-NATTA

The overall kinetic behavior of any catalytic polymerization process results from the overlap of three potentially time-dependent reactions, viz., the

CoordinationAddifion Polymerizafion

761

time dependence of the intrinsic catalyst activity, the time dependence of the overall polymerization reaction, and the time dependence of any diffusion process which mayalso operate. The kinetic behavior of Ziegler-Natta polymerization is no exception to these considerations. Furthermore, the heterogeneous Ziegler-Natta catalysis is complicated by adsorption reactions of monomer,of cocatalyst, of donors (internal and external), and of by-products produced in reactions leading to the formation of active centers. For these reasons the kinetics of Ziegler-Natta polymerization reactions are exceedingly complex. The variety of kinetics reported in the literature for Ziegler-Natta polymerizations is ample testimony to this fact. Early

Kinetic

Models

A numberof simple kinetic models [12,13] have been developed for catalysts that have relatively low activities and are characterized by kinetic rate-time profiles of the type shownin Fig. 9.6 (a to f). These models are based the assumption that the total concentration of active centers, C*, remains constant throughout the polymerization consisting of three steps: chain initiation, chain propagation, and chain transfer. ChainInitiation This step is assumed to be the insertion of the first monomermolecule, M, into a transition metal-carbon bond in an active center, Cat-R, resulting in the formation of a polymerization center Cat-P: Cat--R

+ M -~ Cat--P1

(9.14)

Chain Propagation The chain propagation step is regarded as the insertion of a monomer molecule into a transition metal-carbon bond in a polymerization center, Cat-P : Cat--Pl + M -~ Cat--P2 (9.15) CatmP2

+ M --~

Cat--P3

(9.16)

Cat--Pn + M -~ Cat--Pn+l (9.17) In this scheme, all polymerization centers are regarded as equally active and having the same propagation rate constant, kp, independent of their geometric location and of their degree of polymerization. However,since it is established that all centers are not equally active, the kp in this scheme must be regarded as an average. Chain Transfer Whereno chain transfer agent has been added to the polymerization system three transfer reactions [Eq. (9.9)-(9.12)] are usually considered, viz., chain

762

Chapter 9

transfer with monomer,chain transfer with alkylaluminum, and spontaneous chain transfer : Cat--Pn

+ M ~ Cat--R’

+ Pn

Cat--Pn

+ AIRz ~ Cat--R" + PnaA1Rz_l

(9.18) (9.19)

Cat--P,~ -~ Cat--R" + Pn (9.20) Chaintransferreactions withmonomer[Eq.(9.18)]and withalky]aluminum [Eq.(9.19)]are introduced intothe polymerization scheme because of thewellestablished observations’ thatthemolecular weightof the polymerproduced decreases as the concentrations of monomerand/or alky]aluminum compoundincreases. Thethreenewinitiation centers formed in reactions Eq.(9.18)-(9.20) havedifferent chemical structures. Thusforthepolymerization of propy]ene usingeither A]Et3or A]Et2C] as cocatalyst, R’ = r~-propy], R" = ethyl, and Rm = H. However, for mostcatalyst systemsandundernormalconditions of monomerandaikyla]uminum concentrations, spontaneous chaintransfer reactions [Eq. (9.20)] can be neglected.

Problem9.3 Usingthe kinetic schemegiven above, derive suitable expressions for the rate of polymerization (R~) and average degree of polymerization(DP,). Show howchain transfer constants can be evaluated from measurementof DP,~. Answer: At any instant there will be polymerizationcenters generally representedby Cat--P, (n = 1, 2, .--) and different initiation centers Cat-W , Cat-R" and Cat-R" [cf. Eqs. (9.18)-(9.20)]. Thusthe total concentrationof active centers, C*, is given the sumof the concentration of polymerizationcenters, G’~, and the concentrations of different initiation centers, Ci, at whichinitiation takes place at that particular moment: C* = C; + C~ -~ C~’ + C~" (P9.3.1) Understationary state conditions only the steady state with respect to the concentration of polymerizationcenters need be considered. A steady state thus implies the following conditions:

/ dt = o Z/z~initiation = Z R’transfer

(P9.3.2) (P9.3.3)

Thus for the above reaction scheme, k~’ C~’[M] = k,,,A C~[A]

q" [M]=

(P9.3.4) (P9.3.~) (P9.3.6)

CoordinationAddition Polymerization

763

where [M] and [A] are the concentrations of monomerand alkylaluminum, respectively; k~, k~’, and k~" are the rate constants for chain initiation involvingthe Cat-R’, Cat-R", and Cat-W"species, respectively [cf. Eq. (9.14)]. Combinationof Eqs. (P9.3.4)-(P9.3.6) gives C~ = 1 + kmM/k~ + kmA/k"[M] + k~/k~"[M] Hencethe rate of polymerization is /~ = ~ k, 6’* [Ml = [M] + ktr,M[MI//~+/~,,~/k~’+ ks//~" and the numberaverage degree of polymerization k, [MI D P,~ = ktr,U-t- ktr,A [A]q- ks

or,

(P9.3.7)

(P9.3.8)

(P9.3.9)

1

(P9.3.10) DP,~ kmu/kp + kmA [A]/kv [M] + ks/k v [U] Hencea plot of 1/~--~,~ vs. 1/[M] for polymersamplesprepared with constant [A] shouldbe a straight line havingknr,M/kvas the intercept and (kirk, as the slope. Similarly when 1/DP,~ is plotted vs. [A] for polymer samples preparedwith constant [M], a straight line should be obtained having(ktr.M/k~ ks/kp [MD as the intercept and ktr.z./k ~ [M] as the slope. Consequentlyall chain transfer constants can be evaluatedas ratios of chain transfer rate constants ktr.M, kt~~ and k~ to the propagation rate constant k~. Valueslisted in Table 9.4 are derived in this way.

Somecompounds, particularly hydrogen, are effective chain transfer agents [cf. Eq. (9.13)] that cause reduction of molecular weight. For this reason hydrogen is usually added in the commercial production of polyethylene and po!ypropylene. It is easy to modify Eq. (P9.3.10) include the effect of hydrogen by adding an extra term k, tr,H2 [H~]/kv [M] to the right side of Eq. (P9.3.10). It should be noted that the aluminum alkyls are known to be mainly dimeric in solution, existing in equilibrium with the monomericspecies: A2 ~ 2A [A] = K~/~ [A21Vz (9.21) where the brackets indicate concentrations and Kd is the dissociation equilibrium constant. Because of this dimerizafion, the amount of actual

764

Chapter 9

Table 9.4 Chain Transfer Constants (~,/kp) for Alkene Polymerization with Various Heterogeneous Ziegler-Natta Catalysts Monomer

Catalyst Temp. system (°C) (×10 -4) (×10 -4) Ethylene 6-TiCIa-AIEta 75-80 0.35-0.70 1.7 Ethylene "/-TiCIa-AIEt2CI 40 1.38 0.038 Propylene a-TiCl3-A1Eta 70 5.7-11.5 17-34 Propylene VCIa-AI(Bui)a 60 10 5.6 Source: Data from Refs. 14, 15, and 16.

-4) (×10 1.2-2.5 0.25

monomericA1R3is small. To obtain expressions in terms of dimeric aluminumalkyl concentration, [A] in Eqs. (P9.3.8)-(P9.3.10) must be replaced 1/9. It should be noted that the values of k, tr,A for A1Et3and by K)/2[Az] A1Et2CI listed in Table 9.4 are the effective values and include the equilibriumconstant for dissociation of dimersinto monomeric species, i.e., (]¢¢r,A)real

Problem 9.4 Derive limiting the case where [M] is high.

expressions

= " --/ ~ 1/2 ICtr,A ~I

from Eqs. (P9.3.8)

(9.29.)

and (P9.3.9)

Answer: When[M] is high, Eqs. (’P9.3.7)-(P9.3.9) C~, = 1 q- ktr,M/k’ k~ C* [M] DP,,

= k,/kt,,M

approximate (P9.4.1) (P9.4.2) (P9.4.3)

Further simplification results when ’k~ ~ k~, >> ktr,M. Equations (P9.4.1) and (P9.4.2) then reduce to C~ ~ C* and R~ = kp C* [M]. Thus at high [M], becomes proportional to [M] and DP,~ becomes independent of all polymerization variables.

Adsorption Models As early as 1956Erich andMark[17] pointed out that since most ZieglerNatta catalyst systems were heterogeneousin nature, it wasmost likely

Coordination Addition Polymerization

765

that adsorption reactions were involved in such polymerizations. Since then adsorption processes have featured in many kinetic schemes. A number of reaction schemes have been proposed based on the assumption that the polymerization centers are the adsorbed metal alkyl species. It is assumed that monomer and metal alkyl are reversibly adsorbed on to the surface of a crystalline transition metal halide and that chain propagation occurs between the adsorbed metal alkyl and monomer. In this regaxd the Rideal rate law and the Langmuir-Hinshelwood rate law for adsorption and reaction ’ on solids assume importance (see Problem 9.5).

Problem9.5 Considering reaction between A and B catalyzed by a solid there are two possible mechanismsby which this reaction could occur. The first is that one of them, say A, gets adsorbed on the solid surface and the adsorbed A then reacts chemically with the other component B which is in the gas phase or in solution and is not adsorbed on the surface. The second mechanismis that both A and B are adsorbed, and the adsorbed species undergo chemical reaction on the surface. The reaction rate expression derived for the former mechanism is the Rideai rate law and that for the second mechanism is the Langmuir-Hinshelwood rate law. Obtain simple derivations of these two rate laws. Aaswer: Both the Rideal and Langmuir-Hinshelwood rate laws are based upon the Langmuir adsorption equation, which is applicable for gas-solid as well as liquid-solid systems where diffusion of the sorbate to the solid surface is not rate limiting (generally true). The basic assumption of the Langmuir adsorption is that adsorption occurs at adsorption sites and all these sites are equivalent. For gas-solid systems, the rate of adsorption, r~, of the gas A is proportional to the gas pressure, PA, and the numberof vacant sites, i.e., ~’a = kapA (rl,~ -- hA)

(P9.5.1)

where no is the total number of adsorption sites and nA is the number of sites which are occupied by molecules of A. The rate of desorption, on the other hand, is postulated to be ra A

= kan

(P9.5.2)

At equilibrium these rates are equal and one obtains kapA/ka nA = no (1 + kapA/kd) Defining the fraction of adsorption sites covered by A as 0~ = hA~noand the equilibrium constant for the adsorption equilibrium as KA = ka/ka, the above equation reduces to KAPA OA A ---- 1 +KAp

(P9.5.4)

766

Chapter 9

The term 0A in Eq. (P9.5.4) represents the fraction of total adsorption sites occupied by A. If there are two kinds of molecules, A and B, which are competing for the adsorption sites, one modifies Eq. (P9.5.4)

OA =

KAPA 1 + KA B ÷ KBp

(P9.5.5)

KB PB 1 + KAp A + B KBp where PA and PB are the partial pressures of A and B. ff there is a gas molecule A2 which is adsorbed in the dissociated Eq. (P9.5.4) is modified OB =

(P9.5.6) form, A, (P9.5.7)

OA = KA (Kd PA~) 1/2 1 + KA(K,ipA~)a/2

where K~ is the dissociation equil~rium constant of Equations (P9.5.4)-(Pg.5.7) are applicable for gas-solid adsorption. For liquidsolid adsorption, the partial pressures PA, PB, and PA~ in these equations are replaced by concentrations [A], [B], and [A2], respectively. (a) If the reaction takes place between the adsorbed gas A and the other component B in the gas phase, the rate of reaction is given as RA~ = ks[S]0ar ~ where ks is the surface reaction rate constant and IS] is the concentration of adsorption sites. Substituting Eq. (P9.5.4) into Eq. (P9.5.8) gives Rideal rate law as PAPS RAB = ksKA[S] 1 + K (P9.5.9) ApA For liquid-solid

systems, the corresponding rate law is [A][13] RAB = ks KA IS] 1 + KA [A]

(I’9.5.10)

(b) If bothA and B are adsorbed beforethe chemical reaction occurs, the rate ofreaction is givenas RAB = ks 0A 0S IS] (P9.5.11) Substituting for 0A and 0s from Eqs. (P9.5.5) and (P9.5.5) one obtains Langmuir-Hinshelwood rate law [18] as k, K KB [S] PAPS RAB = (1 +KApaA +KBPB)I/2 (P9.5.12) The corresponding equation for liquid-solid KA Ks [S] [A] [B] 2RAB --¯ ks (1 +KA[A] +Ks[B])

systems is then (P9.5.13)

CoordinafionAddition Polymerization

767

The situation for stereoregular polymerization is quite similar to the cases discussed in Problem9.5, if it is postulated that the dimeric alkylaluminum molecules are adsorbed on TIC13sites to give rise to polymerization centers by the following equilibrium process: (A1R3)2 -[- Active site ~ Polymerization center (PC) (9.23) One is now in a position to apply Eqs. (P9.5.9) and (P9.5.10) stereoregular polymerization. The rate so obtained will be applicable in the stationary zone because steady-state conditions are assumedin deriving Eqs. (P9.5.9) and (P9.5.10). Equation (P9.5.9) is applicable when the PC with the monomermolecules which exist in the mediumof the reaction mass. In this case one has -~ool -- ksx [S] 0AI2R6 [M] (9.24) where [M] is the monomerconcentration in the bulk of the reaction mass and /~1 is the rate under the assumption. Substituting for 0AI2R~ by comparison with Eq. (P9.5.4) one obtains KA [A2] P~ol = ksl [S] [M] 1 + KA [A2]

(9.25)

where KAis the adsorption equilibrium constant and [A2] the concentration of the alkylaluminum dimer. If Henry’s law is assumed for the dissolution of the gaseous monomer in the solution, then

[M] = HpM whereHis the Henry constant and PM is the partial

(9.26) pressure of the

monomer. Equation (9.25) then becomes

K.~[A~] Rool = ks1 [S] HpM1 + KA[A2]

(9.27)

Equations (9.25) and (9.27) are the Rideal rate laws for the Ziegler-Natta polymerization. If the polymerization centers react with the adsorbed monomermolecules then the Langmuir-Hinshelwood rate equation [Eq. (P9.5.13)] should used and one would obtain the rate expression in the stationary zone as KA KM[S] [A2] [M] 2Roo2 = (1 ks2+KA[A2] +KM[M])

(9.28)

where KMis the adsorption equilibrium constant for the monomer. Equations (9.27) and (9.28) have been experimentally verified and values of KMand KAdetermined. Someresults [5,19] are shown in Table 9.5.

768

Chapter 9

Table 9.5 Experimental Values of Adsorption Constants in Langmuir-H_inshelwood and Rideal Rate Laws Temp. Catalyst system (°C) TiCI4 + AIEt3 32 44 57

K A (L/mol) 280 170 60

K M (L/too 0 -

TiCIa + AIEta 50 21.2 VCI3 + AIEta 40 40-60 R: Rideal; L-H: Langmuir-Hinshelwood. Source: Data from Refs. 5 and 19.

0.163 6-7

Rate law used R-type R-type R-type L-H type L-H type

For polymerization in the presence of donor-type impurities such as COS, CS2, H20, H~S, a general rate equation of the following type has been used [20]:

kp KMKAIs] [A2] [M]

~ /~

= (1

q-

KA[A2]

q-

KM[M] q-

KD [D])

(9.29)

where KD is the equilibrium sorption constant and [D] the concentration of donor impurities. Equation (9.29) is identical with Eq. (9.28) except the term KD[D] which is added to the denominator to take into account the donor molecules competing with other species for the sorption sites.

Problem 9.6 Show that under conditions

where KM << KA [A2] (see Table 9.5) the Langmuir-Hinshelwood rate equation becomes indistinguishable from the Rideal expression for the stationary zone rate. Answer: For KM[M] << KA [As], Eq. (9.28) can be simplified R~2 "" ks2 KA [S] [As] [M] 2(1KM + KA[A2])

(P9.6.1)

R~o2-----

(P9.6.2)

or

ks~ KAKMIS] [A2] [M] 1 + 2KA [Az]

where Ki [A~]2 is neglected. If Henry’s law is valid for the dissolution of monomer, Eq. (P9.5.2) becomes R~

~

[S] H KMpM1 ÷ K’ [A2]

which is seen to be of the same form as the Rideal expression, Eq. (9.27).

769

~oordina~ion ~ddi~ion Polymerization

Problem 9.7 Under conditions of high metal alkyl concentrations the propagation rate in Ziegler-Natta polymerization of propylene is found [21] to be given by /~ = k,[M][S] where [S] is the concentration of adsorption sites and is proportional to catalyst weight; [M] is the monomerconcentration and k~ is the rate constant for propagation. At low metal alkyl concentrations, the rate is found to becomedependent on the metal alkyl content. Suggest a reaction scheme to explain these results using Langmuir-Hinshelwood adsorption. Answer: Consider monomeric metal alkyl A and monomer M to be competitors for the adsorption sites S on the catalyst surface. Only a portion of the surface may be covered with either metal alkyl or monomerand adsorption-desorption equil~ria exist : S + A #- S---A (P9.7.1) S + M #- S---M (P9.7.2) where the broken line indicates attachment to the surface site. Considering this as a Langmuir-Hinshelwoodadsorption, the fraction of surface sites covered with monomeric metal alkyl is KA[A] 0A = 1 + KM [M] + KA [A] The fraction

(P9.7.3)

covered with monomeris

KM [M] 0M = 1 + K M[M] +KA[A]

(P9.7.4)

If use is made of Eq. (9.21), then OA =

KA K~/2 [A211/2 1 + KM[M] + KA K~/~ [A2]

(P9.7.5)

These equations are statements of the essentially equilibrium nature of the adsorption. At equilibrium the concentration of the adsorbed monoalkyl is [S]0 A with an analogous expression for M. Assumethat the propagation occurs by reaction of monomer(whether adsorbed or not) with the activated adsorbed alkyl complex: S---A S---AM

+ M ~ + M ----*

S---AM S---AM2,

etc.

The species AM,~is also an aluminum alkyl, differing length of the alkyl group. Thus the propagation rate is /~ A = kp[M] [S]0 kr gA Z~/2 1/2 [M] [S] [A2] = 1 + KM [M] + KA KJ/~ [A2]t/2

(P9.7.6) from A only in the

(P9.7.7)

770

Chapter 9

If -~M [M] is small and KAK~/2[A2]I/2 large compared to unity (i.e., approaches 1), the propagation rates become that found experimentally, /~ = kp [M] IS]

(P9.7.8)

According to Eq. (P9.7.7), at very high monomerconcentrations the propagation rate should tend toward independence of monomerconcentrations; similarly, at low alkyl concentrations it should becomedependent on the metal alkyi content.

Kei et al. [22] have used Langmuir-type adsorption to explain the observed rate behavior both during the initial stage (build-up period) and in the stationary state of the polymerization of propylene with TiC13-AIEh which exhibits a decay-type behavior (curve A in Fig. 9.9) (see Problem 9.8). Similarly the observed rate behavior in the build-up period of the acceleration-type curve (curve B in Fig. 9.9) can be explained.

Problem9.8 Kei et al. [22] have investigated the kinetics of propylene polymerization with TiCI3-AIEta in the temperature range (30-70°C), the pressure range of propylene (100-700 mmHg), and the concentration range of catalysts (2 g TiCl~liter, [A]/[Ti] = 0.4-3.0). The polymerization exhibits a decay-type rate profile (curve A in Fig. 9.9) with the polymerization rate /~ at first increasing rapidly with the polymerization time t (stage I) and then gradually decreasing the stationary value Rco (stage IF). The polymerization rate in stage I in the case whenpropylene is introduced after the addition of (A1Et3)2 is found to agree with the rate expression Rr = klP~

K [A~] 1 + K[A2] t

while the stationary

(P9.S.1)

rate R~ shows a Langmuir-type dependence expressed by

R~ = k~ P K [A2] / (1 + K [A2])

(P9.8.2)

Rationalize the above kinetic behavior assuming Langmuir-type adsorption and formation of polymerization centers from Al2Et6 (represented by A2) and propylene monomeron the surface of TiClz. Answer: Stage I. Stage I. may be considered as the period during which the polymerization center is established on the surface of titanium trichloride from Al~Et6 and the olefin monomer. AI~Et6 adsorbs reversely on the surface S to form a surface complex C. Assuming that the adsorption equilibrium is of the Langmuir type [cf. Eq.(P9.5.4)],

[c], = Is] 1_K +[___A2_] K .’~

(P9.S.31

771

Coordination Addition Polymerization

where A2 is AI2Et~ (dimer). The polymerization center C* may be formed irrevers~ly by the attack on the surface complex C by a propylene monomerM from the solution. Then, the formation rate of the center can be expressed by dt which yields

-- k’, [M] ([C], - [C*])

[c’] = [cle (1 - ~-~,~’~’)

(P9.S.4)

(~9.s.5)

Note that the formation of the polymerization center [Eq. (P9.8.4)] is assumed to be irreversible and that of the surface complex [Eq. (P9.8.3)] to be reversible because the polymerization center is more stable than the surface complex. Initially, Eq. (P9.8.5) can be approximated [C’] = k’p IS] [C], t (Pg.S.6) If the polymerization occurs by the Rideal mechanism, that is, the rate determining step of the polymerization is the attack of a propylene monomerfrom solution onto the polymerization center, the rate P’w can be given by g = k’p IS] [C "] (Pg.S.7) CombiningEqs. (P9.8.3), (P9.8.6) and (P9.8.7), the initial polymerization is represented by

t ’’ [U?IS] 1 +K[A2] g = ~’p~1 K[A=]

Using Henry’s law, [M] = liP, can also be written as g = ~1 ~,=

where li is Henry’s constant,

~ [A=] t 1 + ~ [A~]

(Pg.S.S) Eq. (P9.8.8) (P9.S.9)

where ~ = k~ ~ li= IS]. Stage II. The maximumpolymerization rate in the stationary state is obtained from Eq. (P9.8.7) by substituting, successively, Eq. (P9.8.5) at t ~ to to replace [C*] then Eq. (P9.8.3) to replace [C]~. This yields ~ [A2] = ~’, IS] IS] 1 + ~ [A=] ZC[A~] = ~ e 1 + ~ [A2]

(Pg.S.10)

where k, = ~;li IS] The value of R~ would, however, be lower as compared to /~max due to the reduction in surface site concentration [S] caused by deactivation.

772

Chapter 9

Problem9.9 The kinetic rate behavior of Ziegler-Natta polymerization of ethylene and propylene is given by the empirical equation dt which applies to the build-up period of acceleration-type kinetics (curve B in Fig. 9.9). Showthat this equation can be derived on the basis of a reaction schemewhich assumes that (a) a potential polymerization cenfer is generated when adsorption of an aluminumethylate dimer (AIEta)9. occurs on an active site; (b) the reaction of the first monomermolecule with this potential polymerization center generates a polymerization center; and (c) a polymerization center undergoes propagation reaction with monomermolecules, the polymerization center being regenerated after monomeraddition in each propagation step. Answer: The reaction

scheme can be represented

as:

S + (AIEts)2 ~ ~ S*+ M -~ C

(P9.9.1)

C*+

(P9.9.3)

M ~

(P9.9.2)

C*

where S represents a surface site (TiCI3) for adsorption, S* a potential polymerization center, and C*, a polymerization center. Note that S* is kinetically distinguished from C* in its reaction with monomerM. Reaction in Eq. (P9.9.3) is the propagation reaction. Let [C*]oo be the total concentration of polymerization centers present at t = cx~, i.e., at the stationary state. In the early stages of the reaction, [C*] < [C*]~ and at any time the following mass balance holds (P9.9.4)

IS] + [S*] + [C*] = [S]0

where IS]0 is the total concentration of the active sites at ~ = 0. At the stationarj state, Eq.(P9.9.4) becomes [S]~ + [S*]o~ + [C*]~o = IS]0

(P9.9.5)

Subtracting Eq. (P9.9.5) fi’om Eq. (P9.9.4) one obtains ([S]oo-

IS]) + ([S*]oo- IS*]) + ([C*]~o- [C*])

(P9.9.6)

Since S* is an intermediate entity in the formation of the polymerization center, [S*]~ ----- 0. This follows from the reasoning that after a sufficiently long interval of time, monomermolecules would have reacted completely with all the potential polymerization centers by the irreversible reaction in Eq. (P9.9.2). Moreover, since both [S] and [S]¢~ are large numbers and it may be assumed that only a few of the active sites participate in polymerization, [S]~ - [S] ----- 0. Therefore, Eq. (P9.9.6) reduces [C% - [C]

-[S*]

~--

0

or,

[S ~] = [C*]~o

- [C*]

(P9.9.7)

CoordinationAddition Polymerization The rate of formationof polymerizationcenters [cf. Eq. (P9.9.2)] is given d[C*]/dt = kc [M] [S*I Substituting for [S*] fromEq. (P9.9.7),

(P9.9.8)

(e9.9.9) d[C*]/dt = kc [M] ([C’loo - [C’]) The rates of polymerizationat time t and in the stationary period are given by "] = kp [M] [C P~ (P9.9.10) (R~)~o = kp [M] [C*]~ (P9.9.11) Since [M] is constant, multiplying Eq. (P9.9.9) by kp[M]gives d ~ (kp [M] [C’]) = kc [M] (k~ [M] [C*]~ - k~ Or~

dl~/dt

= k (noo

- t~)

(P9.9.12)

where k = k~ [M].

Models

Involving

Monometallic

Mechanism

Most authors consider that the initiation and propagation of chain growth in the Ziegler-Natta polymerization occur by the same type of mechanism, while there is considerable amountof evidence that the rates of these two processes are very different, the rate of chain initiation being muchlower than that of chain propagation [23]. Thus two types of active sites are envisaged as existing in these polymerization systems, viz., polymerization centers and potential polymerization centers (cf. Problems9.8 and 9.9). The formation of polymerization centers is, however, more likely to involve alkylated transition metal halides as proposed by Cossee [23] and not metal alkyl molecules merely adsorbed onto the surface. Thus, models built on the assumption that alkylated transition metal entities form the polymerization centers have been found to give good agreement for several polymerization systems. The main proposals of one such model proposed by Burfield et al. [24] for the polymerization system VCl3/A1R3/4-methyl-l-pentene are as follows : (a) Chain initiation arises from the following reactions: VC13 + A1R3 f-~ VC12R.A1R2C1

(9.30)

VC12R.A1R2C1 + A1R3 ~ VC12R + A12RsC1

(9.31)

VCI2R + M --~

(9.32)

VCI2P

where VC12Pis the potential polymerization center (C**). The reactions in Eqs. (9.30)-(9.32) are considered to occur with adsorbed rather unadsorbed metal alkyl and monomer.

~hap~er9 (b) Monomerand also alkylaluminum molecules are involved in competitive reversible adsorption reaction with polymerization centers, that is, there is excess alkylaluminum over and above the amount needed to activate the transition metal sites [cf. Eqs. (9.30) and (9.31)]. Thus, VCI2P

+ M ~ VCI2P.M

VCI2P Jr

A ~ VCI2P.A

(9.33) (9.34)

The adsorptions are described by Langmuir-Hinshelwoodisotherms, viz.,

KM[M] 0M = 1 + KM[M] + KA[A] OA =

KA [A]

1 + KM [M] Jr

KA [A]

(9.35)

(9.36)

It should be noted that where the metal alkyl is knownto be dimeric, K1/~[A]1/9 should be substituted for [A], where [A2] is the concentration of metal alkyl dimer and K is the dissociation constant. (c) Chain propagation is a two-stage process, involving initially the adsorption of monomeronto a polymerization center followed by insertion of the adsorbed monomerinto a transition metal-carbon bond, i.e., the process is monometallie: VCI2P + M --~

VCI2P

The overall rate equation, describing the steady-state period is thus given by

= kp0M[c*]

(9.37) polymerization

(9.38)

where kp is the propagation rate constant with respect to adsorbed monomer, and [C*] is the concentration of polymerization centers during this polymerization period. Substituting for 0Mfrom Eq. (9.35) yields kp KM[M] [C*] R,p -- 1 Jr KM[M] Jr KA[A]

(9.39)

(d) Chain transfer reactions take place between adsorbed alkylaluminum and the growing chain attached to the polymerization center and also between adsorbed monomer and the growing polymer chain. The rates of chain transfer with adsorbed metal alkyl, -Rtr,A, and with adsorbed monomer, Rtr,M, are given by (9.40) Rt,,A = ~,,A 0A [C*] C*] Rtr, M = ]¢,tr,MOM[

(9.41)

Coordination AddRionPolymerization

775

(e) The number average degree of polymerization [24] is given for [C*] kvOM[C*] DP,= [c*] + S: hr, M0M + Sodthr,A0A[C*] Inverting, integrating, 1

(9.42)

and removing [C*] gives 1 q-- /¢,tr,M 0Mt -t- ]etr,

A 0A ~;

(9.43)

Substituting for OA,OMand simplifying yields 1

1 /~-,A K~, [A] + kv k~KM[M]t ] KM[M

K, [A] kpKM[M]t

{/~,-,M + 1/~) kp DP. (9.44) From Eq. (9.44) one can predict that DP,~will be: (a) initially dependent on the duration of polymerization, but becoming independent at a later stage of the polymerization, i.e., whent is large; (b) reduced increasing the metal alkyl concentration; (c) dependent on the monomer concentration; (d) independent of the metal halide concentration (since [C*I ~: [VCI3I). The above model not only predicts successfully the observed kinetics and molecular weight dependencies of the VCl3/Al(i-Bu)a/4-methyl-l-pentene system but also appears to have fairly general application to manyother Ziegler-Natta systems. Evaluation of the various kinetic parameters using Eq. (9.39) requires a determination of the concentration of polymerization centers [C*]. This quantity is usually determined from experiments in which the C* sites are quenched (made inactive) with CHaO3H,14CO, or 14CO2 [7,25]. Other methods include the use of number-average molecular weight (combined with polymer yield) and 14C-labeled Group I-III-metal alkyl component. Each of the techniques has limitations that require careful consideration if reliable results are to be obtained. Equation (9.39) has been found applicable to the polymerization systems characterized by kinetic rate-time profile of the type (a) in Fig. 9.6. Comparative values for some of the principal kinetic parameters derived in this way are listed in Table 9.6. The monometallic mechanism of Cossee and Arlman [10] for ZieglerNatta polymerization has found favor in the literature because it is based upon quantum mechanical considerations rather than on agreement with the kinetic data [5]. According to this mechanism,as described earlier and shownin Figs. 9.3 and 9.4, the initiation process involves interaction of aluminumalkyl with an octahedral ligand vacancy around Ti which results +

+

776

Chapter 9

Table 9.6 SomeComparative Values of Kinetic Parameters for the Polymerizationof 4-Methyl-l-pentenewith various Catalysts at 40°C a/~ x 10 K K A M -1) Catalyst system (L/tool) (L/mol) (rain 6-TiCla.0.33AlCla-AI(Bu’)a 16 0.38 1.29 MgClz/EBfriCl4-Al(B ui)a 30 1.5 4.26 VCI3-AI(Bui)a 1.5 0.16 3.10 Source:Data from Ref. 7. in alkylation of the Ti and regeneration of the ligand vacancy. This alkylated Ti with a ligand vacancy constitutes a polymerization center. The process can be schematically represented by ¯ Ti-D + AIEt3 ~ ¯C*--E] + AIEt2C1 (9.45) where C*--~ represents the’polymerization center. The propagation reaction occurs by adsorption of monomerMat the ligand vacancy of the polymerization center followed by monomerinsertion in the polymer chain (via a complex formation) and regeneration of the polymerization center with a ligand vacancy (see Fig. 9.4). The propagation reaction can thus be written schematically as Ik ¯ C*--~l + M(adsorbed) ~---- Monomer complex ~ oC*--12 (9.46) According to Cossee, in the above equation, the polymerization center reacts with the monomermolecules that are adsorbed on the catalyst. It is the adsorbed molecules that migrate on the catalyst surface toward the empty ligand to give rise to the monomercomplex. The monomer molecule in the activated complex is bonded between the alkyl group and the transition metal atoms. Finally, the alkyl group breaks off from the titanium ion giving a vacant ligand in the regenerated polymerization center. Problem9.10 Based on the mechanismof A.rlman and Cossee, derive a suitable expression for the rate of polymerizationand predict the rate behavior that would be expected according to this rate model. Answer: For the propagationreaction described by Eq. (9.46), the rate in the stationary period maybe written as R~ = k~[Monomer complex] (P9.10.1) Since the monomercomplex is in equih~orium with oC*-r-I and the adsorbed monomermolecules,

777

Coordination Addition Polymerization

[Monomer complex]

= K [*C*--E]]

OM surface

(P9.10.2)

where 0M is the fraction of the catalyst covered by the monomer molecules. Therefore, R~o = k~K[.C*-[3]0M = k~[.C*--D]0M (P9.10.3) where kp -- k~K and OMis given by the Langmuir equation [cf. Eq. (P9.5.4)]: (P9.10.4)

0M = KM [M]

1 + KM[M]

Substitution of Eq. (P9.10.4) in Eq. (P9.10.3) then gives KM [M] R~ = k~[,C*-t3]l + KM[M]

(P9.10.5)

The concentration of [,C*-13] can be evaluated by assuming an equilibrium of the initiation reaction in Eq. (9.45), i.e., [.C*-I-I] [AIEt2CI] (P9.10.6) KA = [.Ti-D] [AIEt3] From stoichiometry, one has " [.Ti-13] = [.Ti-U]o ~ [.C*-EI] (P9.10.7) (P9.10.8) [A1Et3] = [AlEtz]o- [.C*-13] (P9.10.9) [A1Et2CI] = [,C~-13] where the subscript o indicates the concentration at t = 0. Substituting these in Eq..(P9.10.6) gives [.C’-13] ~ = KA ([-Ti-E]]o - [-C’-[31) ([AIEt3]0 - [’C~-[3]) (Pg.10.10) For low values of [,C*--[~], one can use the approximation (P9.10.11) [AIEh] ~-- [A1Etz]o and hence

KA[~.t~]0

= [-Ti-E]]0-

[,C*]

(P9.10.12)

Two possible cases may now be considered: (a) WhenA [ AIEt3]o > > [ .Ti-13]0 a nd h ence KA [ AIEtz]0 > > [ .C*-13], t he l eft side of Eq. (P9.10.12) is close to zero and one has [.C*-[3] _~ [.Ti-13]0 and Eq. (P9.10.5) becomes .1~o1

=

kp [oTi-r~]o1KM + [M] KM [M]

(P9.10.13)

(P9.10.1~)

(b) When[~Ti--E1]~ >> KA[AIEt3]o, the term [~C*--E]] in the right side of Eq. (P9.10.12) can be neglected, and Eq. (P9.10.12) reduces

= -- KA[Ti-r~]0[AIEt=]0 [oC*-t3]

778

Chapter 9

or,

[.C*-E]] = [Ti-~]oI"[1]IKA,AIEt~,o/V2 [ [Ti--E]]o J

(Pg.lo.l~)

and hence, from Eq. (P9.10.5), / [°Ti-E]]0

J 1 + KM [M]J The adsorption constant KMin these equations has the same sense as in the modelsconsidered previously and it is small in value (see Tables 9.5 and 9.6). So one should observe that the rates R~ol and Roo~in Eqs. (P9.10.14) (P9.10.16) are proportional to the monomer concentrations. These equations also showthat the stationary rate wouldbe affected by the concentration of A1Et3 at its lower range (KA[AI_ Et~]0/[-Ti-13]0 << 1) and wouldbe independentat its higher range (KA[AlEtz]0/[-Ti--13]0 >> 1), which are in qualitative agreement with experimental observations. The above model based on the Cossee-Arlman mechanismdoes not, however, confirm precisely to all the kinetic features of Ziegler-Natta polymerizations.

Average

Degree

of polymerization

The average degree of polymerization at a given time can be found from the following general relation DP,~ =

Numberof monomermolecules polymerized in time t Numberof polymer molecules produced in time t

(9.47)

The number of polymer molecules polymerized can be obtained by integrating the rate of polymerization. The denominator can, however, be obtained only if the transfer and termination rates are known.If Rt denotes the sum of these rates, then DPn

=

f~ R~dt (9.48) [C*]t + fo* Rtdt where [C*]~ is the concentration of the polymerization centers at time t. One can apply Eq. (9.48) for the stationary state to find DPn.At the stationary state, P~ and -f~t arc both constant and Eq. (9.48) can thus written as 1 /~,~o [C*]~ (9.49)

DPn -

In Eq. (9.49), the contribution to the integrals from the transition zone has been ignored for simplicity. For long durations of time, the second term in right side of Eq. (9.48) tends to zero, and the equation reduces

(9.50)

CoordinationAddition Polymerization

779

To be able to evaluate DPn one has to know the termination and transfer processes. For propylene, for example, these can be described by Eqs. (9.9)-(9.12). The corresponding rates are given Transfer to monomer: Transfer to A1Etz: Spontaneous transfer:

Rtr,M = ktr,M [C*I[M] = a ,A[C*I[AI = [c*l

(9.51) (9.52)

Natta and Pasquon [12] have confirmed this equation experimentally. Transfer agents like H2 may be specifically added to reduce the’molecular weight. The polymers produced by Ziegler-Natta polymerization normally have very wide molecular weight distributions. The polydispersity index PDI (= Mw/M,~)is 5-20 for polyethylene and 5-15 for polypropylene. The cause of the wide dispersity is not precisely known. Someworkers believe that the propagation reaction becomesdiffusion controlled after a few percent conversion and it is this which is responsible for the large dispersity. Some other workers believe that the rate constants are dependent upon the molecular size. Concluding

Remarks

In this section several empirical rate expressions for Ziegler-Natta polymerizations have been presented and attempts to model the polymerization have been described. It is found that several models could be proposed to explain the same rate equations. Ziegler-Natta polymerization systems have been shown to be very complex, and the model of a fixed geometric center that has a defmableidentity and activity invariant with time is far too simplistic. Since most Ziegler-Natta catalyst systems have centers of widely different activities and geometric locations, only limited general agreement of such a model with the observed kinetic behavior or good agreement only in specific conditions could be expected.

SUPPORTED

METAL

OXIDE

CATALYSTS

Oxides of a variety of metals on finely divided inert support materials initiate polymerization of ethylene and other vinyl monomersby a mechanism that is assumedto be similar to that of heterogeneous Ziegler-Natta polymerization; that is, initiation probably occurs at active sites on the catalyst surface [2]. Unlike the traditional Ziegler-Natta two-componentcatalyst systems, the supported metal-oxide catalysts are essentially one-component systems. Amongthe metals that have been investigated for these catalyst

780

Chapter 9

systems are chromium, vanadium, molybdenum~nickel, cobalt, niobium, tantalum, tungsten and titanium. Typical supports include alumina, silica, and charcoal. The most active catalyst is chromiumoxide [7]. Silica (SiOg.) or aluminosilicates (mixed SiO2/Al203) are used as the support material. The sUppv~rt is sometimes modified with titania (TiOg.). The chromiumoxide (Cr IO3) catalyst was originally developed by Phillips Petroleum Company and is referred to as Phillips catalyst. Other metal oxide catalysts were developed primarily at Standard Oil of Indiana, the best known among them being the molybdenumoxide (MoV/O3)catalyst. Catalysts are prepared by one of two methods. The support material is impregnated with the metal ion, then heated in air at a high temperature to form the metal oxide. Alternatively, when the support material is an oxide such as alumina, the two oxides are coprecipitated and dried in air. In such case the catalyst is activated by treatment with a reducing agent such as hydrogen, metal hydride, or carbon monoxide. The function of the support appears to be more than simply providing a large surface area. Sometype of interaction must occur between the metal oxide and support because the oxide alone behaves differently. The supported chromium oxide catalysts can be prepared by impregnating a silica-alumina support with a solution of chromium ions or by coprecipitating the oxides. The preferred impregnating solutions contain dissolved Cr(NO3)3.9H20or CrO3 in nitric acid because catalysts made from chromiumchlorides or sulfates retain some of the anions after calcination. The solid mixture of chromium-silicon-aluminum compounds is calcined in dry air at 400-700°Cor higher to obtain the desired oxide. This probably results in the reaction of surface hydroxy groups in the support material with CrO3 to form chromate (IV) and dichromate (V) species:

o o O=Cr-O- Cr =O OH OH CrO3~, I ~ H2 0 -Si-O -Si

0/ N Ot I

-Si-O-

Si-

(iv)

~

+

~

0 0 t ~ -Si-O-Si-

(v)

The supported chromium oxide catalysts can be activated by carrying out the heat treatment of the .catalyst in a reducing atmosphere of CO, H2, or metal hydride or treatment with AiR.3 or AI(OP,)3. Poisoning the catalyst occurs in the presence of such materials as water, oxygen, or acetylene. Supported molybdena catalysts are prepared by impregnating alumina with ammonium molybdate, calcining in air at 500-600°C to form the oxide,

CoordinationAddition Polymerization and reducing in hydrogen at 430-550°C. Other reducing agents such as CO, SO2, or hydrocarbons can also be used, but hydrogen is preferred. The optimum catalyst contains from 5 to 25% molybdena dispersed on the surface of the support, and polymerization reactions are carried out at 130-325°Cin the presence of an inert solvent. The polymerizations on supported metal oxide catalysts can be carried out by three different processes: solution polymerization, suspension polymerization, and gas-phase polymerization. The solution polymerization occurs in solvents in which the polymer remains dissolved, e.g., cyclohexane. In a continuous process, solvent, monomer,and the catalyst suspension are fed into an agitation vessel at the same rate as the polymer solution is leaving the vessel. At high yields, the polymer can be precipitated out of the specific solvent and there is no need to remove the catalyst. The solvent and unreacted monomerare recycled. The suspension process is quite similar: One takes mixtures of hydrocarbons as solvents, in which the polymer is formed as an insoluble, filterable suspension. Gas phase polymerization may be described as another suspension process in which the monomer,without mixture of solvents, is polymerized directly out of the gas phase. The catalyst is continuously injected into a vertical reactor as gaseous monomeris circulated through the system. Only a small percent of the monomeris polymerized per pass. The polymerization temperature lies below the softening point, and the polymer exists in the reactor as a fine powder, which has the properties of a fluid with regard to its rheologicai behavior. Ethylene is the most important monomerused with supported metal oxide catalysts. In fact, muchof the high-density polyethylene is nowmanufactured this way. Unlike Ziegler-Natta catalysts, which give rise to polymer having primarily saturated end groups, the supported metal oxides yield polyethylene with approximately equal amounts of saturated and unsaturated chain ends. The supported metal oxides, however, are not as active as Ziegler-Natta catalysts, and they do not give rise to a high degree of stereoregularity. Propylene forms partially crystalline polymer, but higher 1-alkenes give amorphous product.

Polymerization

Mechanism

The difference in polymerization mechanism between one-component metal oxide catalysts and traditional Ziegler-Natta two-componentcatalysts seems to exist only in the initiation stage, while the mechanismof continued propagation of polymer chain has manycommonfeatures for all the catalyst systems based on transition metal compounds. Thus most studies of the chromiumoxide catalyst system, for example, deal either with the nature

782

Chapter 9

of the species on the catalyst surface or with the nature of the species responsible for polymerization. Such studies have shown that the formation of a surface chromate takes place by reaction of CrO3with silanol surface groups of the support, as shown by Eq. (9.55), and reduction of this surface chromate by ethylene or hydrogen or carbon monoxide results in the formation of a low-valence chromium center:

O/ \O I I -Si-O-Si-

CH2=CH2 + CO orH2 0"-~

o/Cr\o I

I

Jr

Oxidation products

-Si-O-Si-

The details of initiation mechanismare not understood. There is little doubt, however,that the reaction is a surface catalyzed process requiring the monomerto be adsorbed onto the catalyst surface. Initiation is believed to involve the formation of a metal-carbon ~-bond followed by coordination of an incoming monomermolecule and subsequent insertion into the metalcarbon bond. A mechanism proposed for the formation of the if-bond between the metal and alkyl fragment [26] is shownin Fig. 9.8. The radicals formed in the initiation reactions mayalso participate in the alkylation of the transition metal: M~= + R" ---,M~U

(9.56)

Twogeneral mechanismshave been proposed to explain the formation of polymers with precipitated catalysts: (a) the bound-ion-radical mechanism and (b) the bound-ion-coordinate mechanism. The bound-ion-radical mechanism involves chain growth in a chemisorbed layer of monomermolecules initiated by radicals or ion-radicals bound to the surface of the catalyst, while the coordinate mechanisminvolves chain growth from a complex ionic center in the catalyst. Bound-Ion-Radical

Mechanism

The catalyst is assumed to adsorb monomeron its surface, the first layer being held in a special and uniform fashion [Fig. 9.9(a)]. Initiation occurs when an adsorbed monomeris polarized sufficiently by some constituent on the catalyst surface to convert it to an ion (or a radical or an ion-radical pair) bound to the surface [Fig. 9.9(b)]. Propagation follows along

783

CoordinafionAddition Polymerization

M~I_ I /El

CH2=EH

M~CH= CH /H ¯ 22

ICH2 =CH2

~ 2

M~

CH=CH jr]

M~n

CH2= CH2_.E] M..~ ~CH2-CH 3

+

+ H’

+ CH2=CH"

I

CH2=CH

M~CH2CH3

CH2=CH" Figure 9.8 Mechanismfor the formation of g-bond between the metal and alkyi fragment. (After Ref. 26.) surface [Fig. 9.9(c)] and the polymer chain is eventually terminated and desorbed from the surface, being replaced by fresh monomer. The chain termination may be caused by transfer with monomer or spontaneous transfer or detachment from the surface. The surface layer lines up the monomersso that a polymer with molecular regularity is obtained. In bound-ion-radical mechanismfor ethylene polymerization, initiation can be either through a chemisorbed ethylene molecule (Fig. 9.9) a chemisorbed hydrogen atom. With ethylene as initiator, polymerization occurs simultaneously at two sites, each associated originally with the ends of the double bond in the ethylene molecule adsorbed on an active dual site. With hydrogenas initiator, polymer growth occurs only at one site. In either case, an organometallic bond is formed. The important steps for the polymerization process with the adsorbed hydrogeninitiation are illustrated in Fig. 9.10. Polymerization is initiated by an adsorbed hydrogen atom attaching to a neighboring adsorbed ethylene. Propagation occurs in the adsorbed layer, with the growing chain adding as an ion-radical to a neighboring adsorbed ethylene. Transfer occurs by shift of a hydrogen atom to a neighboring adsorbed monomeror spontaneously to a vacant surface site. In the absence of poisons, termination occurs only by reaction with a chemisorbed hydrogen,which frees a dual site suitable for readsorption of ethylene. With ethylene initiation, the growing polymer is attached at each end (Fig. 9.9) and is converted to single attachment by transfer or termination at one of the ends.

Chapter 9

(a)

(b)

(d)

(c)

Figure 9.9 Bound-ion-radicalmechanismfor polymerization on a catalyst surface showing(a) adsorbedmonomer; (b) initiation; (c), (d), etc., propagation. Ref. 27.)

k~p

Propagation ~i~. ~ Transfer with monomer

M ktr’.---~

Spontaneous transfer

~

Termination

Polymer CH2-CH

-H

+

Polymer

+ Po[ymer

Figure 9.10 Polymerization in adsorbed layrr by adsorbed hydrogen initiation. (After Ref. 28.)

785

Coordination Addition Polymerization

Problem9.11 Metal-oxide catalyzed polymerization of ethylene was carried out in benzene solution in a stirred autoclave with a suspension of hydrogen-reduced molybdena-alumina catalyst [27]. The pressure was maintained nearly constant by repressuring the autoclave with ethylene as it was consumedin the polymerization process. Temperatures of 200-275° were studied. The ethylene concentration in solution was controlled by adjusting the pressure (in the range 625 to 1000 psi) at any particular temperature. The ethylene uptake rate (rate of pressure drop dP/dt) was measured as a function of the catalyst amount (Wear) and ethylene concentration in solution (calculated from ethylene partial pressure) [C~H4]s different, tmperatures. The experimental data plotted as (dP/dt)/(weat vs. [C2H4]sproduce good fit to straight lines whoseslopes decrease at higher temperatures. Further, at higher temperatures, the plot of (dP/dt)/Wcatvs. IC~rhl, fits to straight lines passing through the origin. Derive suitable expressions to explain the aforesaid experimental results, considering that the polymerization takes place in an adsorbed layer of ethylene with initiation by adsorbed hydrogen and transfer and termination processes as illustrated in Fig. 9.10. Answer: It is seen from Fig. 9.10 that chemisorbed ethylene disappears in initiation, propagation, and transfer with monomer.Therefore, dP/dt = k~ [C2I-h],

[H], + (k~ + ktr,

M)[C2H4]~,[Polymer],

fP9.11.1) where [C2H4], is the concentration of ethylene in the concentration of adsorbed hydrogen and ~[Polymer]~ (bound to the surface). Under steady-state conditions, initiation and disappear by spontaneous transfer and state,

adsorbed layer, [H]~ is the that of all growing chains the growing chains arise termination. Thus at steady

/~ [H]= [C2H4]= = (ks + /~[H]=) E.[Polymcr]. or

E,~[Polymer].

= ki [H]. [C2H4]. / (k, + kt [H].)

(P9.11.2)

Substituting in Eq. (P9.11.1), dP/dt = k, [C2H4].[H]. 1 + k. ÷ k~ [H]. [C2H,].

(Pg.11Z)

At low concentrations of ethylene, [C2I-I4],, = KE [S,~] [C2I"I4]s (P9.11.4) where KE is the Langmuir adsorption equilibrium constant, [C~_H4]~is the equilfl> rium concentration of ethylene in solution, and [S~] is the concentration of active dual sites (at which ethylene can be adsorbed with two-point adsorption). [C2H4]~,

Chapfer 9 in turn, is related to the partial pressure of ethylene, PE, in the gas phase by [C2I-I4]s

= K, pE

(P9.11.5)

where Ks is the equilibrium constant for saturation. Substitution of Eq. (P9.11.4) into Eq. (P9.11.3) yields, (dP/dt)

(P9.11.6)

[C2H& which simplifies to

(dR~dr) -- KE k, [H]° + B [C2H4]s (P9.11.7) [S~] [C2H4]s where B is a complexfunction of the various equilibrium and rate constants. Since [S~] is proportional to the Weightof the solid catalyst, a plot of (dR~dr)/(Wcat [C2FI4]s) vs. [C~H4]s should yield a straight line at each temperature, as observed experimentally. Since termination and spontaneous transfer become more important at higher temperatures, the value of the slope B would decrease at higher temperatures. If B is small, Eq. (P9.11.7) simplifies (dP/dt)

[S ~] = KEk, [H] ~ [C~ H4]s (P9.11.8) Thus, a plot of (dP/dt)/Wcat vs. [C2I-I4], at higher temperatures should fit straight lines passing through the origin, as observed experimentally. Note: If ethylene initiation is considered, instead of hydrogeninitiation, the growing polymer is attached at both the ends and is converted to a single attachment by transfer or termination at one of the ends. The equations describing polymerization take exactly the same form as in the case of initiation by hydrogen but are complicated by extra terms dealing with interconversion of the polymer growing from one or both ends.

Problem9.12 Derive an expression for the average degree of polymerization corresponding to the reaction scheme assumed in Problem 9.11. Predict from this relation how the molecular weight of the polymer would be affected by (a) increased amount of catalys t, (b) increased amount of hydrogen adsorbed on the catalyst, (c) increased ethylene concentration, and (d) increased temperature. Answer: The average degree of polymerization is determined as the sum of all the chain growth reactions divided by all the chain transfer and termination reactions: DP~

=

k~ [C~H4]~ E,[Polymer]~ {ktr,M [C2H4]a + ks + kt [H]a} E,[Polymer],

The sum of the growing polymer chains

~,~[Polymer]~

cancels

(P9.12.1) out, and the

CoordinationAddition Polyraerization

787

reciprocal of the simplified DP~becomes -+ DP,~ kp /% [C2H4], Substituting Eq. (P9.11.4) into Eq. (P9.12.2), one obtains

(P9.12.2)

1 = ktr,___~M + k, + 1~ IS], (P9.12.3) DP,~ k, k, KE [S;] [C2H41, Thus the molecularweight should increase slightly with the increased amount of catalyst or increased ethylene concentration, and decrease with an increase in hydrogenadsorbed on the catalyst. (These predictions are in accord with the experimentalobservation.) Becausek~ and [C2I-I4]~ are muchlarger than k, and kt [H]~, especially at lower temperatures, Eq. (P9.12.3) simplifies 1 "~ ktr,M (P9.12.4) k~ DP,~ Expressing the rate constants in the Arrhenlusform, 1 AtoM [AE~ - AEtrM] (P9.12.5) DP,~ = ~exp --R~ ’ J Ordinarily theenergy ofactivation fortransfer withmonomer, AEtr,M, would be greater than the energy of activation for propagation, AEp;so the molecular weight should decrease with increasing temperature.

Bound-Ion-Coordination

Mechanism

The coordinate mechanism is based on earlier proposals that describe the organometallic growth reactions of ethylene with aluminum alkyls alone. The reaction is considered anionic because the negative end of the olefin coordinates with an organometallic complex in the surface. Olefm molecules are inserted one at a time between the metal ions in the complex and the alkyl chain to extend the chain by two carbon atoms (see Fig. 9.11). This mechanismis more satisfying in that the ion pair never becomes widely separated. Addition at an electron-deficient bond bridging the metals in the organometallic complex has also been proposed. In either case, the crucial step in the process is the addition of a monomer molecule held in a fixed orientation at the instant of reaction. This feature is responsible for the stereospecificity of the polymer and can also account for the high rate of reaction, since oriented sorption of the monomercan greatly reduce the activation energy necessary for the propagation step.

Chapter 9

788

~ (b)

(c)

--CH2CH

x

~

CHXCH2CHX

~

~

- - CH2 CHXCH 2 CHX CH2 CHX ~

Figure 9.11 Bound-ion-coordinationmechanismfor polymerization on a catalyst surface with growth from a single active site and replenishment of monomer from the liquid phase. Consecutivepropagation steps are represented in (a), (b), (c). (After Ref. 29.)

Problem 9.13 The coordinate and bound-ion-radical mechanismsalthough apparently quite dissimilar, have manyfeatures in common. If, in the bound-radical hypothesis, the surface involved decreases to the limiting case of three points of contact, the two mechanismswouldappear to be quite similar [28]. Explain this similarity by applyingthe idea of growthon the surface, used in the bound-radical hypothesis, to the coordinate mechanism,considering a surface with only three points of contact. Answer: The schemeis illustrated in Fig. 9.12. The growingpolymermoleculeis in the form of an organometalliccompound at position 1, and adsorbednext to it at positions

~oordin~onAcl~a’~ionPoly~neriza~ion

789

2 and 3 is an olefin molecule. The growing end of the polymer is transferred to the olefin molecule giving a new organometallic compound in position 3. The adsorption of a new olefin molecule in the free positions 1 and 2, followed by transfer of the organometallic compoundback from 3 to 1 in a mannersimilar to the original olefin addition step, then gives an organometalliccompound in the original position but two monomer units longer. This increase can continue back and forth along that portion of the surface that has the proper geometry.Similarity betweenthe surface-coordinate mechanismshownin Fig. 9.12 and the coordinate mechanism shownin Fig. 9.11 is easily seen.

ZIEGLER-NA’I’EA

COPOLYMERIZATION

Randomcopolymers of ethylene and ot-olefins (1-alkenes) can be obtained with Ziegler-Natta catalysts, the most important being those of ethylene and 1-butene (LLDPE) and ethylene with propylene (EPM or EPR EPDM).Representative reactivity ratios are presented in Table 9.7. It is seen from these values that ethylene is much more reactive than tiigher alkenes, and the ratios vary with the nature and physical state of the catalyst. In most instances, rlr2 is close to unity. HeterogeneousZiegler-

C--C C

Figure9.12Idea of growth on surface applied to the coordinate mechanismin the case of a surface consisting of three points of contact. (After Ref. 30.)

790

Chapter 9

Table 9.7 Representative Reactivity Ratios (r) in Ziegler-Natta Copolymerization Monomer 1 Monomer 2 Catalyst ~ Reaction type b rl Ethylene Propylene TiCIa/AIR3 15.72 H C 33_36 TiC14/AIR3 H 5.61 VCI3/AIR3 VCI4/AIR3 C 7.08 Ethylene

1-Butene

Propylene

1-Butene

VCla/AIRa VC14/AIR3

r2 0.110 0.032 0.145 0.088

H C

26.96 0.043 29.60 0.019

VCI3/A1Ra H VCI4/A1Ra C ’~R = C6H~.a;t’H = heterogeneous;C = colloidal. Source:Data from Ref. 2.

4.04 0.252 4_32 0.227

Natta catalysts generally yield a wide range of copolymer compositions, possibly because different active sites maygive rise to different reactivity ratios, or because of the encapsulation of active sites leading to decay of activity. A more homogeneous polymer composition is obtained with soluble Ziegler-Natta catalysts, particularly if monomer composition is carefully controlled to remain relatively constant during polymerization. Commercially important examples of ethylene-propylene copolymer made by this process are EPM(EPR),which contains about 60 parts ethylene to 40 parts propylene, and EPDMwhich is prepared with small amounts of nonconjugated diene to facilitate cross-linking. Typical dienes are ethylidene-norbornene, dicyclopentadiene, and 1,4-hexadiene. Cross-linking of EPM(EPR)is accomplished with peroxides. Both EPM(EPR)and EPDMelastomers have excellent ozone resistance by virtue of having no unsaturation as an integral part of the backbone. A number of block copolymers prepared with Ziegler-Natta catalysts have been reported; however, in most cases the compositions may include significant amounts of homopolymer. The Ziegler-Natta method appears to be inferior to anionic polymerization for synthesizing carefully tailored block copolymers. Nevertheless, bock copolymers of ethylene and propylene (Eastman Kodak’s Potyallomers) have been commercialized. Unlike the elastomeric random copolymers of ethylene and propylene, these are high-impact plastics exhibiting crystallinity characteristics of both isotactic polypropylene and linear polyethylene. They also contain homopolymersin addition to block copolymers.

CoordinationAddition Polymerization METALLOCENE-BASED

Z~_,GLER-NATI’A

7~I CATALYSTS

In contrast to the great successes of Ziegler-Natta catalysts in commercial production of linear polyethylene and isotactic polypropylene and higher cx-olefin polymers, efforts to achieve in-depth knowledgeof the catalysis have not been as successful and manyfundamental questions relating to this process have remained unanswered despite decades of intensive research. A central difficulty is that these catalysts are heterogeneousand function in a ternary gas-polymer-catalyst or liquid-polymer-catalyst system. Ziegler-Natta catalysts, in addition to being heterogeneous, with respect to the number of phases present, are also he, terogeneous with respect to the constitution of the active sites. Multiple sites, each having a different structure and reactivity, are often present and none may be considered to have been characterized completely [31]. More recently, however, homogeneousolefin polymer catalysts have been developed. Although these catalysts are by no means simpler, the fact that much of the chemistry of interest occurs in solution makes possible the application of powerful analytical methodssuch as nuclear magnetic resonance (NMR)spectroscopy and the catalysts can thus be related to the available enormous database of organometallic model compounds and reactions. The first homogeneousZiegler-Natta catalyst was discovered independently by Breslow[32] and Natta [33] in 1957. The catalyst, b/s(cyclopentadienyl)titanium dichloride (Cp2TiC12, Cp = r/S-cyclopentadienyl) activated with alkylaluminum chloride (AIR2CI) exhibited a low polymerization activity for ethylene (~104 g polyethylene/mol Ti-h-atm) and none for propylene. It was found later that small amountsof water increased significantly the activity of the catalyst. The reaction between water and aluminumalkyls was shown to produce alumoxanes. In 1980 Kaminsky and coworkers [34] used oligomeric methyl alumoxane (MAO)with Group IVB metallocene compoundsto obtain ethylene polymerization catalysts having extremely high activities. For instance, a polyethylene productivity of 9.3 x 106 g polyethylene/mol Ti-h-atm is obtained with Cpg.TiC12/MAOat 20°C and 9x107 g polyethylene/mol Zr-h-atm with Cp2ZrCI2/MAOat 70°C. However, these catalysts are non-stereospecific, producing only atactic polypropylene because of the symmetric feature of their active centers. In the early 1980s Brintzinger and coworkers [35,36] synthesized racemic ethylene-bridged bis (indenyl) zirconium dichloride, Et(Ind)xZrC12, racemic ethylene-bridged bt~(4,5,6,7-tetrahydroindenyl)zirconium dichloride, Et(H4Ind)2ZrC12, as well as their titanium analogues, Et(Ind)2TiC12 Et(H4Ind)2TiCl~, which have both meso-and racemic configurations. The Et(Ind)2ZrC12 and Et(H4Ind)2ZrCl2 catalysts activated with MAO catalyzed the stereospecific polymerization of propylene showinghigh productivities. It was the first time that the isotactic polyolefins were made by homoge-

792

Chapter 9

neous Ziegler-Natta polymerization. This finding was immenselysignificant as it demonstrated stereochemical control of the chiral ansa-indenyl ligands (Latin ansa, a handle) on migratory insertion of a vinyl monomer.In contrast, the meso-Et(Ind)2TiCl2/MAOsystem, as predicted, produced only atactic polypropylene. Since then a large group of ansa-metallocene compounds have been developed, each of them having unique catalytic activity and stereospecificlty. These homogeneousmetallocene-based catalysts are of theoretical significance in studies of Ziegler-Natta polymerization. Comparedto conventional heterogeneous Ziegler-Natta systems in which a variety of active centers with different structures and activities usually coexist, homogeneousmetallocene-based catalysts give very uniform catalytically active sites which possess controlled, well-defined ligand environments [37]. Consequently, the polymerization processes in homogeneoussystems are often more simple, and kinetic and mechanistic analyses for these systems are greatly simplified [38]. The metallocene catalysts have been under development for 20 years. Nowadays,there is no doubt that the breakthrough for a technical realization of these catalyst systems has been achieved. This is evident from the announcement of "single-site" catalysts (SSC) by different companies. the core of SSCtechnology are catalysts that permit olefins to react only at single sites on the catalyst molecules. This technology affords unprecedented control over reactivity, and can produce polymers with marked advantages in properties and process conditions. These catalysts are being used to produce tail0r-made high performance polyolefins. Metallocene catalysts now offer possibilities to create novel polymers which have never been produced by conventional Ziegler-Natta catalysts. The recently developed hybrid thermoplastic polyolefins which cover a broad range of products with almost any combination of stiffness/impact are a notable example [39].

Catalyst Composition The main component of homogeneous Ziegler-Natta catalyst systems, the catalyst precursor, is the Group IVB transition metallocenes (titanocenes, zirconocenes, and hafnocenes), which are characterized by two bulky cyclopentadienyl (Cp) or substituted cyclopentadienyl (Cpr) ligands. Twosimple examples of these metallocenes are shownin Fig. 9.13. These molecules have (~2v symmetry. The two Cp rings in the molecules are not parallel and the Cp2Mfragment is bent back with the centroid-metal-centroid angle (0) about 140° due to an interaction with the other two tr bonding ligands [401. The chiral ansa-metallocenes, that is, metallocenes with two Cpr ligands arranged in a chiral way and connected together with chemical bonds by

793

Coordination Addifion Polyraeriza~ion

CP2TiCt 2

CP2 2 Hf (CH3)

Figure9.13 Structures of two metallocenes with C’2~ symmetry. a bridging group were first synthesized by Brintzinger and coworkers [35]. The molecular structures of the two famousBrintzinger catalysts, Et(Ind)2ZrC12and Et(H4Ind)2ZrC12, havingindenyl (Ind) and tetrahydroindenyl (H4Ind) ligands connected with ethylene (Et) bridging groups, depicted in Fig. 9.14. A large numberof ansa-metallocenes have since been synthesized by changing the transition metals (Ti, Zr or Hf) and substituents on the Cprings, as well as the bridging groups. Among a wide variety of Cpligands investigated, the most commonlyused are methylcyclopentadienyl (MeCp),pentamethylcyclopentadienyl (MesCp),indenyl (Ind), tetrahydroindenyl (H4Ind), and fluorenyl (Flu) ligands, while commonly used bridging groups are ethylene (Et, -CHg.CH2-),dimethylsilene [Me2Si,(CH3)2Si=],isopropylidene [i-Pr, (CH3)2C=],and ethylidene (CH3Ch=).

Ct "" Zr--.~ Ct !

E’t(Ind)2 ZrCt2

Zrc Et t2 (H41nd)2

Figure9.14 Structures of Brintzinger catalysts [35].

794

Chapter 9

CH 3

/

CH3\

Pn

Al/ \

I

/

Linear CH~

CycUc

K,

(

Figure 9.15 Possible structures of MAO [41]. The steric interaction of the Cp type ligands surrounding the active metal center with incoming monomerplays a key role in the stereoselectivity of polymerization with metallocene catalysts. Changingthe steric structure of the ligands in the metallocenes leads to changes in steric structures of polyolefin products. The bridging group, which provides a stereorigid conformation for the complex, also dictates the distance between the transition metal atom and the ’Cp ligands and the bending angle 0, thus influencing catalyst activity and stereospecificity (isotactic, syndiotactic, and atactic). Poly(o~-olefins) of any type of stereospecificity can obtained simply by tailoring the stereorigid metallocene (catalyst precursor), basically according to the local symmetxy. While Group IVB transition metallocenes are the main component of homogeneousZiegler-Natta catalyst systems, the most important cocatalyst which activates them is MAO.Before the discovery of the MAO cocatalyst, the homogeneousZiegler-Natta catalyst Cp2TiCI2 was activated with alkylaluminum chloride which led to poor catalyst activity. The use of MAO cocatalyst raised the catalyst activity by several orders of magnitude. MAO is formed by hydrolysis of trimethylaluminum (TMA). As the water source, AI2(SO4)a hydrates are used in order to prevent contamination of MAO from other metal [41]. MAOia an oligomer with 6-20 [-O-AI(Me)-] repeat units. A higher degree of oligomerization of MAO provides a beneficial effect to the catalyst activity. The exact structure of MAO,however, remains a puzzle. While earlier research suggested that MAO might exist in a linear and/or a cyclic form (see Fig. 9.15), later investigation based on 27A1 NMRspectroscopic studies indicated that there appears to be no logical structure for MAO with

CoordinationAddition Polymerization

795

~z > 4, in which all aluminumatoms simultaneously achieve a coordination number of 4. A possible structure as proposed by Suganoet al [42] is shown in Fig. 9.16. The Active

Center

The true active species in metallocene-MAOsystems are believed to be metallocene alkyl cations, i.e., cationic do 14-electron complexesof the type [Cp2M(R)]+ (M = Ti, Zr, Hf). As shown below, the formation of the catalytically active .complexinvolves a series of reactions betweenmetallocenes and MAO [43]. For the halogen-containing metallocenes, a rapid alkylation of metallocene by MAO takes place first, and the active species arises from a methyl transfer reaction between the metallocene alkyls and MAO. Starting from Cp2ZrCI2,for example, the first step of the reactions after mixing the metallocene with MAOis the complexation and alkylation of the former with the latter: Cp2ZrC12 q-- [AI(CH3)-O]n ~ Cp2ZrCI2"[AI(CH3)-O]n CpzZr(CH3)CI AIn(CH3)n_IOnCI ~- Cp2Zr(CH3)CI’[AIn(CH3)n-a’OnCI] In a subsequent alkylation, Cp2Zr(CH3)CI

Cp2Zr(CH3)2is formed:

+ MAO~ CpzZr(CH3)2

+ [Aln(CH3)n-I-OnCI]

The alkylated metallocene, Cp2Zr(CH3)2,further reacts with MAO,forming a compound(A) that features the structural element Zr-O-AI:

CompoundA is believed to be the active’species in the metallocene/MAO systems. Since the Zr-O bond has a polar character and could be ionic in nature, compoundA possibly exists in two different states that are in

796

Chapter 9

equilibrium:

~ + ...O-(’Al-O’~n __ Cp~ZrLR

+ ~ I Cp2(R)Zr L At (CH3)O-

/O’~’At-O~’n I Cp~ Zr\R J

Themetallocenealkyl cation (at one side of the equilibrium) mightbe the true active center. A homogeneous catalyst can be madeas a "single-site," that is, only a single type of active center is present in a homogeneous system,underproper conditions. A "single-site" stereospecific catalyst [44] can producepolymers with sharp melting transitions (Tin) and markedlynarrowmolecularweight distribution (M~/Mn< 2). Polymerization

Mechanism

Therole of the active species in the polymerizationof ce-olefins by homogeneousZiegler-Natta catalysis is nowwell established. Thecationic metallocene alkyls havea strong tendencyto coordinate with olefin molecules, the latter being a weakLewisbase. Oncean olefin coordinatesto the metallocenealkyl, the insertion of the olefin into the alkyl-metalbondwouldthen proceedreadily. Thedriving force for this insertion is the energygained in the transformation of M-Rand M-(C=C)bonds into M-Cand C-R bonds (M= Ti, Zr or Hf; R = alkyl, C=Cis olefin). The insertion leads to the formation of a newdo alkyl complexwhichcan coordinate and then insert another olefin molecule.Successiveoccurrenceof this cycle leads eventually to a polymer. Figure 9.17 showssuch an insertion mechanismby Kaminsky and Steiger [45] for ethylene polymerization with Cp2ZrCI~/MAO.

Figure 9.16 A proposedstructure of MAO with coordination numberof 4 [42].

CoordinationAddition Polymerization

797

I 0 ~A~ ~

0 ~A~ ~

+ CH2-~-CH2

~

I

O--Al--

\ I..o~Zr--CH~.... Ate_

H--- CH--CH 2 [ CH2--Polymer chain

0 ~AI-~ n "~Zr___ClTl.~ I

.AI~

, ~ ’ --CH-,--Polymer CH2--CH2 " chain

Figure 9.17 Kaminsky’smodel for ethylene polymerization [45]. A monometallic mechanism was proposed by Corradini and Guerra [46] in which the active center is a metal-carbon bond and the propagation consists of two stages: the coordination of the olefin to the active site, followed by insertion into the metal-carbon bond through a c/s opening. The + complex. In addition to active species in this model is the cationic Cp2MR aromatic Cp ligands of the precursor metallocenes, an incoming monomer molecule and a growing polymer chain are also coordinated to the metal in the stage preceding monomerinsertion.

Kinetic Models Ewen’s Model The first kinetic model for propagation in homogeneoussystems was proposed by Ewen[47], assuming that the propagation took place as shown in Fig. 9.18. This scheme, shown for Cp2Ti(IV) polymerization of propylene, is representative of the kinetics for all of the polymerizations with Group IVBmetallocenes. In the scheme, species 1 and 4 represent coordinatively unsaturated Ti(IV) complexes that are-formally o 16-electron p seudotetrahedral species, species 2 represents the interacting catalyst/cocatalyst combination, while intermediate 3 is shown with the monomercoordinated

798

Chapter 9

km

CP2TiR2+Cxkc~ (2)

/

CH=CH~ (3)

C’: [-AI.(CH3)-O-]n ~ CP2Ti~ CH2?H_FI +C CH3 Figure 9.18 Ewen’skinetic model[47]. at a molecular orbital. The three non-Cp ligands occupy a commonequatorial plane with the growing chain held between two lateral coordination sites accommodating an unidentified non-Cp anion (Rt) and the monomer. Under pseudo-first-order conditions, the rate of polymerization can be expressed as P~ (9.57) ] ---- kobs[C3H6][MT][C where and

]gobs = kpKcKM/(1 + KM[C3H6] + Kc[C]) KM[C3H6] + Kc[C] << 1 The terms k and K in the above equations represent, respectively, the rate constant and the equilibrium constant. Ewen’smodel is a simplified model, which could explain the experimental results that showed polymerization rates vary linearly with the product of the monomer, metallocene, and alumoxane concentrations at low monomer conversions with [All over a certain range. Chlen’s

Model

Chien’s kinetic model [48,49], unlike Ewen’s model described above, is for the systems in which more than one active species is present. The model assumes the presence of multiple active center types, chain transfer to MAO, chain transfer by fl-H elimination (see p. 801), and first-order deactivation reactions of active centers. Chien applied the modelin the study of ethylene polymerization with Cp2ZrC19./MAO catalyst and propylene polymerization with Et(Ind)2ZrCI2/MAO and Et(H4Ind)2ZrCI2/MAO catalysts.

799

CoordinationAddition Polymerization

For a system with i types of active centers, the polymerization rate of the ith species is

-5,i = kp,dC;l[M]

(9.58)

where kp, i and [C~] are the propagation rate constant and the concentration of the ith active species, respectively. The overall rate of polymerization is obtained by summingEq. (9.58): Rp = E kp,i[C~][M]

(9.59)

or, if [G[] cannot be determined, Rp = kp,avg[G*][M]

(9.60)

where kp is the propagation rate constant and [G*] is the total concentration of active species. The total productivity P can be written as P = [M] ~. kp,i

f[c~i]dt

(9.61)

Assumingthat the catalytic species deactivate according to first order kinetics, the total productivity at time t is P(t) = [M] y~ kp,i[C;]o[1 - exp(-ka,it)] (9.62) the deactivation rate constant. The value of [C*]0 can be taken to be equal to the initial metal-polymer bond concentration, [MPB]0, which can be calculated via the relation between metal-polymer bond concentration, [MPB], and polymerization yield, where led, i is

[MPBIt = [MPB]o q- ktAr~yt/kp[Ml

(9.63)

where ktni r is the rate constant of chain transfer to MAO.Since [MPB] can be determined by the tritium radiolabelling method (by reacting the polymerization mixture with CH303H),[MPB]t can be plotted against Yt and extrapolated to ~ = 0 to obtain [MPB]0---- [C*]0. However, if the polymers produced with ith species can be separated from others (for example, by solvent extraction [49]), [C~]0 can also be obtained from the [MPB]--Yplot for the ith fraction. The propagation rate constants kp,i and ]~p,avg can be calculated according to Eqs. (9.58) and (9.60). In addition, kt~ values can be obtained from the slope of the [MPB]-Yplots. Somerate constant values for propylene polymerization with Et(H4Ind)2ZrCI2/MAO systems, reported by Chien [49], are listed in Table 9.8. Accordingto the data in Table 9.8, it was suggested [49] that the system might have two types active species (I and.II), species I producing n-hexane and n-heptane soluble fractions and species II producing n-pentane and ether-soluble fractions.

800 Table 9.8 Rate Constants for Et(H4Ind)2ZrCi2/MAO aPropylene Polymerizarion [A1]/[Zr]ratio 3500

-1 kp (M-s) 970 1840 1370

PP fraction T Cr C4

-1) k~ (s 0.015 0.015 0.026

8o

c5

0.003

E 130 0.0078 350 T 1480 0.047 C~ 2550 0.027 C~ 2590 0.041 0.0045 Cs 97 275 0.0027 E a[Zr] = 1.0/.tM, T~ = 30°C, [C3H6]= 0.47 M(Pp = 1.7 atm). T: total polymer; Cs: n-pentanesoluble,C~:n-hexanesoluble,C’i’: n-heptane soluble,E: ethersolublefractions. Source:aData fromRef. 49. Besides chain transfer to MAO,chain ti’ansfer by fl-H elimination is another main termination process. These two rates are expressed as follows :

=

(9.64)

Rztr = kff,[C*]

(9.65)

The number-average degree of polymerization is

DPn-- P’~

*] ~/~tr

kp[C*].[M]

(9.66)

-- ~[C*][MAO] + kt~r[C

or

Since kp and kt~r can be determined via the [MPB]--Yplot, the value of kt~r can be obtained from the variation of DP with [M], i.e., from the slope of (1lOP,) (1 /[M]) pl ot. Based on the analyses of experimental results, some researchers have suggested that there are at least two types of active species, which have different activity and stereospecificity, formed in the homogeneouscatalyst systems. Chien [49] indicated that in the case of the Et(H4Ind)2ZrCl2/MAO system, two types of active species (see Table 9.8) coexist in about equal amounts; one has higher selectivity, 10-20 times greater rate constant of propagation, and a factor of 5-15 times faster chain transfer to MAO than the second type of active species. Metallocene complexes with different

CoordinationAddition Polymerization

801

states of coordination with MAO may be responsible for the various active species. The kinetics are muchsimpler in homogeneousmetallocene-based catalyst systems, especially in base-free cationic catalyzed polymerizationsystems, than those in heterogeneous systems. The polymerizations with homogeneous metallocene catalysts are no doubt the best systems for kinetic study of Ziegler-Natta polymerization. The o~-olefin polymerization with these catalysts also offers a good opportunity to study the durability and deactivation of the catalysts, since the polymerization systems remain homogeneous over a considerable long reaction period [50]. Molecular

Weight

and

Chain

Transfer

The molecular weight of a polymer is a result of competition between chain growth and different chain termination reactions:

(9.68)

-~,~ = DP,~. Mo= (P~/ ~_, Rtr,,)

where Mois monomermolecular weight. Several types of transfer reactions which may occur in homogeneous systems terminating polymer chain growth are listed below. /3-Hydrogen Elimination. The metal center abstracts a H atom bonded to the /~-C of the growing polymer chain, forming an M-Hbond (M transition metal) and leaving a polymer with an unsaturated end: CH3

~H3 ) CH2---- CH--~v + Zr--H

Zr --CH ~--CH--vvvv

Chain Transfer by Monomer.r-Hydrogen elimination and monomerinsertion at the active center take place simultaneously without forming the M-H bond:

CH~ CH~CH ZZ~-CH~-CH~

/CH~

) Zr

I ~--~w H

"H~C’""CH

~ 3

CH ~ ---~--)CH~CH--vvvv+ Zr--CH~--CHf-CH~

802

C~apter 9

Chain Transfer to MAO.If MAO is used to activate catalyst, the growing polymer chain attached to an active center can exchange with the methyl group of~a MAOmolecule, forming the Al-terminated polymer chain and the M-CH3bond in the active center: CH 3 Zr--CH~-CH--Vvvv MAO

~H ] ) Al--CH~-CH--VVVV + Zr-CH

H 3C~ CH _vvvv

fl-CH3 Elimination. This chain transfer mechanismhas been discovered in the polymerization of propylene. The metal center abstracts a CH3group, instead of a H atom, at the fl-carbon of the growing chain, thus forming a M-CH3bond at the active center and leaving a polymer with an allyl end : CH 3

CH 3

CH 3

Zr--CH 2--CH-CH/-CH-v~vv----) Zr--CH 3 + CH 2=CH--CH2--CH~ Whereaswith Conventional heterogeneous Ziegler-Natta catalysts molecular weights of polymer from ethylene and propylene polymerizations are rather high, i.e., about 500,000, so that chain terminating agents like hydrogen have to be used, many systems with homogeneousmetallocene catalysts produce only low molecular weight polymers, especially in stereospecific polymerizations. Poor molecular weights for polymer products is thus a major shortcoming with metallocene-based catalysts, limiting them from being fully exploited for practical applications. However,technical breakthroughs have been achieved in that specially designed zirconocene catalysts have led to the production of polymers with molecular weights far above 100,000. A narrow molecular weight distribution (MWD)is a unique, feature of polymer__products of homogeneousmetallocene-based catalysts. Typical values of Mw/Mnare 1.6-2.4 for polyethylenes and 1.9-2.6 for polypropy-

CoordinationAddition Polymerization

803

lenes. The value of Mw/Mn near 2 is indicative of uniform active species in catalysts. Someresearchers have suggested that this is because the catalysts could be "single site," while others have explained that two or more kinds of species can have very similar values for the kp/~r ratio, and therefore polymers with narrow MWD would be obtained Concluding

Remarks

The discovery of homogeneousmetallocene-based Ziegler-Natta catalysts in the 19808 has opened up new and exciting possibilities to explore the mechanismof Ziegler-Natta catalysis and to fine-tune the stereochemical structure of polymer products. A particular significance of this development is that the stereospecificity of the metallocene catalysts can be influenced by controlling the substitution pattern on the Cp ligand. The metallocene catalysts for polyolefins are nowin the early stages of commercialization. The exploding development in this field is reflected by the sharp increase in the number of patents issued in the last few years. The development of metallocene catalysts, or more widely homogeneous catalysts, may be a dominant force in the polyolefin industry through the next two decades of the new millennium. ~

1. G. Natta and E Danusso(eds.), Stereoregular Polymersand Stereospecific Polymerizations, PergamonPress, NewYork (1967). 2. J. Boor, Jr., Ziegler-NattaCatalysts andPolymerizations,Academic Press, New York(1979). 3. D. O. Jordan, The Stereochemistry of Macromolecules,vol. I, A. D. Ketley (ed.), Marcel Dekker, NewYork (1967). 4. Y. Hu and J. C. W. Chien, J. Polym. Sci. Polym. Chem. Ed., 26, 2003 (1988). 5. T. Keii, Kinetics of Ziegler Natta Polymerizations, Halsted Press, NewYork (1973). 6. V. A. Zakharov,G. D. Bukatov, and Y. I. Yermakov,Adv. Polym. Sci., 51, 61 (1983). 7. P. J. T. Tait and N. D. Watkins,"Monoalkene Polymerization: Mechanisms," Chap. 2 in ComprehensivePolymer Science, Vol. 4 (G. C. Eastman, A. Ledwith, S. Russo, and P. Sigwalt, eds.), PergamonPress, Oxford(1989). 8. E Patat and H. Sinn, Angew.Chem., 70, 496 (1958). 9. G. Natta, J. Polym. Sci., 48, 219 (1960); Chim. Ind. (Milan), 42, 1207 (1960). 10. E. J. Arlmanand P. Cossee,J. CataL, 3, 99 (1964).

804

Chapter 9

11. P. Cossee, "The Mechanismof Ziegler-Natta Polymerization. II. QuantumChemical and Crystal-Chemical Aspects," Chap. 3 in The Stereochemistry of Macrolmolecules, Vol. 6 (A. D. Ketley, ed.), Marcel Dekker, NewYork (1967). 12. G. Natta and I. Pasquon, Advan. Catal., 11, 1 (1959). 13. Y. V. Kissin, lsospecific Polymerization of Olefins, Springer-Verlag, Berlin (1985). 14. E Erich and H. E Mark, J. Colloid Sci., 11,748 (1956). 15. V. A. Zakharov, N. B. Chumaevsky, G. B. Butakov, and Y. I. Yermakov, Kinet. Katal., 18, 848 (1977). 16. B. M. Grieveson, Makromol. Chem., 84, 93 (1965). 17. A. P. Firsov, V. I. Tsvetkova, and N. M. Chirkov, Dokl. Akad. Nauk SSSR, Ser I~im, 11, 1956 (1964). 18. J. H. Espenson, Chemical Kinetics and Reaction Mechanisms, McGraw-Hill, NewYork (1981). 19. K. Vesley, Pure Appl. Chem., 4, 407 (1962). 20. K. Vesley, J. Ambroz, R. Vilim, and O. Hamrick, J. Polym. Sci., 55, 25 (1961). 21. W. M. Saltman, J. Polym. Sci., 46, 375 (1960). 22. T. Keii, K. Soga, and N. Saiki, J. Polym. Sci., Part C, 16, 1507 (1967). 23. P. Cossee, TetrahedronLett., 17, 12 (1960). 24. D. R. Burfield, I. D. Mackenzie, and P. J. T. Tait, Polymer, 13, 302 (1972). 25. P. Vozka and J. Mejzlik, Makromol. Chem., 19, 589 (1990). 26. N. Yermakovand V. Zakharov, Adv. in Catalysis, 24, 173 (1975). 27. H. N. Friedlander and K. Oita, lnd. Eng. Chem., 49, 1885 (1957). 28. H. N. Friedlander, J. Polym. Sci., 38, 91 (1959). 29. G. Natta, Makromol. Chem., 16, 213 (1955). 30. H. N. Friedlander and W. Resnick, "Solid Polymers from Surface Catalysts," in Advances in Petroleum Chemistry and Refining, Vol. I (K. A. Kobe and J. J. McKetta, eds.), Interscience, NewYork (1958). 31. A. R. Siedle, W. M. Lamanna, R. A. Newmark,J. Stevens, D. E. Richardson, and M. Ryan, Makromol. Chem., Macromol. Syrup., 66, 215 (1993). 32. D. S. Breslow and N. R. Newburg, J. Am. Chem. Soc., 79, 5072 (1957). 33. G. Natta, P. Pino, G. Mazzanti, and R. Lanzo, Chim. Ind. (Milan), 39, 1032 (1957). 34. H. Sinn and W. Kaminsky, Adv. Organomet. Chem., 18, 99 (1980). 35. E R. W. P. Wild, M. Wasincionek, G. Huttner, and H. H. Brintzinger, J.. Organomet. Chem., 288, 63 (1985). 36. E R. W.P. Wild, L. Zsolnai, G. Huttner, and H. H. Brintzinger, J. Organomet. Chem., 232, 233 (1982). 37. P. J. T. Tait, Transition Metal and Organometallics as Catalysts for Olefin Polymerization (W. Kaminsky and H. Sinn, Eds.), p. 315, Springer, Berlin (1988). 38. J. Huangand G. L. Rempel, Prog. Polym. Sci., 20, 459 (1995).

~oordina~ion Addition Polymerization

805

39. A. Guyot, Makromol. Chem., Macromol. Symp., 66, 1 (1993). 40. J. W. Lauher and R. Hoffmann, J. Am. Chem. Soc., 98, 1729 (1976). 41. W. Kaminsky, H. Sinn, and R. Woldt, Makromol. Chem. Rapid Commun., 4, 417 (1983). 42. T. Sugano, K. Matsubara, T. Fujita, and T. Takahasi, J. Mol. Catal., 82, 93 (1993). 43. W. Kaminsky, A. Bark, and R. Steiger, J. MoLCatal., 74, 109 (1992). 44. J. C. W. Chien and M.-W. Tsai, MakromoLChem., Macromol. Syrup., 66, 141 (1993). 45. W. Kaminskyand R. Steiger, Polyhedron, 7(22/23), 2375 (1988). 46. P. Corradini and G. Guerra, Prog. Polym. Sci., 16, 239 (1991). 47. J. A. Ewen, J. Am. Chem. Soc., 106, 6355 (1984). 48. J. C. W. Chien and B. P. Wang, J. Polym. Sci., Polym. Chem. Ed., 28, 15 (1990). 49. J. C. W. Chien and R. Sugimoto, J. Polym. Sci., Polym. Chem. Ed., 29, 459 (1991). 50. J. Herwig and W. Kaminsky, Polym. Bull., 9, 464 (1983).

EXERCISES 9.1. Describe each of the following terms used to represent the structure of stereoregular polymers. Give example of a polymer having each structure: (a) isotactic, (b) syndiotactic, (c.) atactic, (d) diisotactic, and (e) disyndiotactic. 9.2. Account for the fact that atactic structures commonlypredominate in freeradical polymerization in contrast to ionic and Ziegler-Natta polymerizations. 9.3. Explain why it is possible to synthesize two diisotactic but only one syndiotactic polymer structure.

polymer structures,

9.4. What are the basic modes of addition of 1,3-dienes (CH2 =CR-CH=CH2) to a growing polymer chain ? Howmany stereochemical arrangements are possible for each mode ? 9.5. Howmanydifferent structures are possible, in principle, from isoprene polymerization ? Whydoes no 1,2-polymerization occur with this monomer? Mention some catalysts for stereospecific polymerization of isoprene. 9.6. Certain Ziegler-Natta polymerization systems showing stable kinetic ratetime profiles of the type (a) in Fig. 9.6 agree well with Eq. (9.39) rate behavior. Describe how kinetic experiments would be planned and the results analyzed in order to derive the kinetic parameters of this equation.

806

Chapter 9

9.7. In the build-up period of the decay-type Ziegler-Natta polymerization of propylene, the rates are found to be different when propylene is introduced after TiCla and A1Eta are allowed to equilibrate from the case when A1Eta is added after the gas in introduced. The rates in the former case are given by the expression K[AIEta] t kiP1 + K[AIEta] where P is the propylene gas pressure. It is assumedthat the following equilibrium exists in the former case: S +AIEt3 ~--- S* where S represents an active site (i.e., a Ti atom) and S* a potential polymerization center. The concentration of S* would be given by the Langmuir equation. On introduction of the gas, polymerization centers, C*, are formed. Show that a proper balance of C* yields the aforesaid equation. 9.8. Consider now the case where propylene is introduced before AIEta is added. Then, all the reactions in Eqs. (P9.9.1)-(Pg.9.3) would occur multaneously. Derive the following rate expression for small extents of time Ro = kP[AIEta]t where R0 refers to the rate for small intervals of time, P is the propylene gas pressure, [AIEt3] is the concentration of AIEta, and. t is the time of the reaction. 9.9. For the measurement of polymerization rate of a gaseous monomer with a heterogeneous Ziegler-Natta catalyst, the catalyst may be mixed with a suitable solvent for the gaseous monomer, and the monomer supplied at constant pressure through a gas meter to the catalyst suspension. Any dependence of the rate of polymerization on the stirring speed would then indicate diffusion control of the propagation reaction. Using the above type of experimental setup for propylene pol.vmerization with TiCIa-AIEta in n-heptane, the rate of polymerization was measured [5] at different speeds of stirring and constant propylene pressure. The results obtained indicated that there were two different steady-state rate curves for the stirring speeds of 400 and 600 rpm. In each case, a steady bulk monomer concentration was reached in about 3-4 hours. Showhow the overall process at steady state can be modelled to show dependence of the polymerization rate on stirring speed and to enable determination of both the mass transfer rate constant and polymerization rate constant from rate measurements at different stirring speeds. Hint: Defining [M]0 = monomer concentration at the gas-liquid interface, I’M] = average concentration of monomer in solution, [cat] = concentration of catalyst, k,~ -- mass transfer constant (dependent on stirring speed), and ~ = polymerization rate constant, the mass balance for the

807

Coordination Addition Polymerization

monomergives d[M]/dt = k,~([M]0 - [M] - kp [cat] [M]. At the steady state, d[M]/dt = 0. One may assume k,,~ >> kp[cat] at very high stirring speeds and hence the rate of polymerization, P~ = kp[cat][M]0, while at low stirring speeds, k,~ <( k~[cat] so that P~ = k,~[M]0. 9.10. Consider the reaction scheme described by Eqs. (9.30)-(9.41). Derive expression for the rate of polymerization at steady state and show how it can be used to evaluate the initiation rate constant k~. 9.11. What predictions can be made from Eq. (9.44) about the dependence the number-average degree of polymerization on the various parameters ? 9.12. Using Eq. (9.39) for the rate of polymerization, derive the following MayoWalling equation for Ziegler-Natta polymerization: 1

k~ k~KA k~K u [M] k~K M [U] Howwould this equation be modified if polymerization is carried out in the presence of H2 used as a terminating agent ? DP,~

k~,M k~,

9.13. Hydrogenis used as the chain length regulator in Ziegler-Natta polymerization. A proposed mechanismpostulates that there exists a pre-established equilibrium of dissociative adsorption of hydrogen on TiCI3 catalyst surface as H2 ~ 2Hads It is this adsorbed hydrogen that participates in the following reaction: Cat-P + 2Had s :~ Cat-H + PH The Cat-H reacts with monomermolecules at a different rate in the following fashion : Cat-H + monomer ==~ Cat-P Derive an expression for the rate of polymerization. 9.14. Ethylene was polymerized [A. Schindler, J. Polym. ScL Part C, 4, 81 (1963)] in n-heptane with a catalyst system consisting of diisobutylaluminum hydride (1.5 mmol/L) and titanium tetrachloride (2.5 mmol/L). The catalyst components were reacted at these concentrations at 0°C and the mixture was then aged under a nitrogen blanket for 15 min. Ethylene was fed at 770 mmHg pressure and the temperature of the catalyst suspension maintained at 40°C. It was found that the experimental data for the rate of polymerization could be represented best by assuming a mixed first and second-order dependence of the form Rr = KV~/(1 + K’pE) where FE is the ethylene pressure.

(E9.11.1)

Consider a generalized mechanismbased on the following assumptions: (a) monomeris added to an organometallic compoundforming a polymerization

808

Chapter 9 center, (b) chain growth proceeds thereafter through the addition of further units, and that the growth of an individual polymer chain can be terminated either (c) by a transfer reaction with the monomeror (d) by a spontaneou.s termination-reaction involving the transfer of a hydride ion. Showthat this reaction scheme leads to a rate expression similar to Eq. (E9.11.1) if one considers the process (c) as a true transfer reaction and only (d) termination step. Howwould Eq. (E9.11.1) be modified if the experiments were conducted with hydrogen addition ? Namea few solid-support catalyst systems and the monomersfor which they can be used. Howis the stereoregularity achieved in these polymerizations ? Comparepolymerizations with conventional Ziegler-Natta catalysts and metallocene-based Ziegler-Natta catalysts in respect of (a) reaction mechanism, (b) stereospecificity of polymer product, and (c) polymer molecular weight and distn’bution.

Chapter

10

Ring-Opening

Polymerization

INTRODUCTION In addition to step and chain polymerizations, another modeof polymerization is of importance. This is the ring-opening polymerization of cyclic monomerssuch as cyclic ethers, esters (lactones), amides (lactarns), and siloxanes. Examples of commercially important types are given in Table 10.1. Of those listed, only the polyalkenes are composedsolely of carbon chains. Those that have enjoyed the longest history of commercial exploitation are polyethers prepared from three-membered ring cyclic ethers (epoxides), polyamides from cyclic amides (lactams), and polysiloxanes cyclic siloxanes. Cyclic compoundsconstitute a potentially polymerizable class of monomers. The difunctionality criterion for polymerizability is achieved by a ring-opening process as shown below for ethylene oxide: X CH2--CH2 ~ -~--CH2-CH2-O-]-x(10.1) /~O Cyclic monomersshould, therefore, be capable of being polymerized provided a suitable mechanismfor opening the ring is available (kinetic factor). The ease of polymerization of a cyclic monomer,however, depends on both thermodynamicand kinetic factors. The single most important factor that determines whether a cyclic monomercan be converted to linear polymer is the thermodynamic factor, that is, the relative stabilities of the cyclic monomerand linear polymer structure [1,2]. Table 10.2 shows the semiempirical enthalpy, entropy, and free-energy changes for the conversion of cycloalkanes to polymethylene in all cases. The lc (denoting liquid-crystalline) subscripts of AH, AS, and AGindicate that the values are those for the polymerization of liquid monomerto crystalline polymer. The data in Table 10.2 show that polymerization is favored thermodynamically (/k(~ is negative) for all except the 6-memberedring. King-opening 809

810

Chapter 10

Table 10.1 Commercially Important Polymers Prepared by Ring-Opening Polymerization Monomertype

Monomer struc’(ure

Cyctoatkene

Trioxane

Repeating unit

Polymer type

-’~ CH=CH (CH2)x-3- Potya!.kene 70\ CH2 CIH2 I

-E CH20~-

Potyether

-E(cH2) x- O--}-

Potyether

O"cH~O a Cyctic ether

(CH2) x O

Lactone

/.._~c~O (CH2) x 1 x.._~o

o I1_

-~ {CH2)x CO-J- PoIyester

o

Lactam

(CH2) × I ’<~.~,/NH

Aziridineb Cyclic x

jNH\ CH2- CH 2

II -Fz(CH2)×CNH-]- Po[yarnide

-~CH2CH2NH ~ Polyamine

si[oxane Gi (CH3)2] Polysiloxane -i- ISiOT

LCH3] Hexachtoro cyctotriphosphazeneC

Cl ~ /Cl N~ P ~N Cl~ I II/CI ct/P~

N/P~c[

aEpoxide (x=2).; oxetane (x=3) bAlso catted atkyleneimine Cphosphonitritic chloride trimer

CI -E

i =N~Ct

Potyphosphazene

811

PJn~Openin~ Polymerizafion Table 10.2 Thermodynamicsof Polymerization of Cycloalkanes at 25°C (J/mol-deg) (kJ/mol) (CH2)~ 3 -69.1 -113.0 4 -105.1 -55.3 5 -21.2 -42.7 - 10.5 6 +2.9 7 -21.8 -15.9 7 -34.8 -3.3 Source:Datafrom Refs. 1 and 2.

(kJ/mol) -92.5 -90.0 -9.2 +5.9 -16.3 -34.3

polymerization of 6-memberedrings is generally not observed. Based on the values of AGtc, the order of thermodynamicfeasibility is 3,4>8>5,7. The principal factor determining the magnitude of the free-energy change is the existence and extent of ring strain. Ring strain is a thermodynamicproperty caused by either forcing the bonds between ring atoms into angular distortion or by steric interaction of substituents on the ring atoms. It is the release of ring strain by polymerization that provides the principal driving force for the polymerization of cyclic monomers.Considering the thermodynamic relation AGlc = AHlc - TASte, where T is the temperature (°K), one notes that AHlc is the major factor in determining AGtc for the 3- and 4-memberedrings, while AStc is very important for the 5- and 6-membered rings. For larger-sized rings the enthalpy and entropy factors AHlc and ASlc contribute equally. As ,might be expected from considerations of ring strain, the most reactive monomersare usually those containing 3- or 4-memberedrings. Fiveand six-memberedrings are virtually free of angle strain; however,torsional strain arising from conformational eclipsing of C-H bonds is the major factor responsible for the lower thermodynamicstability of cyclopentane relative to cyclohexane. Incorporation of heteroatoms into the ring influences ring stability in waysthat are not always predictable. Minor perturbations in the physical conditions and chemical structure can have a markedeffect in causing the sign of the free-energy change to be in favor or against polymerization. Thus, whereas 5-memberedcyclic ethers such as tetrahydrofuran exhibit negative free-energy change and are consequently polymerizable, the five-memberedcyclic esters (7-butyrolactone) exhibit positive free-energy change and are not polymerizable. In contrast, 6-memberedcyclic ethers do not polymerize while the corresponding cyclic esters do. With other cyclic systems such as imides and anhydrides both 5- and 6-memberedrings are polymerizable. Cyclic ethers, which exhibit ring strain similar to that encounteredin cycloalkanes, exhibit the following order of reactivity in terms of ring size: 3 > 4 > 8 > 7 > 5 > 6. For rings of all sizes the presence of substituents decreases the thermodynamicfeasibility for polymerization. Substitution in the ring tends to

812

Chapter 10

make the free-energy change more positive, thereby decreasing polymerizability. Thus, whereas tetrahydrofuran is polymerizable, 2-methyl tetrahydrofuran is not. Since both AHts and ASlc are negative, AGlc becomes less negative with increasing temperature. Above some temperature (the ceiling temperature) AGtc becomes positive, and polymerization is no longer favorble. Ceiling temperatures are often quite low in ring-opening polymerizations compared with vinyl polymerizations, particularly where fiveor six-membered rings are involved. Although ring-opening polymerization is thermodynamically favored for all cycloalkanes except the 6-memberedones (see Table 10.2), polymerization of cycloalkanes has been achieved in practice only in very few cases, almost exclusively with cyclopropane derivatives--and only oligomers are obtained. This shows that a favorable thermodynamicfactor alone does not guarantee actual polymerization of a cyclic monomer.Since polymerization requires a kinetic pathwayfor the ring to open, kinetic factors should also be favorable for polymerization to occur. The cycloalkanes do not have a bond in the ring structure that is prone to attack by an initiator. This is in marked contrast to the cyclic monomerssuch as lactones, lactams, cyclic ethers, acetals, and manyother cyclic monomersthat have a heteroatom in the ring that provides a site for nucleophilic or electrophilic attack by an initiator species, resulting in initiation and subsequent propagation. These monomersthus polymerize because both thermodynamic and kinetic factors are favorable. POLYMERIZATION

MECHANISM

AND KINETICS

The overall process of ring-opening polymerization of cyclic compoundscan be schematically represented by the reaction: A--B A--B

)z

~ --(--A-B---~

,

(10.2)

Two main differences from the two other general methods of preparing linear polymers--condensation and addition polymerizations--can be emphasized. First, in contrast to condensation reactions, ring-opening polymerization does not result in the loss of small molecules. Second, unlike olefin polymerization for which the loss of unsaturation is a powerful driving force, ring-opening polymerization does not involve a loss of multiplebonding enthalpy. Ring-opening polymerizations are primarily initiated by ionic initiators (including coordinate ionic) as well as initiators that are molecular species, e.g., water. This later class of initiators are generally only effective for the more reactive cyclic monomers.Ionic initiators are usually more reactive and typically are the same as those described previously for the cationic and anionic polymerizations of monomerscontaining carbon-carbon and carbon-oxygen double bonds.

813

Ring-Open’.mgPolymerization

Mechanisms of ring-opening polymerization vary according to monomer type and initiator, but in most instances they fit one of the two general forms : 1. Monomeris attacked by some ionic or coordination species (designated X*) at the functional group (designated G) that causes ring opening. This is followed by the attack of the ring-opened monomeron another cyclic unit, and so on:

x ® o*

(-’q

etc.

e.g., anionic polymerization of lactones with initiation

®

(10.3)

by methoxideion:

9 ~ + O/’C\R ~ CH30-CO-R-OCH30 -CO -R-O-CO-R-O(10.4) The typical anionic ring-opening polymerization involves the formation and propagation of anionic centers. Reaction proceeds by nucleophilic attack of the propagating anion on monomer. 2. Monomeris attacked by X* to form a coordination species (most frequently a cation) that undergoes reaction with a second monomermolecule to open the ring, and so on, CH30-

(10.5)

X-- OvwwOww~O~ etc. e.g., cationic polymerization of ethylene/mine:

Most cationic ring-opening polymerizations of cyclic ethers involve the formation and propagation of oxonium ion centers. Reaction involves the nucleophilic attack of monomeron the oxonium ion, e.g., for 1,2-epoxides (oxiranes) : cn 3-c~-°Xc~i2 / CH3CH~CH2 + HA ~ CH3CH-z--_CH2

)

~ A + ~H /HO-CH2~

H~O CHa A- CH~_

(10.7/

Ionic ring-opening polyerizations show most of the characteristics described in Chapter 8 for cationic and anionic polymerizations of vinyl monomers.Thus they show effects of solcent and counterion, propagation by different species (covalent, ion pairs, and free ions), and association phenomenaanalogous to those discussed in Chapter 8.

814

Chapter 10

While ionic or coordination compounds are used most commonly to initiate ring-opening poymerization, molcular compounds(water, alcohols, amins, etc.) are also used. In the latter cases, the initiator (XY)serves open the ring, then polymerization prceeds by step growth. ~ ~ XwwvG--Y ~ XwwvGvvwvG--Y ~ etc., (10.8) For example, polymerization of e-caprolactam to nylon-6 is carried out by heating the monomerin the presence of 5-10% water to temperatures of 250-270°C: O II (CH2)5/

~NH ~ HO2C(CH2)sNH2 ( 2)~.-NNH HO2C(CH2)5NHCO(CH2)5NH2 --~

etc.

(10.9)

The nature of the growth processes shown above [Eqs. (10.3)-(10.9)] bears a superficial resemblance to that in chain polymerization, namely that only monomeradds to the growing chains in a propagation step and species larger than monomerdo not react with each other. However, this is not necessarily the case. Epoxide polymerization under anionic conditions, for example, exhibits characteristics of both step growth and chain growth depending on choice of initiator. Lactams may polymerize by a combination of step-growth and chain-growth processes with a single initiator. The classification of a ring-opening polymerization as a chain or step polymerization can be made on the basis of two criteria: (a) the experimentally observed kinetic laws that describe polymerization and (b) the relationship between polymer molecular weight and conversion. The second criterion marks the major distinction between step-growth and chain-growth polymerizations. High-molecular-weight polymer is formed throughout the course of a chain-growth polymerization in contrast to the slow build-up of polymer molecular weight in step-growth polymerization. Most, but not all, ring-opening polymerizations behave as step polymerizations in that the polymer molecular weight increases relatively slowly with conversion. The rate constants for ring-opening reactions of cyclic monomers,such as ethers, amines, amides, esters, and siloxanes have values muchcloser to those for the reactions of step polymerization (e.g., amidation and esterification) than for chain polymerization (viz., addition of radical, carbocation, or carboanion to C=C). However, irrespective of whether a particular ring-opening polymerization is a step reaction or a chain reaction with respect to molecular weight build-up, its kinetics are usually described by equations resembling those of chain-growth polymerization since only monomeradds to growing chains [see Eqs. (10.3)-(10.9)]. As explained earlier, many ring-opening polymerizations are also complicated by the occurrence of polymerizationdepolymerization equilibria. Various situations will be described in this chapter.

815

Rin~-OpeningPolymerization

Anionic Polymerization Reaction

of Epoxides

Characteristics

The anionic polymerization of epoxides such as ethylene and propylene oxides can be initiated by hydroxides, alkoxides, oxides, and metal alkyls and aryls, including radical-anion species such as sodiumnaphthalene. Thus, -, e.g., K+(ButO-), involves the polymerization of ethylene oxide by M+A initiation : /O\ H2C~CH2 + M+A followed by propagation: A-CH2CH20-M

+ ~ A-CH2CH20-M

(10.10)

+ + H2C--CH2 ~ + A- CH2CH2OCH2CH20- M

( i0. I 1

which may be generalized as + + H2C--CH2 ~ Aff--CH2CH20)nCH~CH20-M A_(_CH~CH20~(n+---I)12H~CH~O- + ( 10.12) A number of initiators can cause epoxide polymerization to proceed through anionic coordination mechanism. These initiators include adducts such as Zn(OCHa)~and ([Zn(OCHa)2]2.[C2HsZnOCHa]6) derived action of dialkylzinc with alcohol, a ferric chloride-propylne oxide adduct C1Fe[OCH(CH3)CH~CI]2 (referred to Pruitt-Baggen ini tiator) andmetatporphyrin derivatives of zinc, aluminum, and manganese. Propagation in these systems involves a concerted process in which the epoxide monomer is inserted into a metal-oxygen bond: (10.13)

w~vCH2CH20,~CH 2 ~ wwvCH2CH2OCH2CH20 /

\

This propagation reaction can be visualized as involving the formation of an incipient alkoxide anion on cleavage of the oxygen-metal bond in the propagating chain--hence the name anionic coordination. Many anionic coordination polymerizations proceed with stereochemical consequences. Polymer molecular weights are low for anionic polymerizations of propylene oxide (< 5000) since polymerization is severely limited by chain transfer to monomer.Chain transfer to monomercan take place by proton abstraction from the methyl group attached to the epoxide ring:

816

Chapter 10 O

CH3 wwvCH2--

I

+

CH-O-Na

q:n

+ wwvCH2--CH--OH

+ CH~-CH~CH2

~

/o\ (10.14)

+ H2C--CH-CH~’Na

Chaintransfer to monomer is muchlessprevalent forpolymerizations withmostof theanioniccoordination initiators. Muchhighermolecular weights arethuspossible in thesepolymerizations. Forexample, molecular weightsof theorderof 105 arereported forpropy]ene pol~eH~tion by an initiator derived ~omdipheny]tin sulfide andbis(3-dimethylaminopropy])zin

Kinetics Most epoxide polymerizations have the characteristics of living polymerizations, that is, the ability to polymerize successive monomercharges forming block copolymers. The expressions for the rate and degree of polymerizations are essentially those used in living chain polymerizations (see Chapter 8). The polymerization rate is given by Rp = ]~;[M-I[M] (10.15) where k~ is the overall apparent rate constant and [M-] is the total concentration of all living anionic propagating centers (free ions and ion pairs); [M] is the concentration of monomer. The decrease of poymerization at time t in the reaction is given by the concentration of monomerthat has reacted divided by the initial initiator concentration :

DPo = ([M]0- [M]t)/[I] where [M]0 and [M]t are the monomerconcentrations

(10.16) at times 0 and t.

Problem 10.1 The polymerization of ethylene oxide was studied by Gee et al. [3] in solution in 1,4-dioxane, catalyzed by solutions of sodiumin a small excess of a simple alcohol, ROH.(An excess of alcohol is added to increase the solubility of the catalyst.) Their experimentssupport the suggestionthat there is no termination reaction and that both the ion pair, RO-Na*,and the ion, RO-, are catalysts. Derive a simple expressionfor the rate of polymerizationunder the experimentalconditions. Answer: Makinga simplifying assumptionthat all the polymerization steps have the same bimolecular rate constant, that is, k~- or k~, de.pending on whetherfree ion or ion pair is involved,

= [ak,; + (1- a)k~] [M-][C_¢I%O](1’10.1.1)

~~

Rin~-Openin~Pol~meriza~ion

where[M-] is the total concentration of all anionic propagatingcenters and ~ is the degree of dissociation. So long as the degreeof ionization remainsconstant, no error will be introduced by using an overall rate constant k~ in place of [~k~- -t- (1 - o0k~]. Moreover, [M-]can be replaced by Co, the total initial alkoxide concentration. Thus, (mO.l.~) i% = ~[C~H,O] where (P10.1.3) k; = ak; + (1- a)k~ Note: A series of proton exchangereactions (see later) arise from the presence excess alcohol. If these exchangereactions are muchfaster than the polymerization, the effect on the kinetics of polymerizationwill be negligible unless the alcohol addeddiffers markedlyin acid strength from the polyether alcohols formedby the exchangereactions.

Polymerization of epoxides occurs readily under the influence of strong bases in both protic and aprotic solvents, propagation involving stepwise growth of alkoxide ions. Dimethyl sulfoxide (DMSO)is the most useful of the dipolar aprotic solvents and shows a marked ability to solvate cations (especially +) whilst l eaving a nions e ssentially u nsolvated. A s a consequencenucleophilic reactivity of anions is greater in solvents such as DMSO. There is a special interest in the use of metal alkoxides in DMSO because of a+ rapid proton transfer leading to equilibria of the type ButO-K + CHaSOCHa ~K CHsSOCH~_K+ + ButO H (10.17) Methyl sulfinyl carbanion (CHsSOCH~) has been given the trivial name dimsyl ion and is involved in the majority of the base-catalyzed reactions in DMSO, in spite of the fact that the equilibrium lies far to the left (K = 1.5 x 10-7). The equilibrium of reaction (10.17) is established quickly and even though the dimsyl ion is present in very low concentrations, it is orders of magnitude more reactive than the t-butoxide ion [4]. Chemical analysis of the lower molecular weight polymers produced by polymerization of epoxides with metal alkoxides in DMSO shows that sulfur is present [5] in low concentrations (< 0.4 percent) and this would arise from initiation involving DMSO, that is, pre-initiation [Eq. (10.17)] followed by CH3-~-CH~’K

o

+ + + H2C~CH2 ~ CH3-.S,

CH2CH~CH20-K

6

Propagation involves the reaction

/o\

CH~oCH~CH2-CH~O-K+

+ H2C~CH 2

--~

+ CH3~CH2(CH2CH20)~K

818

Chapter10

followed by successive monomeradditions which are assumdto be kinetically indistinguishable and have the same rate constant

Problem10.2 Polymerization of ethylene oxide with potassium tert-butoxide in DMSO was followed [5] by conventional dilatometry, using special procedures to eliminate zero time errors consequenton rapid initiation reactions. Givenbelow are someof the data obtained for this systemat 50°C: -1 Concentration, mol L Initial rate (Rr z) × 10 Expt. MonomerInitiator ×103 mol L-1 -1 s 1 3.14 6.84 9.61 2 3.43 7.81 6.51 3 3.45 1.36 0.99 4 4.81 7.30 7.50 5 2.42 7.60 5.00 6 1.54 7.90 3.65 Calculate an estimate .of the apparent rate constant for propagation. Whichof the systems given in the table wouldyield the highest molecular-weightpolymer at 90%conversion ? Answer: Beside the metal oxide, dimsyl ions formed by reaction of metal alkoxide with DMSO [Eq. (10.17)] take part in the initiation of polymerization. Withpotassium as gegenionboth alkoxide and dimsyl ion pairs are completelydissociated in DMSO at salt concentrations less than 0.1 M[5]. The initial rate of polymerizationmay be given by the expression[cf. Eq. (10.15)]: R°~ = -d[Ml/dt Io = k~,PP [M]o [ButO-K+]o (P10.2.1) The values of k~PPcalculated for the six cases from Eq. (P10.2.1) are 0.23, 0.24, 0.21, 0.21, 0.27, and 0.30 L mo1-1s-~, -~ yielding an averagevalue of 0.24 L tool -1. S

Accordingto Eq. (10.16), the highest molecularweight at a given conversion will be obtainedfor the case whichhas the highest value of the [Monomer]/[Initiator], that is for expt. 3 with [Ml0= 3.45 tool L-~ and [I]~ = 1.36x10-~ molL-1, giving -1 L (0.90)(3.45 mol = 2286 DP~ = -z -1) (1.36 x 10 tool L The polymerization of unsymmetrical epoxide such as propylene oxide involves the possibility of two different sites (at carbons 1 and 2 or o~ and/9) on the epoxide ring for the nucleophilic ring-opening reaction. Two different propagating species are then possible, one involving the alkoxide of a primary alcohol [Eq. (10.20)] and the other that of a secondary alcohol [Zq. (10.21)1:

819

Ring-Opening Polymeriza¢ion CH3

vww-CH-CH2-O-K + (10.20) vw~vO-K + + CH3-CH--CH 2 vwwCH2-CH--O-K + (10.21) CH3 However,the polymer has a predominantly head-to-tail structure with propagation occurring almost exclusively by attack at the fl-carbon-the less sterically hindered site [Eq. (10.21)], that is, an SN~attack. Exchange

Reactions

Epoxide polymerizations taking place in the presence of protonic substances such as water or alcohol are always accompaniedby exchange reactions. The presence of water or alcohol is often necessary in polymerizations initiated by metal alkox/des and hydroxides in order to produce homogeneoussystem by solubilizing the initiator. In the presence of alcohol the exchangereaction R-(--OCH2CH2-),n-O-Na + + ROH,---~ R-(-OCH2CH2-)rrOH

+ RO-Na + (10.22)

between a propagating chain and the alcohol is possible. Similar exchange reactions are also possible between the newly formed polymeric alcohol in Eq. (10.22) and other propagating chains: R-(--OCH2CH2--)n-OH + R-xt-OCH2CH2--)wrO-Na + ,~R-{-OCH2CH2--)~-O-Na + + R-(--OCH2CH2--)m-OH

(10.23)

Since these reactions leave the total ion concentration unchanged, they cannot affect the observed overall rate if equal reactivity of all ions can be assumed. (The exchange reactions thus appear equivalent to chain transfer reactions, but they are not. Any polymeric alcohol formed via exchange is not dead but simply dormant since all alcohol and alkoxide molecules in the reaction system are in a dynamic equilibrium.) The alcohol will, however, affect the molecular weight and since each alcohol molecule contributes equally with an initiator species to determining the numberof propagating chains, the number-average degree of polymerization will be given by [cf. Eq. (10.i6)] DPn - [I]

+.[ROH]

(10.24)

This counts each initiator and alcohol molecule as a potential chain. The exchange reaction places an upper limit on the polymer molecular weight for polymerizations performed in the presence of alcohols or other

Chapter 10

820

protonic solvents. Polymerizations initiated by alkoxides and hydroxides in aprotic polar solvents do not have this limitation and so also polymerizations initiated by other initiators such as metal alkyls and aryls and the various coordination initiators, since the latter initiators are soluble in aprotic solvents such as benzene or tetrahydrofuran. However, the addition of alcohol or other protic substance is useful for control of polymer molecular weight. Equation (10.24) can be used to calculate the amount of added alcohol or other substance required to achieve a desired molecular weight.

Problem10.3 Discuss the effect of exchangereaction on polymerizationsinitiated by metal alkoxides in alcoholic solution, for the eases wherethe addedalcohol is (a) equally acidic, (b) moreacidic, and (c) less acidic than the polymericalcohol formedby the exchangereaction. Whatwouldbe the result of the use of HCIor RCOOH in place of ROH(or H20)? Answer: (a) The exchangereaction will occur throughoutthe course of the polymerization, if the acidities of the two alcohols are approximatelythe same.The polymerization rate will be unaffected while the molecularweight will decrease [Eq. (10.24)], but the molecular weight distribution (MWD) will be Poisson. (b) If the added alcohol ROHis muchmore acidic than the polymeric alcohol, most of it will undergoreaction with the first-formed propagatingspecies R’OCH2CH20-Na+ + ROH -~ R’OCH2CH2OH+ RO-Na-~ (P10.3.1) ÷ before polymerization begins. Since ROH is more acidic, reinitiation by RO-Na wouldbe usually slower, resulting in a decreased polymerization rate and a broadeningof the molecular weight. (c) For the case in whichROH is less acidic than the polymericalcohol, the rate of polymerization will be relatively unaffected during most of the polymrization and exchangewill occur in the later stages of reaction with a broadeningof the MWD. Whenprotonic compoundssuch as HCIor RCOOH take part in the exchange reaction, the result is not exchange,as occurs with ROH (or H20), but inhibition or retardation, since an anion such as C1- or RCOOpossesses little or no nucleophilicity. Reinitiation does not occur or is very slow and the polymeric alcohols are no longer dormantbut are dead. Both the polymerization rate and polymermolecular weight thus decrease along with a broadening of the polymer molecular weight.

Cationic

Polymerization

of Cyclic Ethers

Propagation in the cationic polymerization" of cyclic ethers is generally considered as proceeding via a tertiary oxonium ion, for example, for the polymerization of 3,3-b/s(chloromethyl) oxetane (R = CH2C1):

Pdng-OpeningPolymeriza¢ion

:O

wwvOCH2CR~.CH2-O N ,

--~ R R

+ R

R

A~

"R

where A- is the counterion. ~e a-carbon of the oxo~umion is electrondeficient bemuseof the adjacent positively charged o~gen. Propagation is a nucleophilic attack of the o~gen of a monomermolecule on ~e a-carbon of the oxoniumion. A vafie~ of initiator systems of the ~pes used in the ~tionic polymerization of alkenes (Chapter 8) can be used to generate the terfia~ oxonium ion p~oagating species. Strong protonic acids such as sul~ric, tfifluoroacetic, fluorosulfonic, and tfifluoromethanesulfonic (triflie) acids initiate pol~eri~tion via the initial formation of a seeond~ oxonium ion: R R (10.26) H+A + O~ ~ H~ R A ~ R which reacts with a second monomermolecule to form the tertia~ oxonium ion: A~

R

R

N~

R (~0.~

This type of initiation is limited by the nucleophilici~ of the anion A-derived from the acid. For acids other than the ve~ strong acids, such ~ fluorosulfonic and triflic acids, the anion is su~ciently nucleophilic to compete with monomer for either the proton or seconda~ and tertia~ oxonium ions, and consequently, only ve~ low,molecular-weight products are possible. Water, often present as impuri~, ~n also reduce the molecular weight significantly since its nucleophilicity allows it to competewith monomerfor the oxonium ions. Lewis acids such as BFa and SbCIs, initiate polyme~zation of cyclic ethers. Used almost always in conjunction with water or some other protogen, Lewis acids form an initiator-coinitiator complex [e.g., BFa.H;O, H+(SbCI~)-], which acts as a proton donor in an initiation sequen~ similar to Eqs. (10.26) and (10.27). Under ce~ain conditions, polyme~zations of cationic cyclic ethers show the characteristi~ of living polymerizations in that the propagating species are long-lived and narrow MWDs are obtained. The rate and degree of polymerizations are then given by expressions pre~ously described [Eqs. (10.15) and (10.16)]. Living polyme~zations occur when inflation is fast relative to propagation and there is an absence of te~ination processes. Such conditions are found for polymerizations initiated Mth a~lium (I) and

822

Chapter 10

1,3-dioxolan-2-ylium (III) salts containing very stable counterions such AsF~-, PF~-, and SbCI~-. These initiators are obtained either in situ or as isolable salts; for example, O -4aCCI + SbC15 ----+

O q~C+(SbCI~)

(i) ~3C+(SbF6)

-

+ O~O (II)

~ q~3CH

(10.28)

H + O~O (SbFr)

(10.29)

(III)

where qb = phenyl group (-C6H5). Reaction (10.29) involves hydride abstraction which is facile with 1,3-dioxolane (II). Transfer reactions may occur by a variety of reactions, some of which are analogous to those in the cationic polymerization of alkenes. Chain transfer to polymer is a commonmode by which a propagating chain is terminated. The reaction (Fig. 10.1) involves nucleophilic attack by the ether oxygen in a polymer chain on the oxoniumion propagating center (the same type of reaction that is involved in propagation) to form the tertiary oxonium ion (IV). Subsequent nucleophilic attack on (IV) by monomer yields (V) and regenerates the propagating specie s. The kinetic chain is thus unaffected and the overall effect is an exchange of polymer chain segments with a broadening of the MWDfrom the narrow MWDfor a living polymerization. Intramolecular chain transfer to polymer (but not intermolecular transfer), which becomes progressively more important at lower monomerconcentrations, results in the formation of cyclic oligomers, that is, the chain ends in (V) are connected to each other. Termination also occurs to varying degrees by combination of the propagating oxonium ion with either the counterion or an anion derived from the counterion; for example,

wwvOCH2CH2-;~"-~IA (BF3OH)-

~ vwwOCH2CH2OCH2CH2OH + BF3 (10.30)

Transfer of an anion from the counterion occurs to varying degrees depending on the stability of the counterion. Thus, counterions such as (PF6)and (SbC16)- have little tendency to bring about termination by transfer of a halide ion, while counterions of aluminum and tin show appreciable transfer tendencies; others such as (BF4)- and (FeC14)- are intermediate in behavior. Termination may also occur by chain transfer with a deliberately added chain transfer agent. Hydroxyl and amine end groups are obtained by using, respectively, water and ammoniaas chain transfer agents.

Ring- OpeningPolymerization

T~N"~

823

"(CH2)z’ ~ ÷ j(CH2)4""~ ~ ¢,,,.,., . ~ O ( C H ~)4"-O (C H o),’.-(IV;" ~ ~(C H2)4"~

~

,.,,.,,,~O(CH2)4--O(CH2)~-O(CH2)4"""

+ ~A_’-(CH2)4""~

(v) Figure 10.1 Terminationby chain transfer to polymerin cationic polymerization of tetrahydrofuran. Kinetics The rate equations that describe the cationic ring opening polymerizations of cyclic ethers take several forms. Somepolymerizations, where there is little or no termination, can be described by kinetic expressions similar to those used in living polymerizations of alkenes (see Chater 8), for example, Pqo = kp[M][M*] (10.31) where [M*] is the total concentration of propagating oxonium ions of all sizes. Reversible ring-opening polymerizations that take place without termination are described in a different manner. However, a treatment similar to that for other reversible polymerizations such as those of alkenes (pp. 532-541 ) is applicable. The propagation-depropagation equilibrium can thus be expressed by kp . . Mn + M = M (10.32) n which is analogous to Eq. (6.187). The rate of polymerization is given by the difference between the rates of the propagation and depropagation reactions : P~ = -d[M]/df~ = kv[MI[M* ] -- kd:o[M*] (10.33) At equilibrium, the rate of polymerization is zero and Eq. (10.32) thus becomes kp[M]e = kdp (10.34) where [M]e is the equilibrium monomerconcentration, as in Eq. (6.189) for equilibrium alkene polymerization considered in Chapter 5. The derivations

824

Chapter10

given there for [M]e and ceiling temperature Tc as a function of AS° and °, viz., Eqs. (6.195) and (6.198) are also applicable to the present AH system. Combination of Eqs. (10.33) and (10.34) gives the polymerization as

- dN]/dt = kp[M*]- N]e)

(lo.a5)

which can be integrated to yield

where [M]0 is the initial monomerconcentration. Equations (10.33) and (10.35) ~n be used to dete~e the propagation rate constant. The equilibrium monomerconcentration [M]e is obtained by direct analysis or as the intercept of a plot of polymefi~tion rate versus initial monomerconcentration (see Problem 10.4). ~e polymerization data are then plotted, in accordance with Eq. (10.36), ~ the left side of ~at equation versus time to yield a staright line where slope is k~[~]. Since [M~] for a living pol~er can be obtained from measurements of the number-average molecular weight, one can determine the propagation rate constant.

Problem10.4 (a) The kinetics of polymerization of tetrahydrofuran was studied -, as initiator [6] with the use of triethyloxoniumtetrafiuoroborate, (C2Hs)aO+BF~ and dichloromethaneas solvent. Conversionversus time was measuredat 0°C with initial catalyst concentration [I]0 = 0.61 x 10-2 mol/Land monomer concentration [M]0varying from 3 to 9 mol/L. The initial rates, P~, determinedIrom these data are given in Table A. (b) In another series of experiments, all with [M]0 = 6.1 mol/L and [I]0 3.05x10-2 mol/L, the monomerconversion (p) was measured as a function time (t) and the numberaverage molecular weight (M~)was determined yielding the data given in Table B. Table A -~ [M]0, mol L 9.15 8.00 7.00 6.10 5.00 4.06 3.05

/~, mol L-1 rain -~ 0.00930 0.00820 0.00680 0.00500 0.00350 0.00170 0.00037

Table B t, min 24 48 84 162 210 398

p 0.085 0.207 0.310 0.443 0.490 0.565

M~ 1405 2795 4078 -

Determine(a) the equilibrium monomerconcentration and (19) the propagation rate constant kp at 0°C.

825

PJn~- Openin~ Polymerization Answer: (a) The polymerization (P10.4.2). Initiation

proceeds in the manner shown in Eqs.

R

(P10.4.1) and

R

ISoIF¯

R)O

(P10.4.1)

+ R--O

Propagationldepropagation R-OCH2CH2CH2CH2-+O_~

(P10.4.2)

BF 4 In these equations, ki, kr, and kap are the specific rate constants of initiation, propagation, and depropagation reactions, respectively. The data of column 2 arc plotted against the data of column 1 in Fig. 10.2. From the intercept at /~ = 0, [M]~ = 2.65 mol/L. (b) Monomerconversion,

p --

[M]0

or [M] = [M]0(1 - p) (P10.4.3)

With [M]~ = 2.65 mol/L and [M] calculated from Eq. (P10.4.3), the left side Eq. (10.36) is evaluated and plotted against t in Fig. 10.3. The slope gives k~[M*] = 9.5×10-3 -1. rain Assumingthat the termination of the cationic polymer chain is brought about by the addition of water, the polymer has the formula C2Hs(OCH2CH2CH2CH2)=OH and molcular weight = 46 + 72x. For M~= 1405, DP,~ = x =

1405 - 46

= 19 72 Similarly, for Mn = 2795, DP,~ = 38 and for Mn = 4078, DP, = 55. Since for the living polymerization, DP,~ will be given by DP,~ = ([M]o - [M])/[M*], [M*] can be calculated from [M] and DP,~. With [M]0 = 5.1 tool/L, this yields p 0.085 0.207 0.310

[M], mol/L 5.58 4.84 4.21

DP,~ 19 38 56

[M*], mol/L -2 2.74 x 10 -2 3.31 x 10 -2 3.37 × 10 Since the value of [M*] is nearly constant at monomer conversions p > 0.20, it may be assumed with fair approximation that the initiator is completely reacted. Since k~[M*]= 9.5x 10-a, as obtained above, (9.5 x 10-3 -1) min k; = (3.37×10 -2 -1) tool L = 0.28 Lmo1-1 min -~ (_= 4.7×10 -3 -~s-~). Lmol

Chapter 10

826

10

’, 6

x

I

0

2

6

6

8

10

FM30mol/L

Figure 10.2 Determination

of the equilibrium monomer concentration [M]e rate (/?.p) versus initial monomerconcentration ([M]0) data. (Problem

from initial

10.4.)

2.0

0

Figure 10.3 10.4.)

I 40

I I 80 120 Time,rain

i 160

I ,, 200

Plot of Eq. (10.36) for the determination

of kp[M’]. (Problem

827

Rin~- Openin6Polymerizafion

The availability of reliable kinetic and thermodynamic data is far less for ring-opening polymerizations than for step and chain polymerizations. However, a comparison of the available data on propagation rate constants clearly reveals the general similarity of ring-opening and step polymerizations. For various oxirane, oxetane, tetrahydrofuran, and 1,3-dioxepane polymerizations, kv is in the range 10-1-10-a L/mol-s [791. These values can be compared to the corresponding values for step and chain polymerizations (Tables 5.3 and 6.7). The kv values for cyclic ether and acetal polymerizations are seen to be close to the rate constants for polyesterification and muchsmaller than those for various chain polymerizations. Degree of Polymerization The quantitative dependence of the degree of polymerization on various reaction parameters has been described [10,11] for an equilibrium polymerization involving initiation. Suppose that we had an initiator XYwhich brought about the polymerization of a cyclic monomerM in accordance with the following equilibria: [XMY] XY ifM ~ XMY, (10.37) Ki -- [XY][M] XMY q--

M ~ XM2Y,

XM2Y + M ~ XM3Y, XMnY q-M ~ XMn+IY ,

Kp-

[XM2Yl (lO.38)

[XMY] [M]

Kp- [XM2Y] [XM3Y] [M (10.39) l Kp-

[XM,,+Cv] (10.40)

[XMnY][M ]

where XYis the initiator (which can be both ionic or nonionic), M the monomer, and Ki and Kv the equilibrium constants for initiation and propagation, respectively. One obtains the following expression for [XMnY]: n-1 [XMnY] = Ki [XY] [M] (Kv[M]) (10.41) By summing [XMnY] over all species from n = 1 to n = c~, one obtains the total concentration, [N], of polymer molecules of all sizes as IN]

= E XMnY --

Ki[XY] [M] (10.42) 1- K~[M] Similarly, one can obtain by summation the total concentration of monmer segments incorporated into the polymer ~:hain. Denoting this quantity by [W], one has

Chapter 10

828 o~

Ki [XY] [M] 2 (1K~,[M]) ,~=1 The average degree of polymerization DP,~is given by W/N: 1 [W] DPn IN] 1 - Kp[M] The initial concentration of monomer[M]0 is clearly given by

[w] = Z ~[x~v] =

[M]o

(10.44)

= [M] + [W] K,[X~] [,]

{1

+ (1

S~])~}

= {1 + The initial

(10.43)

concentration of initiator

(10.45) [XY]~is Nven by

[xv]0= [xv] + [XY]1 + (1 ---~p[M]) = [XY] {1 + Ki[MI~-ff,,

(10.46) } Combinationof Eqs. (10.44) through (10.46) yields the following (otherwise obvious) relationship: [M]0 -- [M] DP,~ = [XY]0- [XY] (10.47) The quantities [XY] and [M] in the foregoing equations represent the equilibrium concentrations of unreacted initiator and unreacted monomer. The various relationships derived above show the dependence of the degree of polymerization on the initial nd equilibrium concentrations of monomerand initiator and on the equilibium constants Ki and Kr~. The polymer molecular weight increases with decreasing Ki and [XY]0 and increasing Kp and [M]0. The quantities that are experimentally measurable are [XY]e, [M]e, [XY]0, [M]0, and DPr~. To evaluate the constants Kp and Ki one might proceed as follows: Kp is obtained from Eq. (10.44) using the experimen{ally determined values of [M] and DPn; Ki is obtained from Eq. (10.45) using the experimentally determined values of [M], [XY], and DPn. Problem10.5 The experimentaldata available in the literature on the equil~rium polymerization of caprolactam (CL) are of the type: a knownmole ratio m H20to CLis charged in a vessel; the vessel is brought to a definite temperature and polymerization is carried out to equih"orium. Whenequilibrium is reached, the value of DP,~and [M] (or conversion p) are determined.

829

Rin~-Openin~ Pol~merizafion

Two sets of data [11] obtained over a sufficiently wide range of variables (temperature and initiator concentration) are given below: Temp. (°C) 221.5

Run m DP, p 1 0.060 0.9390 120 2 0.192 0.9359 60 253.5 0.060 0.9210 100 1 2 0.132 0.9188 60 ° values for the initiation and propagation steps of the equiDetermine the AH librium polymerization of caprolactam. Answer: The polymerization of caprolactam is accompanied by a volume contraction; therefore, the use of concentration units of moles per kilogram is preferred over the application of moles per liter. It is also necessary to be able to compute [M]0 and [XY] from the initial mole ratio m. This is accomplished from the following relationship : [XY]0 = (1000m)/(18m [M]00

[Mlo = [M]~ [Xy]oo ~

+ 113)

rxvl = 8.85 -

(P10.5.1) 0.1594[xY1o

(P10.5.2)

where the molecular weight of H20 is 18 and that of CL is 113; [M]0o is equal to the moles per kilogram of pure caprolactam (= 8.85) and [XY]00 is equal to the moles per kilogram of pure water (= 55.5). From a set of values of DP,~, p, [XY]0, and [M]0 the constants Ki and K v can be computed by the following steps (a) [M] is calculated from [M] = [M]0(1 (b) Kv is calculated from Eq. (10.44): (DP, - 1) gp - [M] ~-P. (c) [XY] is calculated from Eq. (10.47): [XY] = [XY], [MIo - [M] DP, (d) Ki is calculated from Eqs. (10.46) and (10.47): K = [XY]o-[XY] = [M]0-[M] [XY] [M] DP, [XY] [M] [It should be noted that in the theory outlined above leading to Eqs. (10.44)-(10.47), the presence of cyclic oligomers has not been taken into consideration. They are present in the equilibrium mixture of caprlactam polymerization to the extent of at most 5%.] ° values can be calculated from- the equilibrium constant values at The AH several temperatures. Only one set of values for [XY]0, [M]0, [M], and DP, is necessary in principle to calculate the equilibrium constants; however, both sets at each temperature may be taken to determine the constancy of Ki and K~. The

Chapter10

830 values are T1 = 221.5°C (-= 494.5°K) Run 1 Ki =0.0023 Kp = 1.855 Run 2 K~ =0.0027 Kp= 1.787 Average Ki =0.0025 Kp = 1.821 Fromvan’t Hoff’s equation,

T2 = Run 1 Run 2 Average

253.5°C (= 526.5~K) Ki =0.0025 Kp= 1.430 Ki =0.0031 K~= 1397 Ki = 0.00285 K~ = 1.4135

In (K,)2 _ AH° (T~_ (Ki)I R \ TIT2

or

ln(0.002SS [(526.- 494.5) 0.00:50/(1.987tool-1 °K

Solving, AH~= 2118 cal/mol. Similarly, from K~ values, AH~= -4095 caYmoL Thus, the results obtained are ° = 2118 cal/mol CL + H~O ~ H(CL)OH AH H(CL)OH + CL ~ H(CL)~OH AH° = -4095 cal/mol H(CL),,OH + CL ~ H(CL)~+~OH

Polymerization

of Lactams

Lactams are cyclic amides formed by the intramolecular amidation of amino acids. The polymerization of lactams [Eq. (10.48)] O C (CH2)/m-----~-~NH

~ -{--NH(CH~)mmCO--]-n--

(10.48)

can be initiated by bases, acids, and water. Initiation by water (see Problem 10.5), referred to as hydrolytic polymerization, is the most often used method for industrial polymerization of lactams. Anionic initiation is preferred when polymerization is done in molds to directly produce objects from monomer.Cationic initiation is, however, not useful because the conversions and polymer molecular weights are significantly lower. Hydrolytic

Polymerization

Hydrolytic polymerization [12,13] of e-caprolactam to form nylon-6 [m = 5 in Eq. (10.48)] is carried out commercially in both batch and continuous processs by heating the monomer in the presence of 5-10% water to temperatures of 250-270°C for periods of 12 hr to more than 24 hr. In the first step, the lactam is hydrolyzed to e-aminocaproic acid:

(CH2) 5" \NH + H20 ~ H2N(CH2)sCO2H (M) (W)

(10.49)

831

Ring-OpeningPolymerization This is followed by step polymerization of the amino acid with itself w,,wCOOH

+ H2NHwvw ~ vvvwCONI+av~+

(Sn)

H20 (10.50)

(W)

(Sn+.0

(Sr~)

and initiation of ring-opening polymerization of lactam by the amino acid, in which the COOH group of the amino acid protonates the lactam followed by nucleophilic attack of amine on the protonated lactam: OH -OOCRNH-(CORNH-~CORNH~

+ H R ,--

(sn)

(M)

HO0 CRNH-(CORNH-)-n+ 1-C0 RNH2

(S,~+l)

(10.51)

The propgation process follows in the same manner. The initial ring-opening [Eq. (10.49)] and subsequent propagation steps [Eqs. (10.50) and (10.51)], which include both condensation and stepwise addition reactions, constitute the principal mechanismof the polymerization and may be represented by the following three equilibria: [$1]

M+ w Sn

+ Sm ~ Sn+m

M + Sn

q-

W

~ Sn+l

K2- [Sn+m][W]

g3- [Sn+I]

(10.52)

(10.53) (10.54)

Employingthe usual simplifying assumption, that the reactivity of the end groups are equal and independent of the chain length of the respective molecules, K2 my be expressed as K~- [Sn+~][W] [sd[sl]

(10.55)

and it becomes obvious that Ka = K~Ku (10.56) Applyingthe principle of equal reactivities one can write a more generalized expression for K~ = [NHCO][W]

(10.57)

and since [NH2] = [C00H] = [S],

= [waco][w]/[s]’-

(lo.5s)

Chapter 10

832 where

[NH2] = Concentration of amino end groups [COOH]= Concentration of acid end groups [NHCO]= Concentration of amide linkages [S] = Concentration of polymeric chains Values for the equilibrium constants, K1,/(2, and K3, may be calculated from Eqs. (10.52), (10.58), and (10.56), respectively, by substituting various terms in these equations with quantities that are experimentally obtainable. Kinetics The significant quantities upon which the formulation of kinetic equations may be based are the concentrations of caprolactam (M), polymeric chains (S), and amide groups in linear chains (Z). Neglecting amide groups cyclic oligomers, the relationship between these quantities is given by [Z]

= 1 -[M]-

[S]

(10.59)

from which it follows that d[Zl/dt

= -d[Ml/dt

- diS]/dr

(10.60)

Werecall that the three principal chemical equations for the considered mechanism[cf. Eqs. (10.52)-(10.54)] where Ring opening (RO) M

+

W~

(10.61)

Po~condensath)n(PC) Sn

-t-

Sn+,~

Sm

+ W, K2 = k2/k-~

(10.62)

or, in general, NH2 (S)

+ C00H (S)

~-

NHCO (Z)

+ H20 (W)

Polyaddition (PA) Sn + M ~ Sr~+~,

K3 = k3/k-3

(10.63)

(s) The kinetic equations representing the contribution of the individual reactions are RO: -d[Ml/dt

= kl

([M][W]

- [S1]/K1)

(10.64)

833

t~ng-Open~ngPolyrneHzaf~on PC: d[sl/dt

"--

/gl

([MI[W] -- [S1]/K1)

-- k2 [S] { [S] [W](1-[M]K2IS] PA: -d[M]/dt

= k3[S] {[M] -- (1 -- [S1]/[S])

Since [Wl = [W]o - [S], it follows that -d[M]/dt -- kt {[M] ([W]o- IS]) q-/~3[S]

{[M] --

--[SI]/Kt}

(1 -[S1]/[S])/K3}

- 1¢1 {[M] ([Wlo -- Is]) -- [Sll/K1}

(10.67)

(10.68)

Since it has been assumedthat the reactivities of all carboxyl and amino groups are equal (that is, independent of the chain length of the molecule), the equations derived for linear macromolecules may also be applied to aminocaproic acid (S0. For this case, Eqs. (10.52)-(10.54) may be written as M + W ~ S1, Sn + S1 ~ Sn+l + W, M + S1 ~ $2 The following rate equation may then be written for St : dt

-- kl{[Ml([W]o -- [S])

(10.69) Note that in the last term of this equation, it has been assumed for the sake of simplicity that [S2] ~ [$t]. The set of differential equations comprising Eqs. (10.67)-(10.69) suitable for the evaluation of experimental data of [M], [S1, and [$1], obtained as a function of time, for the estima!ion of values for kt, k2, and

Anionic

Polymerization

Strong bases as alkali metals, metal hydrides, metal araJdes, and organometallic compoundsinitiate the polymerization of a lactam. The initiation involves the formation of a lactam anion, e.g., for caprolactam with a metal:

(CH2)~

\NH + M ~ (CH2)5’

+ + ½H 2 (10.

70)

834

Chapter 10

or with a metal derivative

(

+ 2)5--NH

+ B-M

~

/C\ (CH2)~N-M

+ +

BH (10.71)

The lactam anion (VI) reacts with monomerin the second step of the initiation process by a ring-opening transamidation [Eq. (10.72)]:

9 /c\ (CH2)5~N-M

,o, + HN--(CH2)5

Slow

+ (CH2)s--N-CO(CH2)~-N-M (VII) The primary amine anion (VII), unlike the lactam anion (VI), is stabilized by conjugation with a carbonyl group and therefore this reaction is energetically unfavorable and very slow. For the same reason, (VII) highly reactive and, once it is formed, it undergoesa rapid proton-abstraction reaction with a caprolactam monomerproducing an imide dimer (VIII), N-caproylcaprolactam, and regenerating the lactam anion [Eq. (10.73)]. O

(CH2)~N-CO(CH2)5-N(VII)

+

+ (CH2)5--NH

Fast

O (CH2)f-"N-

CO (CH2)sNH 2 + (CH2)(~NN-M (VIII)

+ (10.73)

This lactam anion attacks the carbonyl group attached to the nitrogen in the N-acyllactam (VIII), as shownin Eq. (10.74), followed by fast proton exchange with monomer[Eq. (10.75)] to regenerate the lactam anion and the propagating N-acyllactam (X). The imide dimer (VIII) has been isolated and is the actual initiating species for the onset of polymerization [14]. The slowness of reaction in Eq. (10.72) accounts for the occurrence of an initial induction period of low reaction rate in lactam polymerization. The imide dimer is necessary for polymerization because the amide linkage in the lactam is not

835

Ring-OpeningPolymerization

sufficiently reactive (i.e., not sufficiently electron deficient) towardtransamidation by lactam anion [cf. Eq. (10.72)]. The presence of the exo-carbonyl group attached to the nitrogen in N-acyllactam (VIII) increases the electron deficiency of the amide linkage. This increases the reactivity of the amide ring structure toward nucleophilic attack by the lactam anion [Eq. (10.74)]. Propagation follows through a sequence of reactions similar those shownin Eqs. (10.74) and (10.75).

,9 (CH2)s

O

~ N- +

+ (CH2)5/-~N-CO(CH2)sNH2

(viii)

O (10.74)

~N-GO(GH~)5--N~-GO(GH~.)s-NH~

(IX)

0

0

/\ M* (CH2)g-~N-CO(CH2)5--N~-CO(CH2)5-NHg. (IX)

+ (CH2)5~NH

,o, /\ C (CH2)~N-CO(CH2)~-NHCO(CH2)5-NH~

(x)

C / \ + (CH2)g~N-M

(10.zs)

It may be noted that the propagating center in the above reaction mechanism is the cyclic amide linkage of the N-acyllactam. Monomerdoes not add to the propagating chain; it is the monomeranion (lactam anin), often referred to as activated monomer, which adds to the propagating chain. The rate of propagation depends on the concentrations of lactam anion and N-acyllactam, both of which are determined by the concentrations of lactam monomer and base. Addition

of N-Acyllactam

The use of strong base alone for aninic polymerization of lactam is limiting. As noted previously, the polymerization is characterized by induction periods and, moreover, only the more reactive lactams, such as e-caprolactam and 7-heptanolactam (~-enantholactam), readily undergo polymerization, while the less reactive lactams, 2-pyrrolidinone, and 2-piperidinone, are

836

Chapter 10

are much more sluggish. Both these limitations are overcome by forming an imide by reaction of lactam with an acylating agent such as acid chloride or~ anydride, isocyanate, and others. Thus, e-caprolactam is readily converted to an N-acylcaprolactam (XI) by reaction with an acid chloride [Eq. (10.76)]. The N-acyllactam can be synthesized in situ by this reaction preformed and then added to the polymerization system. O (CH2)5~NH

O + RCOCI

--~

(CH2)~---2N-CO-I~ (XI)

+ HCl (10.76)

Initiation consists of the reaction [Eq. (10.77)] of the N-acyllactam with activated monomer(lactam anion) followed by fast proton exchange with monomer [Eq. (10.78)]. Species (XII) and (XIII) correspond species (VII) and (VIII) for polymerization in the absence of an lating agent. The acylating agent achieves facile polymerization of many lactams by substituting the slow reaction in Eq. (10.72) by the faster reaction in Eq. (10.77). Induction periods are thus eliminated, polymerization rates are significantly increased, and lower reaction temperatures can be used. Polymerizations in the presence and absence of an acylating agent are often referred to as assisted (or activated) and nonassisted polymerizations, respectively. O

O

(CHe)5~N-M

+ (CH2)s---N-CO-R O

(CH2)5~N-CO(CH2)5--N=--CO-R

(10.77)

(xIr)

O

O

(CH2)5~N-CO(CH2)5--Nz--CO-R O (CH2)(-\N-CO(CH2)5-NH--CO--R (XIII)

+ (CH2)5//NXNH O + + (CH?)5/-NN-M (10.78)

Propagation follows in the same manner as for propagation of species (VIII) through sequence of reactions similar to those in Eqs. (10.74)

Ring- OpeningPolymerization

837

(10.75) except that the propagating chain has an acylated end group instead of an amine end group. The rate of assisted lactam polymerization is dependent on the concentration of base and N-acyllactam, which determine the concentrations of activated monomerand propagating chains, respectively. The degree of polymerization increases with conversion and with increasing concentration of monomeror decreasing N-acyllactam concentration. These characteristics are qualitatively similar to those of living polymerizations, but lactam polymerizations seldom are living [15,16].

REFERENCES 1. H. R. Allcock,/. Macromol.Sci. Re, vs. Macromol.Chem.,C4, 141 (1970). 2. H. Sawada, Thermodynamicsof Polymerization, Chap. 6, Marcel Dekker, NewYork (1976) 3. G. Gee, W.C. E. I-Iigglnson, and G. T. Men:all, J. Chem.Sac., 1345(1959). 4. A. Ledwithand N. R. Mcfarlane, Proc. Chem.Soc., 108 (1964). 5. C. E. Bawn,A. Ledwith, and N. Mcfarlane, Polymer, 10, 653 (1969). 6. D. Vofsi and A. V. Tobohky,J. Polym.Sci., Part A, 3, 3261(1965). 7. J. C. W. Chien, Y.-G. Cheun, and C. E Lillya, Macromolecules, 21, 870 (1988). 8. E Mijangosand L. M. Leon, J. Poem.Sci. Polym.Lett. Ed., 21,885 (1983). 9. S. Penczekand P. Kubisa, "Cationic Ring-OpeningPolymerization: Ethers," Chap. 8 in ComprehensivePolymer Science, Vol. 3 (G. C. Eastmond, A. Ledwith,S. Russo, and P. Sigwalt, eds.), PergamonPress, London(1989). 10. A. V. Tobolsky,J. Polym.Sci., 25, 220(1957); 31, 126(1958). 11. A. V. Tobolskyand A. Eisenberg, Y. Am.Chem.So¢, 81, 2302 (1959); 82, 289(1960). 12. G. Bertalan, L Rusznak, and E Anna, Makromol.Chem., 185, 1285 (1984). 13. H. Sekiguchi, "Lactams and Cyclic Amides," Chap. 12 in Ring Opening Polymerization, Vol. 2, (K. J. Ivin and T. Saegusa, eds.), Elsevier, London (1984). 14. H. K. Hall, Jr., J. Am.Chem.Soc., 80, 6404(1958). 15. M. Kuskova,J. Roda, and J. Kralicep, Makromol.Chem., 179, 337 (1978). 16. J. Sebenda, ’~knionic Ring-OpeningPolymerization: Lactams," Chap. 35 in ComprehensivePolymerScience, Vol. 3 (G. C. Eastmond,A. Ledwith, S. Russo, and E Sigwalt, eds.), PergamonPress, London(1989).

EXERCISES 10.1. Salts of carbazoleare excellent initiators for ethyleneoxide polymerization, giving living polymersof prdieted molecularweights. Fromconductivity measurements at 20°C, th dissociation constant of carbazylpotassium(NK)was found [E Sigwalt and S. Boileau, J. Polym. Sci. Polym. Syrup., 62, 51 (1978)] to be 7.0×10-9 in THF, 1.1xl0 -5 in THF+[2.2.2]cryptand and 7.5× 10-2 in hexamethylphosphoramide (HMPA).,(a) Calculate the fraction

838

Chapter 10

of flee ions in the thre cases with NKconcentration 10-3 mol/L. (b) Calculate the molecular weights of polyethylene oxide initiated by NKin the three cases at 90%conversion of the monomerof initial concentration 1.4 mol/L. (c) What is the number of initiator residue per molecule of the polymer formed ? [Ans. (a) THF0.1303, THF+[2.2.2] 0.11, HMPA 0.99; (b) 55,440; (c) Oneinitiator residue per molecule.] 10.2. The initiation of ethylene oxide polymerization by sodium naphthalene involves direct addition of the monomerto the radical anion and reduction of the adduct by sodium naphthalene producing a dianion. Suggest two tests to support this mechanism. 10.3. Consider the following monomersand initiating systems: (a) Monomers:Propylene oxide, trioxane, oxacyclobutane, a-pyrrolidone (’7butyrolactam), a-piperidone ((5-valerolactam, and ethyleneimine. (b) Initiating

system: n-C4HgLi, H20, BF3 + H20, NaOC2Hs,H:~SO4.

Which initiating system(s) can be used to polymerize each of the various monomers ? Show the mechanism of each polymerization by chemical equations. 10.4. Explain why the polymerization of an epoxide by hydroxide or alkoxide ion is often carried out in the presence of an alcohol. Discuss how the presence of alcohol affects both the polymerization rate and the degree of polymerization. 10.5. Explain the following observations: (a) Anionic polymerization of propylene oxide is usually limited to producing a relatively low-molecular-weight polymer. (b) A small amount of epichlorohydrin gretaly increases the rate of polymerization of tetrahydrofuran by BF3 even though epichlorohydrin is much less basic than tetrahydrofuran. (c) The addition of small amounts of water to the polymerization of oxetane by BF3 increases the polymerization rate but decreases the degree of plymerization. (d) In the presence of an acylating agent, the anionic polymerization !actams occurs without an induction period. 10.6, In an equilibrium polymerization of e-caprolactam initiated by water at 220°C, [I]0 = 0.352 mol/L, [M]0 = 8.79 mol/L, and [M]~ = 0.484 tool/L, DP,~ = 152. Calculate the values of/~i and Kp at equilibrium. [Ans. 2.51×10-3; 2.07] 10.7. An equilibrium polymerization of tetrahydrofuran is carried out with an initial monomerconcentration [M]0 = 12.1 mol/L, [M]~ = 2.0 x 10-3 mol/L. Calculate the initial polymerization rate if [M], = 1.5 mol/L and/~ = 1.3x10-2 l_Jmols. What is the polymerization rate at 20%conversion ? [Ans. (Rr)0 = 2.76x10-4 tool/L-s; Ph0 = 2.13×10-4 mol/L-s.]

Appendix

1

Conversion

SI

UNITS

AND CONVERSION

Physical quantity Length Mass Time Force Pressure Energy Power

Name of SI unit Meter Kilogram Second Newton Pascal Joule Watt

Physical quantity Length Mass Force

Customary unit in. lb dyne kgf lbf 2dyne/cm

Pressure

of Units

atm mm Hg lbf/in. 2 or psi

FACTORS

Symbol for SI unit m kg s N Pa J W

SI unit m kg N N N -2 Pa or -2 Pa or -2 Pa or -2 Pa or

Nm N-m Nm Nm

Definition of SI unit Basic unit Basic unit Basic unit kg m s -2 (= J -I) kg m-1 s -2 (= N -2) kg m2 -2 s kg m2 s -a (= J -1) To convert from customary unit to SI units multiply by -2 2.54 x 10 -1 4.535 923 7 x 10 -s 1 x 10 9.806 65 4.448 22 -1 1 x 10 1.013 25 × 105 1.333 22 x t02 a6.894 76 × 10

839

840 SI

Appendix I UNITS

Physical quantity Energy

ml’ea

AND CONVERSION

Customary unit erg Btu ft-lbf cal eV 2in. 2ft

FACTORS

SI unit J J J J J 2m -~ m

To convert from customary unit to SI units multiply by -7 1 x 10 1.055 056 x 103 1.355 82 4.187 -19 1.602 × 10 -4 6.451 6 x 10 -2 9.290 304 x 10

Viscosity

alb/ft "~) poise (dyne s/cm

-a kg m kg m~1 -1 s Nsm-2

1.601 846 3 x 10 -1 1 x 10

Viscosity, kinematic

Stoke (cme/s)

m2 $-1

10-4

Ibf/ft dyne/cm

-1 Nm -1 Nm

14.59 -3 10

Density

Surface tension

Additional

Conversion

Units

1 ~. = 10-8 cm = 10 -1° m

1 kgf/cm ~ ~ = 9.807 x 104 N/m

1 atm =76cm Hg (at 0°C) = 14.696 psi 1 eV = 1.602 x 10-12 erg 1 #= 10 -4cm = 10 -8 m

1 t-IP = 550 ft lbf/s = 2545 Btu/h = 746 W °C= °F t (1.8t+32) t ° F = (5/9)(t

- ° C

Appendix

2

Fundamental

Constant

Constants

CGS system

Acceleration of gravity (g) (standard value) Normal atmospheric pressure Volume of 1 mole of ideal gas at at 1 atm and 0°C Avogadro’s number Atomic mass unit Universal gas constant (R) Boltzmann costant (k)

S! system

-2 980.665 cm s

-2 9.8066 m s

-2 1,013,250 dyne cm

1.01325 x l0 s 2N m

22.4136litre -1 6.0220 × 1023 mole 1.6604 × 10-24 g

22.41136 × 10-3 m3 -1 mole z~ -1 6.0220 × 10 mole

1.9872 cal deg-1 -1 mole -is 1.3807 × 10 erg deg-1 -1 molecule

Faraday constant

1.6604 × 10-27 kg 8.3143 J deg-1 -1 mole -23 1.3807 × 10 J deg-1 -~ molecule -1 96,487.0 Coulombmole

6.6262 × 10-2~ ergs

6.6262 × 10TM Js

2.99792 × 10rooms--1 4.80325 × 10-l° esu

2.99792 × 10s -1 ms -19 1.60219 × 10 Coulo’mb

Proton rest mass

9.1095 × 10-2a g 1.67265 × 10-24 g

9.1095 × 10-31 kg 1.67265 × 10-27 kg

Neutron rest

1.67482 × I0 -24 g

1.67482 × 10-27 kg

Planck constant (h) Velocity of light in vacuum (c) Electronic charge (e) Electron rest mass

mass

Source: E. R. Cohenand B. N. Taylor, J. Phys. Chem.Ref. Data, 2 663 (1973). 841

Index

Absolute rate constants, determination of, 480 Acetophenone, 464 Actinometry, 468 Addition polymerization, 11 Addition polymers, table of, 12 Aging, 32 AIBN, 454 Alfin catalysts, 742 Al|ylic transfer, 509 Alternating copolymers, 7, 580 Amorphous state, 43 Anionic polymerization, 659-753 alkali metal initiation, 662-664 block copolymers, 699-703 copolymerization, 695-669 degree of polymerization, 673, 676-680 excess counterion effect, 686-690 experimental methods, 672 kinetic chain length, 670-682 kinetics, 669-676 living polymers, 665, 676 polydispersity index, 680, 681 reaction media, 682

Ansa metallocenes, 793 Arborols, 416 Arithmetic mean, 230, 237 Atactic isomer, 74 Atactic polymer, 74 Atactic polypropylene, 52 Athermal mixing, 145 Athermal solvent, 175 Atomic mass unit, 5 Autoacceleration effect, 477, 518-522 Autoinhibition, 509 Autoinitiation, 708 Avrami equation, 88 Azeotropic copolymer, 592 Azeotropic copolymerization, 592 Azeotropic feed composition, 592 Beer’s law, 468 Benzophenone, 464 Bernoullian distribution, 580, 587 Bernoullian model, 82 Bernoullian statistics, 76-82 Bifunctional initiation, 701, 702 Bisphenol-A, 201 Block copolymerization, 590, 641-644 843

844 Block copolymers, 7, 580 coupling agents, 702 star-shaped, 703 Boiling point elevation, 240 Bondrotation, angle of, 45, 46, 47 Branch units, types of, 380 Branched polymer, 21 Branching, 71 Branching probability, 374, 376 Brintzinger catalysts, 793 Bulk polymerization, 553, 554 Butane, potential energy of, 46 n-Butyllithium, 661, 664, 700 Cage of solvent, 455 Carboxyl-terminated polybutadiene, 667 Carother’s equation, 331, 371 Cationic polymerization, 704-755 absolute rate constants, 727-731 chain termination, 713, 718 chain transfer, 713-718 copolymerization, 735 degree of polymerization, 723-725 kinetics, 718-723 Lewis acid, 707 molecular weight distribution, 732-734 propagation, 711-713 protonic acid initiation, 706 steady-state assumption, 726 Ceiling temperature, 533-541 Cellulose, 4 Chain length, 488-490 Chain polymerization, 11 Chain termination: by combination, 437 by disproportionation, 437, 438 mode of, 490 Chain transfer, 435, 493-510 constants, 499-510 effect on DP,~, 96-499 to monomer, 493, 494

Index [Chain transfer] to polymer to solvent, 494, 495 Chemical potential, 141 Cis conformation, 44, 46, 47 Cloud point curve, 192 Cohesive energy, 199 Coiled conformation, 51 Coiled polymer molecules, 59 Cold drawing, 34, 35 Colligative properties, 159 Combination, chain termination by, 13, 491, 492 Combinatorial entropy, 145 Compact molecules, 179 Condensation polymerization, 313-424 branching, 366 breadth of MWD,351 dosed system, 333 control of molecular weight, 338 cross-linking, 369 degree of polymerization, 330 equilibrium consideration, 332 molecular weight distribution, 347, 367 multichain polymer, 366 nonlinear polymer, 366 nonstoichiometric A-A plus B-B, 355 open driven system, 334 recursive approach, 394-415 stoichiometric A-Aplus B-B, 353 type I, 314 type II, 314 Condensation polymers, table of, 19, 20 Configurational entropy, 145 Configurational isomerism, 67, 72 ¯ Configurations, 44, 67 Conformational changes, 44-58 Conformations, 44, 48, 49, 50, 51 gauche, 45, 47 in crystals, 53

Index [Conformations] planar cis, 44, 46, 47 planar trans, 45 Constitutional isomerism, 67-71 Coordination catalysts, see Ziegler-Natta catalysts Coordination polymerization, see ZieglerNatta polymerization Copolymer equation, 586 Copolymerization, 579-646 composition equation, 590-600 effect of cross-linking, 633-640 multicomponent, 625-629 penultimate effect, 631, 632 rate of, 620-625 sequence distribution, 615-619 types of, 587-590 Copolymers, 7, 8 Coupling, termination by, 544-546, 550 Critical branching coefficient, 374 Critical solution temperature, 197 Critical temperatureof miscibility, 190, 193 Cross-linked polymer, 21 Cross-linking, 23, 369 oxidative, 32 Cross-propagation, 582 Cryoscopy, 240 Crystallinity of polymers, 83 Crystallites, 90 Curing, 27 Dead-end polymerization, 477-480 DeGennes theory, 123 Degradative chain transfer, 509, 513 Degree of polymerization, 6, 233, 488, 489 effect of temperature, 531 Dendrimers, 416 Dendritic polymers, 416-424 Dendritic unit, 417

845 Depolymerization, 533 Diblock copolymers, 7 Dicarbanion, 664 Diels-Alder dimer, 475 Diffusion coefficient in bulk polymers, 125 Diglycidyl ether, 201 Dilatometer, 448, 449 Dilute solution viscometry, 285-295 nomenclature, 287 terminology, 286 Diol, cleavage of, 69 Diphenylpicrylhydrazyl radical, 458 Dispersion forces, 206 Disproportionation, 491, 492 termination by, 14, 543-547 Dissymmetrycoefficient, 274, 283, 284 Dissymmetry method, 281 Dissymmetry of scattering, 272 Doolittle equation, 108 DPPHradical, 438 Drawing, 34 Dyad, 77 Ebulliometry, 240 Eclipsed conformation, 44, 46, 47 Eclipsed state, 44, 46 Elastomers, 27-29 Emulsion polymerization, 556-569 constant rate, 561 detergents, 557 kinetics of, 562, 568 qualitative picture of, 551-561 End groups, 6 End-group analysis, 240 End-to-end distance, 58 Epoxy curing, 29 Epoxy resins, 29 Excess Rayleigh ratio, 265 Excluded volume, 65, 178, 179 theories of, 177-181 Excluded volume parameter, 182

~d~ Expansion factor, 185, 215, 217 Extinction coefficient, 467 Fibers, 27-29 Fiducial mark, 289 Fineman-Ross method, 604 First moment, 235 First-order Markov, 580, 581 First-order transition, 91, 92 Fisher projections, 75 Flory constant, 213, 214 Flory-Fox equation, 214, 215, 217 Flory-Huggins equation, 156 Flory-Huggins model, 150-159 Flory-Huggins theory, 145, 150-159 modification of, 168-176 Flory-Krigbaum theory, 176-178 Flory parameter, 213 Flory temperature, 186 Fold period, 86 Fold plane, 86 Fox equation, 119, 120 Free-draining coil, 210 Free-draining molecule, 210 Freely jointed chains, 58-61 Free volume, 121 theory of, 103-106 Freezing point depression, 240 Functionality, 9 Gauche conformation, 45, 46, 47, 48 Gaussian coils, 214 Gaussian distribution, 213 Gegenion, 654 Gelation process, 370 model for, 383, 384 molecular size distribution in, 384 recursive approach for, 399--415 Gel effect, 477 Gel-permeation chromatography, 295-302 apparatus, 297

[Gel-permeation chromatography] calibration, 298, 300 column used, 299 experimental arrangement, 296 universal calibration, 300, 302 Geometrical isomerism, 72 G~bs free energy, 185 Glass transition temperature (Tg), 89-120 effect of branching, 114 effect of copolymerization, 117 effect of cross-linking, 115 effect of diluents, 116 effect of molecular weight. 112 factors affecting Tg, 98-120 relation with T.~, 102, 103 secondary, 91 theoretical treatment, 103-107 Glyptal resin, 32 Graft copolymerization, 644-646 Graft copolymers, 580 Hafnocenes, 792 Hagen-Poiseuille equation, 291 Hansen parameter, 206-210 H-bonding group, 204 HDPE, 22 Head-to-head linkage, 67-71 Head-to-tail linkage, 67-71 Heterotactic triad, 79 High-conversion polymerization, 552 High-density polyethylene, 22 High-mileage catalysts, 747 High-performance polymers, 20 High-speed membrane osmometer, 252 Hildebrands, 202 Hole theory, 111 Homopropagation, 582 Hyd_ride elimination, 668 Hydride ion shift, 711 Hydrodynamic volume, 304 Hydrogen bond; 57

847

~ndex Hydroquinone, retardation by, 524 Hydroxyl-terminated polybutadiene, 667 Hyperbranched polymers, 416-424 applications of, 421-424 by polycondensation, 418-419 generations of, 420, 421 synthetic approaches for, 420 Ideal copolymers, 588, 591 Ideal solution, 173 Ideal solvents, 215 Impermeable coil, 211 Induced dipoles, 148 Induction period, 522 Inherent diluent, 99 Inherent viscosity, 287 Inhibition, 522-527 kinetics of, 525 Inhibition constant, 525 table of, 529 Inhibition period, 522 Inhibitors, 522 Initiator chain transfer constant, 502 Initiator efficiency, 454-458 determination of, 457 Initiator-monomer complex, 512 Initiators, 452--460 thermal decomposition of, 478 Instantaneous copolymer composition, 594 Interaction energy, 148, 185 Interaction parameter, 149, 159-168 critical value of, 189 from osmometry, 255 from osmotic pressure, 163 from vapor pressure, 161 from virial coefficient, 167 Intermittent illumination, 483 Intraparticle interference, 272 Intrinsic viscosity, 212, 214, 215, 287 measurement of, 289 Ionic chain polymerization, 653-735 polymerizability of monomers, 656-659

Ionizing radiation, 469 initiation by,469 Iso-free-volume state, 105 Isomefization polymerization, 711 Isomers, 67 Isotactic form, 53, 55 Isotactic polymer, 74-76 Isotactic polypropylene, 53, 55, 56 Isotactic triad, 79 1-UPACnomenclature, 37

Kelen-Tudos method, 605 Kevlar, 34 Kinetic chain length, 488-490 Kinetic energy correction, 292

Ladder polymers, 25 Lambert-Beer’s law, 468 Lamellae, 86 Lattice theory, 145, 146 LCST, 197, 198 LDPE, 22 Lifetime of radical, 481, 492 Light scattering method, 216, 262 end-to-end distance from, 276, 280 instrumentation of, 282 radius of gyration from, 276 Zimmplots, 278-283 Limiting viscosity number, 287 Linear polymer, 21 Living polymers, 665, 676 London forces, 206 Long-range interactions, 215 Low-conversion polymerization, 542 Low-density polyethylene, 22 Maltese cross pattern, 87, 88 Mark-Houwinkconstants, 215, 290 Mark-Houwinkequation, 176, 215 Mark-Houwink-Sakurada equation, 215 calibration, 288 Maxwell model, 110

848 Mean field approximation, 152, 156 Mean square end-to-end distance, 214 Me-an square radius of gyration, 214 Melt viscosity, dependence on chain length, 122 Melting temperature (T,~), 101-102 factors affecting T,,~, 101-102 relation with Tg, 102-103 Membrane osmometer, 248 Membrane osmometry, 240 Mer, 2 Meso placement, 76 Metallocene catalysts, 791-803 active center, 795, 796 catalyst composition, 792-795 chain transfer, 801, 802 Chien’s model, 798-801 Ewen’s model, 797, 798 kinetic models, 797 polymerization mechanism, 796, 797 Methylalumoxane cocatalyst, 794, 795 Metallocenes, 792 Methylalumoxane, 794, 795 MHSconstants, 304 Miscibility of polymers, 192 Molar attraction constants, 200, 201 Molar mass, 5 Molecular weight, 5 Molecular weight averages, 230-239 determination of, 239-306 in terms of moments, 235-238 Molecular weight determination: by cryoscopy, 243-244 by ebulliometry, 242-243 by end-group analysis, 240-242 by membrane osmometry, 244-257 by vapor pressure osmometry, 258-261 Monodisperse polymer, 235 Monofunctional initiation, 700

/~dex Monomerchain transfer constant, 499-501 Monomerreactivity ratios, 600 table of, 606-607 Monomer, 1, 2 Multiblock copolymers, 7 Network polymer, 22 Newmanprojections, 46 Newtonian flow, 291 Nomenclature of polymers; 37 Nondraining molecule, 210, 212 Nonsteady-state kinetics, 481 Normalized distribution, 231, 237 Norrish-Smith effect, 518 Nuclear magnetic resonance, 76 Nucleophilic attack, 660--662 Number-average molecular weight, 231 Numberdistribution, 231, 236 Nylon, 35, 36, 39, 40 hydrogen bonds in, 57 Oil~soluble initiators, 555 Oligomer, 2 Opalescence, 190 Optical activity, 73 Osmometry, 244-257 practical aspects, 250-257 Osmotic pressure, 163, 240, 245 reduced, 255 Ostwald viscometer, 289, 291 Oxidative cross-linking, 32 Partial molar G~bs free-energy change, 142 Partial molar property, 140, 141 Particle scattering factor, 277 Pentaerythritol, 26 Perturbation theories, 182 Pert_urbed dimension, 186-188 Phase equilibria, 176 in polymer solvent mixtures, 188-198

Index Phenol-formaldehyde resins, 28 Phenylenediamine, 26 Photochemical initiation, 462-468 Photoinitiation, 462-468 Photopolymerization, 466-468 rate of, 466 Photosensitization, 464, 465 Photosensitizers, 464-466 Planar trans, 46, 47 Plastic, 5 Plastics, 27-29 Polyamide, 33 Polybenzimidazopyrrolone, 25 Polybutadiene, isomers of, 70 Polycaprolactam, 38 Polychloroprene, isomers of, 71 Polycondensation reaction, 314 Poly(dimethyl siloxane), Polydispersity index, 235 Polyester resin, unsaturated, 30 Polyesterification, 319-330 acid-catalyzed reaction, 319, 320, 323-325 kinetic parameters, 329 uncatalyzed reaction, 320-321 Polyethylene: crystal structure of, 54 orthorhombic, 54 zigzag form, 49 Poly(ethylene terephthalate), Poly(hexamethylene adipamide), 39-40 Polyimide,26 Polyisobutylene: conformations, 50 Newmanprojections, 50 Polyisoprene (c/s), 102 (trans), 102 isomers of, 71 Polymer coils, 181 Polymerization processes, 11,553-569 Polymerization-depolymerization equilibrium, 532-541

849 Poly(methyl methacrylate), NMRspectra of, 81 Poly(p-phenylene), Poly(phenylene oxide), Polypropylene: atactic form, 52 isotactic form, 53, 55, 56 Newmanprojections, 52 syndiotactic form, 55 Polyspiroketal, 26 Polyurethane, 31 Poly(vinyl acetate), hydrolysis of, 69 Poly(vinyl alcohol), cleavage of, 69 Poly(p-xylylene), Poor solvent, 176, 185 Positional isomerism, 67-71 Post-gel relations, 392-394, 408-415 Prepolymers, 27, 369 Primary radical, 426 Primary radical termination, 511, 512 Primary radicals, recombination of, 455 Primary termination, 511 Pseudoasymmetrie carbon, 73 Pseudochiral carbon, 73 Pyromellitic dianhydride, 25, 26 Q- e schem~ 612-615 Quantum yield, 464 Quencher, 467 Quinone, 524 Radiation-induced polymerization, 469-474 free-radical chain initiation in, 472, 473 initiation of, 469-472 ionic chain initiation in, 474 Radical chain polymerization, 435-570 catalysts for, 452

850 [Radical chain polymerization] dilatometry for, 447-454 experimental, 447--452 initiation in, 452-454 integrated rate of, 443, 444 rate of, 441 Radical reactivity, 609 Radical-monomer reactions, 607-611 polar effects, 610 resonance effects, 607 steric effects, 610 Radius of gyration, 60, 71 Ramanscattering, 263 Random copolymers, 7, 580, 588 Random-alternating copolymerization, 589 Random polymer coil, 275 Raoult’s law, 147, 150 Rate of polymerization, effect of temperature, 527-530 Ratio of moments, 237 Rayleigh equation, 263 Rayleigh ratio, 262, 264 Reactivity of functional groups, 315 Real polymer chains, 61-66 Recursive method, 394-415 for linear step-growth, 395-398 for nonlinear step-growth, 399-415 Redox catalysis, 458 Redox initiation, 458-462 Reducedvariables shift factor, 110 Reduced viscosity, 287, 292 Relative viscosity, 287, 292 Relaxation processes, 121-126 Relaxation time, 123 Repeating unit, 3 Reptation model, 123-126 Resin, 5 Retardation, 522-527 kinetics of, 525 Retarders, 522 Rigid-rod polymer, 276

Index Ring-opening polymerization, 809-837 addition of N-acyllactam, 835-837 anionic polymerization, 833--835 assisted polymerization, 836 cationic mechanism of, 820-823 degree of polymerization, 827-830 exchange reactions, 819, 820 hydrolytic polymerization, 830-833 kinetics, 816-818, 823-826 lactams, 830-837 mechanism, 812-837 nonassisted polymerization, 836 polymers made by, 810 RMSdistance, 60 RMSend-to-end distance, 276, 280 RMSradius of gyration, 276 Rotational isomeric states, 62 Run number, 618, 619 Saw horse projection, 46 Scattering factor, 275 Second moment, 236 Second-order Markov, 580 Second-order transition, 91, 92 Secondvirial coefficient, 180, 254 Semicrystalline polymers, 90 Semiladder structure, 26 Sequence length distribution, 615-619 Shift factor, 110 Short-stopping of polymerization, 672 Silicones, 38 Single-site catalysts, 792 Size of polymer, 58 Skeist equation, 598 Sodium naphthalenide, 663 Solubility behavior, 198-209 of alternating copolymers, 205 Solubility parameter, 199-210 of mixtures of liquids, 205 Solution polymerization, 554 Solvent chain transfer constant, 503-506

85I

Index Specific viscosity, 212, 287 Spherical polymer, 276 Spherulites, 87-89 Spiro-polymers, 26 Spontaneous termination, 668 Staggered conformation, 46, 47 Starburst, 416 Statistical copolymer, 580 Steady-state assumption, 441 Step polymerization, 15-18 Step-growth polymerization, 313-424, see Condensation polymerization Stereoisomerism, 52, 73-82 Stereoisomers, 73 Steric parameter, 63 Stoichiometric imbalance, 339-347 Stopped-flow technique, 672, 674 termination reactions, 665-669 Stress-strain behavior, 35 Styragel, 296 Supported metal oxide catalysts, 779-790 bound-ion coordination mechanism, 787-788 bound-ion radical mechanism, 782-787 polymerization mechanism, 781, 782 Suspended-level viscometer, 291 Suspension polymerization, 554, 555 Syncatalytic system, 709 Syndiotactic form, 55, 74 Syndiotactic polypropylene, 55, 56 Syndiotactic triad, 79 Tacticity, 74--82 by NMR, 78-82 Telechelic polymers, 667 Terpolymerization, 579 Tetrablock copolymers, 7 Tetrad, 77

Tetraminobenzene, 25 Thermal initiation, 475 Thermal transitions, 89-120 thermodynamics of, 810--812 Thermoplastic elastomers, 699, 701 Theta conditions, 172, 185, 214 Theta solutions, 169 Theta solvent, 66, 185, 215, 254 Theta state, 175 Theta temperature, 169-175 Titanocenes, 792 p-ToluenesuLfinate, 465 Torsional mobility, 98 Trans conformation, 45, 49 Triad, 77-79 Triblock copolymers, 7 Tromsdorff effect, 518 Turbidity, 266, 267 Type I condensation, 314, 347, 353, 359 Type I! condensation, 314, 347, 353, 354 Ubbelhode viscometer, 289, 291 UCST, 197, 198 Unimolecutar micelles, 422 Unnormalized distribution, 237 Unperturbed dimension, 184, 186-188, 218 Unsaturated polyester, 30 Unzipping, 541 Urea-formaldehyde resins, 28 Valence angle model, 61 van Laar model, 145-150 Vapor-phase osmometry, 258-261 practical aspects, 259 Vapor pressure, relative, 161 Vapor pressure lowering, 240 Vapor pressure osmometry, 257 Vinyl acetate, 4 Vinyl esters, 27

852 Virial coefficients, 165 Virial equations, 166, 248 Virial expansion, 168 Viscoelastic behavior, 93-94 Viscometry, 285-295 Viscosity: intrinsic, 212 of polymer solutions, 211-219 of suspensions, 211, 212 specific, 212 Viscosity-average molecular weight, 286-288 Viscous flow, 292 Volumeexclusion, 184 Vulcanization, 23 WAXSmethod, 83, 85 Weight-average molecular weight, 234 Weight distribution, 234, 236 Wide-angle x-ray scattering, 83 WLFequation, 107-110 Woodequation, 119, 120 Ziegler-Natta catalysts, 742-755 catalyst composition, 742 high-mileage, 747 metallocene-based systems, 791 titanium-aluminum systems, 746 Ziegler-Natta polymerization, 741-803 adsorption models, 764-773 anionic coordination, 748 bimetallic mechanism, 749, 750 catalyst site control, 748 catalysts, 742-755 cationic coordination, 748 copolymerization, 789-790 degree of polymerization, 778, 779 enantiomorphic site control, 748 kinetic models, 760-764, 773-778 kinetics, 755-778 mechanism, 747-755 monometallic mechanism, 749-751

Index [Ziegler-Natta polymerization] stereoregulation, 754 stereospecific placement, 748 Zimmplots, 278-283 Zirconocenes, 792