viii Preface to the Ist Edition The most subjective of the words which (I hope)describe this book is "interesting."The fascinating behavior of polymers themselves,the clever experiments of laboratory researchers,and the elegant work of the theoreticians add up to an interesting total.I have tried to tell about these topics with clarity and enthusiasm,and in such a way as to make them intelligible to students.I can only hope that the reader agrees with my assessment of what is interesting. This book was written with the student in mind.Even though "student"encompasses persons with a wide range of backgrounds,interests,and objectives;these are different than the corre- sponding experiences and needs of researchers.The following features have been included to assist the student: 1. Over 50 solved example problems are sprinkled throughout the book. 2. Exercises are included at the end of each chapter which are based on data from the original literature. 3. Concise reviews of pertinent aspects of thermodynamics,kinetics,spectrophotometry,etc.are presented prior to developing applications of these topics to polymers. Theoretical models and mathematical derivations are developed in enough detail to be comprehensible to the student reader.Only rarely do I "pull results out of a hat,"and I scrupulously avoid saying "it is obvious that..." Generous cross-referencing and a judicious amount of repetition have been included to help unify a book which spans quite a wide range of topics. 6.SI units have been used fairly consistently throughout,and attention is paid to the matter of units whenever these become more than routine in complexity. The book is divided into three parts of three chapters each,after an introductory chapter which contains information that is used throughout the book. In principle,the three parts can be taken up in any order without too much interruption in continuity.Within each of the parts there is more carryover from chapter to chapter,so rearranging the sequence of topics within a given part is less convenient.The book contains more material than can be covered in an ordinary course.Chapter 1 plus two of the three parts contain about the right amount of material for one term.In classroom testing the material,I allowed the class to decide- while we worked on Chapter 1-which two of the other parts they wished to cover;this worked very well. Material from Chapter 1 is cited throughout the book,particularly the discussion of statistics.In this connection,it might be noted that statistical arguments are developed in less detail further along in the book as written.This is one of the drawbacks of rearranging the order in which the topics are covered.Chapters 2 through 4 are concerned with the mechanical properties of bulk polymers,properties which are primarily responsible for the great practical importance of poly- mers.Engineering students are likely to have both a larger interest and a greater familiarity with these topics.Chapers 5 through 7 are concerned with the preparation and properties of several broad classes of polymers.These topics are closer to the interests of chemistry majors.Chapters 8 through 10 deal with the solution properties of polymers.Since many of the techniques described have been applied to biopolymers,these chapters will have more appeal to students of biochem- istry and molecular biology. Let me conclude by acknowledging the contributions of those who helped me with the preparation of this book.I wish to thank Marilyn Steinle for expertly typing the manuscript.My appreciation also goes to Carol Truett who skillfully transformed my (very)rough sketches into effective illustrations.Lastly,my thanks to Ron Manwill for preparing the index and helping me with the proofreading.Finally,let me acknowledge that some errors and/or obscurities will surely elude my efforts to eliminate them.I would appreciate reports about these from readers so that these mistakes can eventually be eliminated. Paul C.Hiemenz
The most subjective of the words which (I hope) describe this book is ‘‘interesting.’’ The fascinating behavior of polymers themselves, the clever experiments of laboratory researchers, and the elegant work of the theoreticians add up to an interesting total. I have tried to tell about these topics with clarity and enthusiasm, and in such a way as to make them intelligible to students. I can only hope that the reader agrees with my assessment of what is interesting. This book was written with the student in mind. Even though ‘‘student’’ encompasses persons with a wide range of backgrounds, interests, and objectives; these are different than the corresponding experiences and needs of researchers. The following features have been included to assist the student: 1. Over 50 solved example problems are sprinkled throughout the book. 2. Exercises are included at the end of each chapter which are based on data from the original literature. 3. Concise reviews of pertinent aspects of thermodynamics, kinetics, spectrophotometry, etc. are presented prior to developing applications of these topics to polymers. 4. Theoretical models and mathematical derivations are developed in enough detail to be comprehensible to the student reader. Only rarely do I ‘‘pull results out of a hat,’’ and I scrupulously avoid saying ‘‘it is obvious that . . .’’ 5. Generous cross-referencing and a judicious amount of repetition have been included to help unify a book which spans quite a wide range of topics. 6. SI units have been used fairly consistently throughout, and attention is paid to the matter of units whenever these become more than routine in complexity. The book is divided into three parts of three chapters each, after an introductory chapter which contains information that is used throughout the book. In principle, the three parts can be taken up in any order without too much interruption in continuity. Within each of the parts there is more carryover from chapter to chapter, so rearranging the sequence of topics within a given part is less convenient. The book contains more material than can be covered in an ordinary course. Chapter 1 plus two of the three parts contain about the right amount of material for one term. In classroom testing the material, I allowed the class to decide– while we worked on Chapter 1–which two of the other parts they wished to cover; this worked very well. Material from Chapter 1 is cited throughout the book, particularly the discussion of statistics. In this connection, it might be noted that statistical arguments are developed in less detail further along in the book as written. This is one of the drawbacks of rearranging the order in which the topics are covered. Chapters 2 through 4 are concerned with the mechanical properties of bulk polymers, properties which are primarily responsible for the great practical importance of polymers. Engineering students are likely to have both a larger interest and a greater familiarity with these topics. Chapers 5 through 7 are concerned with the preparation and properties of several broad classes of polymers. These topics are closer to the interests of chemistry majors. Chapters 8 through 10 deal with the solution properties of polymers. Since many of the techniques described have been applied to biopolymers, these chapters will have more appeal to students of biochemistry and molecular biology. Let me conclude by acknowledging the contributions of those who helped me with the preparation of this book. I wish to thank Marilyn Steinle for expertly typing the manuscript. My appreciation also goes to Carol Truett who skillfully transformed my (very) rough sketches into effective illustrations. Lastly, my thanks to Ron Manwill for preparing the index and helping me with the proofreading. Finally, let me acknowledge that some errors and/or obscurities will surely elude my efforts to eliminate them. I would appreciate reports about these from readers so that these mistakes can eventually be eliminated. Paul C. Hiemenz Hiemenz/ Polymer Chemistry, 2nd Edition DK4670_C000 Final Proof page viii 5.11.2007 8:20pm Compositor Name: JGanesan viii Preface to the Ist Edition
Contents 1 Introduction to Chain Molecules....... 1.1 Introduction.. 1 1.2 HOw BIsbl.......................................................................3 1.2.1 Molecular Weight.. 3 12.2Spatial Extent..5 1.3 Linear and Branched Polymers,Homopolymers,and Copolymers........................... 1.3.1 Branched Structures........... 1.3.2 Copolymers.… 7 9 1.4 Addition,Condensation,and Natural Polymers.11 1.4.1 Addition and Condensation Polymers...... 1.4.2 Natural Polymers.. .13 1.5 Polymer Nomenclature. 18 1.6 Structural Isomerism.. 20 1.6.1 Positional Isomerism........... .20 1.6.2 Stereo Isomerism.… 21 1.6.3 Geometrical Isomerism...22 1.7 Molecular Weights and Molecular Weight Averages.24 1.7.1 Number-,Weight-,and z-Average Molecular Weights.5 1.7.2 Polydispersity Index and Standard Deviation.6 1.7.3 Examples of Distributions28 1.8 Measurement of Molecular Weight..31 1.8.1 General Considerations.31 1.8.2 End Group Analysis...32 18.3 MALDI Mass Spectrometry35 1.9 Preview of Things to Come................... 37 l.l0 Chapter Summary… .38 Pt0 blems… .38 References.… .41 Further Readings............ …41 2 Step-Growth Polymerization............43 2.1 Introduction. 43 2.2 Condensation Polymers:One Step at a Time............. .43 2.2.1 Classes of Step-Growth Polymers......... …43 2.2.2 First Look at the Distribution of Products... .44 2.2.3 A First Look at Reactivity and Reaction Rates …46 ix
Contents 1 Introduction to Chain Molecules .................................................................1 1.1 Introduction.................................................................................................................... 1 1.2 How Big Is Big? ............................................................................................................ 3 1.2.1 Molecular Weight ..............................................................................................3 1.2.2 Spatial Extent.....................................................................................................5 1.3 Linear and Branched Polymers, Homopolymers, and Copolymers ............................. 7 1.3.1 Branched Structures...........................................................................................7 1.3.2 Copolymers ........................................................................................................9 1.4 Addition, Condensation, and Natural Polymers.......................................................... 11 1.4.1 Addition and Condensation Polymers.............................................................11 1.4.2 Natural Polymers .............................................................................................13 1.5 Polymer Nomenclature ................................................................................................ 18 1.6 Structural Isomerism.................................................................................................... 20 1.6.1 Positional Isomerism........................................................................................20 1.6.2 Stereo Isomerism .............................................................................................21 1.6.3 Geometrical Isomerism....................................................................................22 1.7 Molecular Weights and Molecular Weight Averages................................................. 24 1.7.1 Number-, Weight-, and z-Average Molecular Weights..................................25 1.7.2 Polydispersity Index and Standard Deviation .................................................26 1.7.3 Examples of Distributions ...............................................................................28 1.8 Measurement of Molecular Weight............................................................................. 31 1.8.1 General Considerations....................................................................................31 1.8.2 End Group Analysis.........................................................................................32 1.8.3 MALDI Mass Spectrometry ............................................................................35 1.9 Preview of Things to Come......................................................................................... 37 1.10 Chapter Summary ........................................................................................................ 38 Problems .................................................................................................................................. 38 References................................................................................................................................ 41 Further Readings...................................................................................................................... 41 2 Step-Growth Polymerization......................................................................43 2.1 Introduction.................................................................................................................... 43 2.2 Condensation Polymers: One Step at a Time ............................................................... 43 2.2.1 Classes of Step-Growth Polymers .....................................................................43 2.2.2 First Look at the Distribution of Products ........................................................44 2.2.3 A First Look at Reactivity and Reaction Rates ................................................46 Hiemenz/ Polymer Chemistry, 2nd Edition DK4670_C000 Final Proof page ix 5.11.2007 8:20pm Compositor Name: JGanesan ix
X Contents 2.3 Kinetics of Step-Growth Polymerization. 49 2.3.1 Catalyzed step-Growth Reactions..50 2.3.2 How Should Experimental Data Be Compared with Theoretical Rate Laws?....52 2.3.3 Uncatalyzed Step-Growth Reactions....53 2.4 Distribution of Molecular Sizes.............. 55 2.4.1 Mole Fractions of Species.... 56 2.4.2 Weight Fractions of Species .58 2.5 Polyesters 60 2.6 Polyamides. …64 2.7 Stoichiometric Imbalance. 67 2.8 Chapter Summary....... 71 Problems....... 71 References.... 76 Further Readings......... 76 3 Chain-Growth Polymerization..77 3.1 Introduction......... 77 3.2 Chain-Growth and Step-Growth Polymerizations:Some Comparisons.................77 3.3 nitiati0n............. .79 3.3.1 Initiation Reactions................ 80 3.3.2 Fate of Free Radicals....... 81 3.3.3 Kinetics of Initiation. 82 3.3.4 Photochemical Initiation.84 3.3.5 Temperature Dependence of Initiation Rates.....85 3.4 Termination. .86 3.4.1 Combination and Disproportionation...... 86 3.4.2 Effect of Termination on Conversion to Polymer...... 88 3.4.3 Stationary-State Radical Concentration............. .89 3.5 Propagation .90 3.5.1 Rate Laws for Propagation....... 91 3.5.2 Temperature Dependence of Propagation Rates... .92 3.5.3 Kinetic Chain Length.................. .94 3.6 Radical Lifetime........... .96 3.7 Distribution of Molecular Weights..99 3.7.1 Distribution of i-mers:Termination by Disproportionation.................99 3.7.2 Distribution of i-mers:Termination by Combination............102 3.8 Chain 3.8.1 Chain Transfer Reactions..105 3.8.2 Evaluation of Chain Transfer Constants..106 3.8.3 Chain Transfer to Polymer.108 3.8.4 Suppressing Polymerization.109 3.9 Chapter Summary. .110 Problems… .110 References.… .114 Further Readings.... .115
2.3 Kinetics of Step-Growth Polymerization ...................................................................... 49 2.3.1 Catalyzed Step-Growth Reactions.....................................................................50 2.3.2 How Should Experimental Data Be Compared with Theoretical Rate Laws? .....................................................................................52 2.3.3 Uncatalyzed Step-Growth Reactions.................................................................53 2.4 Distribution of Molecular Sizes .................................................................................... 55 2.4.1 Mole Fractions of Species .................................................................................56 2.4.2 Weight Fractions of Species..............................................................................58 2.5 Polyesters ....................................................................................................................... 60 2.6 Polyamides ..................................................................................................................... 64 2.7 Stoichiometric Imbalance .............................................................................................. 67 2.8 Chapter Summary .......................................................................................................... 71 Problems .................................................................................................................................. 71 References................................................................................................................................ 76 Further Readings...................................................................................................................... 76 3 Chain-Growth Polymerization....................................................................77 3.1 Introduction.................................................................................................................... 77 3.2 Chain-Growth and Step-Growth Polymerizations: Some Comparisons....................... 77 3.3 Initiation......................................................................................................................... 79 3.3.1 Initiation Reactions............................................................................................80 3.3.2 Fate of Free Radicals.........................................................................................81 3.3.3 Kinetics of Initiation..........................................................................................82 3.3.4 Photochemical Initiation....................................................................................84 3.3.5 Temperature Dependence of Initiation Rates....................................................85 3.4 Termination.................................................................................................................... 86 3.4.1 Combination and Disproportionation ................................................................86 3.4.2 Effect of Termination on Conversion to Polymer ............................................88 3.4.3 Stationary-State Radical Concentration ............................................................89 3.5 Propagation .................................................................................................................... 90 3.5.1 Rate Laws for Propagation ................................................................................91 3.5.2 Temperature Dependence of Propagation Rates...............................................92 3.5.3 Kinetic Chain Length.........................................................................................94 3.6 Radical Lifetime ............................................................................................................ 96 3.7 Distribution of Molecular Weights................................................................................ 99 3.7.1 Distribution of i-mers: Termination by Disproportionation .............................99 3.7.2 Distribution of i-mers: Termination by Combination.....................................102 3.8 Chain Transfer ............................................................................................................. 104 3.8.1 Chain Transfer Reactions ................................................................................105 3.8.2 Evaluation of Chain Transfer Constants .........................................................106 3.8.3 Chain Transfer to Polymer ..............................................................................108 3.8.4 Suppressing Polymerization ............................................................................109 3.9 Chapter Summary ........................................................................................................ 110 Problems ................................................................................................................................ 110 References.............................................................................................................................. 114 Further Readings.................................................................................................................... 115 Hiemenz/ Polymer Chemistry, 2nd Edition DK4670_C000 Final Proof page x 5.11.2007 8:20pm Compositor Name: JGanesan x Contents
Contents xi 4 Controlled Polymerization..........117 4.1 Introduction.. .117 4.2 Poisson Distribution for an Ideal Living Polymerization...............18 4.2.1 Kinetic Scheme............. .119 4.2.2 Breadth of the Poisson Distribution..... .122 4.3 Anionic Polymerization........... 126 4.4 Block Copolymers,End-Functional Polymers,and Branched Polymers by Anionic Polymerization......... .129 4.4.1 Block Copolymers... 129 4.4.2 End-Functional Polymers........ 133 4.4.3 Regular Branched Architectures.135 4.5 Cationic polymerization137 4.5.1 Aspects of Cationic Polymerization..138 4.5.2 Living Cationic Polymerization.140 4.6 Controlled Radical Polymerization142 4.6.1 General Principles of Controlled Radical Polymerization.........................142 4.6.2 Particular Realizations of Controlled Radical Polymerization.....................144 4.6.2.1 Atom Transfer Radical Polymerization (ATRP).......................144 4.6.2.2 Stable Free-Radical Polymerization (SFRP)............................145 4.6.2.3 Reversible Addition-Fragmentation Transfer(RAFT) Polymerization... .146 4.7 Polymerization Equilibrium.. .147 4.8 Ring-Opening Polymerization (ROP)............................ .150 4.8.1 General Aspects .150 4.8.2 Specific Examples of Living Ring-Opening Polymerizations.....................152 4.8.2.1 Poly(ethylene oxide)....152 4.8.2.2 Polylactide.... 153 4.8.2.3 Poly(dimethylsiloxane)......... .154 4.8.2.4 Ring-Opening Metathesis Polymerization(ROMP) .155 4.9 Dendrimers............ .156 4.10 Chapter Summary. .160 Problems .... .161 References.... .163 Further Readings...... .163 5 Copolymers,Microstructure,and Stereoregularity...........................165 5.1 Introduction. .165 5.2 Copolymer Composition...... …166 5.2.1 Rate Laws ..166 5.2.2 Composition versus Feedstock.............. .168 5.3 Reactivity Ratios............... .170 5.3.1 Effects of r Values..... ..171 5.3.2 Relation of Reactivity Ratios to Chemical Structure.. .173 5.4 Resonance and Reactivity............ ..175 5.5 A Closer Look at Microstructure ............ …179 5.5.1 Sequence Distributions.......... .180 5.5.2 Terminal and Penultimate Models.... .183
4 Controlled Polymerization .......................................................................117 4.1 Introduction................................................................................................................ 117 4.2 Poisson Distribution for an Ideal Living Polymerization ......................................... 118 4.2.1 Kinetic Scheme ..............................................................................................119 4.2.2 Breadth of the Poisson Distribution ..............................................................122 4.3 Anionic Polymerization ............................................................................................. 126 4.4 Block Copolymers, End-Functional Polymers, and Branched Polymers by Anionic Polymerization........................................................................ 129 4.4.1 Block Copolymers .........................................................................................129 4.4.2 End-Functional Polymers ..............................................................................133 4.4.3 Regular Branched Architectures....................................................................135 4.5 Cationic Polymerization ............................................................................................ 137 4.5.1 Aspects of Cationic Polymerization..............................................................138 4.5.2 Living Cationic Polymerization ....................................................................140 4.6 Controlled Radical Polymerization ........................................................................... 142 4.6.1 General Principles of Controlled Radical Polymerization............................142 4.6.2 Particular Realizations of Controlled Radical Polymerization.....................144 4.6.2.1 Atom Transfer Radical Polymerization (ATRP)...........................144 4.6.2.2 Stable Free-Radical Polymerization (SFRP).................................145 4.6.2.3 Reversible Addition-Fragmentation Transfer (RAFT) Polymerization ...............................................................................146 4.7 Polymerization Equilibrium....................................................................................... 147 4.8 Ring-Opening Polymerization (ROP) ....................................................................... 150 4.8.1 General Aspects .............................................................................................150 4.8.2 Specific Examples of Living Ring-Opening Polymerizations......................152 4.8.2.1 Poly(ethylene oxide)......................................................................152 4.8.2.2 Polylactide......................................................................................153 4.8.2.3 Poly(dimethylsiloxane) ..................................................................154 4.8.2.4 Ring-Opening Metathesis Polymerization (ROMP) .....................155 4.9 Dendrimers................................................................................................................. 156 4.10 Chapter Summary ...................................................................................................... 160 Problems ................................................................................................................................ 161 References.............................................................................................................................. 163 Further Readings.................................................................................................................... 163 5 Copolymers, Microstructure, and Stereoregularity................................165 5.1 Introduction................................................................................................................ 165 5.2 Copolymer Composition............................................................................................ 166 5.2.1 Rate Laws ......................................................................................................166 5.2.2 Composition versus Feedstock ......................................................................168 5.3 Reactivity Ratios........................................................................................................ 170 5.3.1 Effects of r Values.........................................................................................171 5.3.2 Relation of Reactivity Ratios to Chemical Structure....................................173 5.4 Resonance and Reactivity.......................................................................................... 175 5.5 A Closer Look at Microstructure .............................................................................. 179 5.5.1 Sequence Distributions ..................................................................................180 5.5.2 Terminal and Penultimate Models ................................................................183 Hiemenz/ Polymer Chemistry, 2nd Edition DK4670_C000 Final Proof page xi 5.11.2007 8:20pm Compositor Name: JGanesan Contents xi
xii Contents 5.6 Copolymer Composition and Microstructure:Experimental Aspects.185 5.6.1 Evaluating Reactivity Ratios from Composition Data..........................185 5.6.2 Spectroscopic Techniques.....188 5.6.3 Sequence Distribution:Experimental Determination.......190 5.7 Characterizing Stereoregularity193 5.8 A Statistical Description of Stereoregularity......................96 5.9 Assessing Stereoregularity by Nuclear Magnetic Resonance. 44444444444444444 200 5.10 Ziegler-Natta Catalysts.. 444444444444444444 205 5.11 Single-Site Catalysts............. 208 5.12 Chapter Summary… 211 Problems........ 212 References..... 216 Further Readings......... 216 6 Polymer Conformations................... 217 6.1 Conformations,Bond Rotation,and Polymer Size.... .217 6.2 Average End-to-End Distance for Model Chains... 219 Case 6.2.1 The Freely Jointed Chain.................... 220 Case 6.2.2 The Freely Rotating Chain........... 221 Case 6.2.3 Hindered Rotation Chain......... .222 6.3 Characteristic Ratio and Statistical Segment Length 223 6.4 Semiflexible Chains and the Persistence Length..... 225 6.4.1 Persistence Length of Flexible Chains..... .227 6.4.2 Worm-Like Chains.... .228 6.5 Radius of Gyration. 230 6.6 Spheres,Rods,and Coils. 234 6.7 Distributions for End-to-End Distance and Segment Density..............235 6.7.1 Distribution of the End-to-End Vector..236 6.7.2 Distribution of the End-to-End Distance..239 6.7.3 Distribution about the Center of Mass.240 6.8 Self-Avoiding Chains:AFirst Look.241 6.9 Chapter Summary 242 Problems… 242 References...... 244 Further Readings.… 245 7 Thermodynamics of Polymer Solutions................... .247 7.1 Review of Thermodynamic and Statistical Thermodynamic Concepts 247 7.2 Regular Solution Theory. 249 7.2.1 Regular Solution Theory:Entropy of Mixing....249 7.2.2 Regular Solution Theory:Enthalpy of Mixing......................................5.1 7.3 Flory-Huggins Theory254 7.3.1 Flory-Huggins Theory:Entropy of Mixing by a Quick Route....................2.55 7.3.2 Flory-Huggins Theory:Entropy of Mixing by a Longer Route...55 7.3.3 Flory-Huggins Theory:Enthalpy of Mixing....................................257 7.3.4 Flory-Huggins Theory:Summary of Assumptions...............258
5.6 Copolymer Composition and Microstructure: Experimental Aspects...................... 185 5.6.1 Evaluating Reactivity Ratios from Composition Data .................................185 5.6.2 Spectroscopic Techniques..............................................................................188 5.6.3 Sequence Distribution: Experimental Determination ...................................190 5.7 Characterizing Stereoregularity................................................................................. 193 5.8 A Statistical Description of Stereoregularity ............................................................ 196 5.9 Assessing Stereoregularity by Nuclear Magnetic Resonance................................... 200 5.10 Ziegler–Natta Catalysts ............................................................................................. 205 5.11 Single-Site Catalysts .................................................................................................. 208 5.12 Chapter Summary ...................................................................................................... 211 Problems ................................................................................................................................ 212 References.............................................................................................................................. 216 Further Readings.................................................................................................................... 216 6 Polymer Conformations ...........................................................................217 6.1 Conformations, Bond Rotation, and Polymer Size..................................................... 217 6.2 Average End-to-End Distance for Model Chains ....................................................... 219 Case 6.2.1 The Freely Jointed Chain .......................................................................220 Case 6.2.2 The Freely Rotating Chain .....................................................................221 Case 6.2.3 Hindered Rotation Chain ........................................................................222 6.3 Characteristic Ratio and Statistical Segment Length.................................................. 223 6.4 Semiflexible Chains and the Persistence Length........................................................ 225 6.4.1 Persistence Length of Flexible Chains............................................................227 6.4.2 Worm-Like Chains ..........................................................................................228 6.5 Radius of Gyration....................................................................................................... 230 6.6 Spheres, Rods, and Coils ............................................................................................. 234 6.7 Distributions for End-to-End Distance and Segment Density.................................... 235 6.7.1 Distribution of the End-to-End Vector............................................................236 6.7.2 Distribution of the End-to-End Distance.........................................................239 6.7.3 Distribution about the Center of Mass ............................................................240 6.8 Self-Avoiding Chains: A First Look ........................................................................... 241 6.9 Chapter Summary ........................................................................................................ 242 Problems ................................................................................................................................ 242 References.............................................................................................................................. 244 Further Readings.................................................................................................................... 245 7 Thermodynamics of Polymer Solutions .................................................247 7.1 Review of Thermodynamic and Statistical Thermodynamic Concepts ..................... 247 7.2 Regular Solution Theory ............................................................................................. 249 7.2.1 Regular Solution Theory: Entropy of Mixing.................................................249 7.2.2 Regular Solution Theory: Enthalpy of Mixing ...............................................251 7.3 Flory–Huggins Theory................................................................................................. 254 7.3.1 Flory–Huggins Theory: Entropy of Mixing by a Quick Route ......................255 7.3.2 Flory–Huggins Theory: Entropy of Mixing by a Longer Route ....................255 7.3.3 Flory–Huggins Theory: Enthalpy of Mixing ..................................................257 7.3.4 Flory–Huggins Theory: Summary of Assumptions ........................................258 Hiemenz/ Polymer Chemistry, 2nd Edition DK4670_C000 Final Proof page xii 5.11.2007 8:20pm Compositor Name: JGanesan xii Contents